A term of commutative algebra

Published by the Worldwide Center of Mathematics, LLC; 2012 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License Version of September 3, 2012 A Term of Commutative Algebra By Allen ALTMAN and Steven KLEIMAN Contents Preface . . . . . . . . . . 1. Rings and Ideals . . . . . . 2. Prime Ideals . . . . . . . 3. Radicals . . . . . . . . . 4. Modules . . . . . . . . . 5. Exact Sequences . . . . . . 6. Direct Limits . . . . . . . 7. Filtered Direct Limits . . . . 8. Tensor Products . . . . . . 9. Flatness . . . . . . . . . 10. Cayley–Hamilton Theorem . . 11. Localization of Rings . . . . 12. Localization of Modules . . . 13. Support . . . . . . . . 14. Krull–Cohen–Seidenberg Theory 15. Noether Normalization . . . Appendix: Jacobson Rings . . 16. Chain Conditions . . . . . 17. Associated Primes . . . . . 18. Primary Decomposition . . . 19. Length . . . . . . . . . 20. Hilbert Functions . . . . . Appendix: Homogeneity . . . 21. Dimension . . . . . . . 22. Completion . . . . . . . 23. Discrete Valuation Rings . . 24. Dedekind Domains . . . . 25. Fractional Ideals . . . . . 26. Arbitrary Valuation Rings . . Solutions . . . . . . . . . 1. Rings and Ideals . . . . 2. Prime Ideals . . . . . 3. Radicals . . . . . . . 4. Modules . . . . . . . 5. Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 1 6 10 14 20 26 33 37 43 49 55 61 66 71 75 80 82 87 91 97 101 107 109 115 122 127 131 136 141 141 143 145 148 149 6. Direct Limits . . . . . . . 7. Filtered direct limits . . . . . 8. Tensor Products . . . . . . 9. Flatness . . . . . . . . . 10. Cayley–Hamilton Theorem . . 11. Localization of Rings . . . . 12. Localization of Modules . . . 13. Support . . . . . . . . . 14. Krull–Cohen–Seidenberg Theory 15. Noether Normalization . . . 16. Chain Conditions . . . . . 17. Associated Primes . . . . . 18. Primary Decomposition . . . 19. Length . . . . . . . . . 20. Hilbert Functions . . . . . 21. Dimension . . . . . . . . 22. Completion . . . . . . . 23. Discrete Valuation Rings . . . 24. Dedekind Domains . . . . . 25. Fractional Ideals . . . . . . 26. Arbitrary Valuation Rings . . References . . . . . . . . . . . Index . . . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 156 158 159 161 164 167 168 171 174 177 179 180 183 185 188 190 193 197 199 200 202 203 Preface There is no shortage of books on Commutative Algebra, but the present book is different. Most books are monographs, with extensive coverage. There is one notable exception: Atiyah and Macdonald’s 1969 classic [2]. It is a clear, concise, and efficient textbook, aimed at beginners, with a good selection of topics. So it has remained popular. However, its age and flaws do show. So there is need for an updated and improved version, which the present book aims to be. Atiyah and Macdonald explain their philosophy in their introduction. They say their book “has the modest aim of providing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts on Commutative Algebra . . . The lecture-note origin of this book accounts for the rather terse style, with little general padding, and for the condensed account of many proofs.” They “resisted the temptation to expand it in the hope that the brevity of [the] presentation will make clearer the mathematical structure of what is by now an elegant and attractive theory.” They endeavor “to build up to the main theorems in a succession of simple steps and to omit routine verifications.” Their successful philosophy is wholeheartedly embraced below (it is a feature, not a flaw!), and also refined a bit. The present book also “grew out of a course of lectures.” That course was based primarily on their book, but has been offered a number of times, and has evolved over the years, influenced by other publications and the reactions of the students. Their book comprises eleven chapters, split into forty-two sections. The present book comprises twenty-six sections; each represents a single lecture, and is self-contained. Atiyah and Macdonald “provided . . . exercises at the end of each chapter.” They “provided hints, and sometimes complete solutions, to the hard” exercises. Moreover, they developed a significant amount of the main content in the exercises. By contrast, in the present book, the exercises are integrated into the development, and complete solutions are given at the end of the book. There are well over two hundred exercises below. Included are nearly all the exercises in Atiyah and Macdonald’s book. Included also are many exercises that come from other publications and many that originate here. Here the exercises are tailored to provide a means for students to check, solidify, and expand their understanding of the material. The exercises are intentionally not difficult, tricky, or involved. Rarely do they introduce new techniques, although some introduce new concepts and many statements are used later. Students are encouraged to try to solve each and every exercise, and to do so before looking up its solution. If they become stuck, then they should review the relevant material; if they remain stuck, then they should study the given solution, possibly discussing it with others, but always making sure they can eventually solve the whole exercise completely on their own. In any event, students should read the given solution, even if they think they already know it, just to make sure; also, some exercises provide enlightening alternative solutions. Finally, instructors are encouraged to examine their students, possibly orally at a blackboard, on a small iii iv Preface randomly chosen subset of exercises that have been assigned for the students to write up in their own words over the course of the term. Atiyah and Macdonald explain that “a proper treatment of Homological Algebra is impossible within the confines of a small book; on the other hand, it is hardly sensible to ignore it completely.” So they “use elementary homological methods — exact sequence, diagrams, etc. — but . . . stop short of any results requiring a deep study of homology.” Again, their philosophy is embraced and refined in the present book. Notably, below, elementary methods are used, not Tor’s as they do, to prove the Ideal Criterion for flatness, and to relate flat modules and free modules over local rings. Also, projective modules are treated below, but not in their book. In the present book, Category Theory is a basic tool; in Atiyah and Macdonald’s, it seems like a foreign language. Thus they discuss the universal (mapping) property (UMP) of localization of a ring, but provide an ad hoc characterization. They also prove the UMP of tensor product of modules, but do not use the term this time. Below, the UMP is fundamental: there are many canonical constructions; each has a UMP, which serves to characterize the construction up to unique isomorphism owing to one general observation of Category Theory. For example, the Left Exactness of Hom is viewed simply as expressing in other words that the kernel and the cokernel of a map are characterized by their UMPs; by contrast, Atiyah and Macdonald prove the Left Exactness via a tedious elementary argument. Atiyah and Macdonald prove the Adjoint-Associativity Formula. They note it says that Tensor Product is the left adjoint of Hom. From it and the Left Exactness of Hom, they deduce the Right Exactness of Tensor Product. They note that this derivation shows that any “left adjoint is right exact.” More generally, as explained below, this derivation shows that any left adjoint preserves arbitrary direct limits, ones indexed by any small category. Atiyah and Macdonald consider only direct limits indexed by a directed set, and sketch an ad hoc argument showing that tensor product preserves direct limit. Also, arbitrary direct sums are direct limits indexed by a discrete category (it is not a directed set); hence, the general result yields that Tensor Product and other left adjoints preserve arbitrary Direct Sum. Below, left adjoints are proved unique up to unique isomorphism. Therefore, the functor of localization of a module is canonically isomorphic to the functor of tensor product with the localized base ring, as both are left adjoints of the same functor, Restriction of Scalars from the localized ring to the base ring. There is an alternative argument. Since Localization is a left adjoint, it preserves Direct Sum and Cokernel; whence, it is isomorphic to that tensor-product functor by Watts Theorem, which characterizes all tensor-product functors as those linear functors that preserve Direct Sum and Cokernel. Atiyah and Macdonald’s treatment is ad hoc. However, they do use the proof of Watts Theorem directly to show that, under the appropriate conditions, Completion of a module is Tensor Product with the completed base ring. Below, Direct Limit is also considered as a functor, defined on the appropriate category of functors. As such, Direct Limit is a left adjoint. Hence, direct limits preserve other direct limits. Here the theory briefly reaches a higher level of abstraction. This discussion is completely elementary, but by far the most abstract part of the book. The extra abstraction can be difficult, especially for beginners. Below, filtered direct limits are treated too. They are closer to the kind of limits treated by Atiyah and Macdonald. In particular, filtered direct limits preserve Preface v exactness and flatness. Further, they appear in the following lovely form of Lazard’s Theorem: in a canonical way, every module is the direct limit of free modules of finite rank; moreover, the module is flat if and only if that direct limit is filtered. Atiyah and Macdonald handle primary decomposition in a somewhat personal and dated fashion. First, they study primary decompositions of ideals in rings. Then, in the exercises, they indicate how to translate the theory to modules. The decompositions need not exist, as the rings and modules need not be Noetherian. Associated primes play a secondary role: they are defined as the radicals of the primary components, and then characterized as the primes that are the radicals of annihilators of elements. Finally, they prove that, when the rings and modules are Noetherian, decompositions exist and the associated primes are annihilators. To prove existence, they study irreducible modules. Nowadays, associated primes are normally defined as prime annihilators of elements, and studied on their own at first; sometimes, as below, irreducible modules are not considered. There are several other significant differences between Atiyah and Macdonald’s treatment and the one below. First, the Noether Normalization Lemma is proved below in a stronger form for nested sequences of ideals; consequently, for algebras that are finitely generated over a field, dimension theory can be developed directly without treating Noetherian local rings first. Second, in a number of results below, the modules are assumed to be finitely presented, rather than finitely generated over a Noetherian ring. Third, there is a rudimentary treatment of regular sequences below and a proof of Serre’s Criterion for Normality. Fourth, below, the AdjointAssociativity Formula is proved over a pair of base rings; hence, it yields both a left and a right adjoint to the functor restriction of scalars. The present book is a beta edition. Please do the community a service by sending the authors lists of comments, corrections, and typos. Thanks! 1. Rings and Ideals We begin by reviewing basic notions and conventions to set the stage. Throughout this book, we emphasize universal mapping properties (UMPs); they are used to characterize notions and to make constructions. So, although polynomial rings and residue rings should already be familiar in other ways, we present their UMPs immediately, and use them extensively. We close this section with a brief treatment of idempotents and the Chinese Remainder Theorem. (1.1) (Rings). — Recall that a ring R is an abelian group, written additively, with an associative multiplication that is distributive over the addition. Throughout this book, every ring has a multiplicative identity, denoted by 1. Further, every ring is commutative (that is, xy = yx in it), with an occasional exception, which is always marked (normally, it’s a ring of matrices). As usual, the additive identity is denoted by 0. Note that, for any x in R, x · 0 = 0; indeed, x · 0 = x(0 + 0) = x · 0 + x · 0, and x · 0 can be canceled by adding −(x · 0). We allow 1 = 0. If 1 = 0, then R = 0; indeed, x = x · 1 = x · 0 = 0 for any x. A unit is an element u with a reciprocal 1/u such that u·1/u = 1. Alternatively, 1/u is denoted u−1 and is called the multiplicative inverse of u. The units form a multiplicative group, denoted R× . For example, the ordinary integers form a ring Z, and its units are 1 and −1. A ring homomorphism, or simply a ring map, φ : R → R′ is a map preserving sums, products, and 1. Clearly, φ(R× ) ⊂ R′× . We call φ an isomorphism if it is ∼ R′ . We call φ an endomorphism if R′ = R. bijective, and then we write φ : R −→ We call φ an automorphism if it is bijective and if R′ = R. If there is an unspecified isomorphism between rings R and R′ , then we write R = R′ when it is canonical; that is, it does not depend on any artificial choices, so that for all practical purposes, R and R′ are the same. (Recognizing that an isomorphism is canonical provides insight and obviates verifications.) Otherwise, we write R ≃ R′ . A subset R′′ ⊂ R is a subring if R′′ is a ring and the inclusion R′′ ֒→ R a ring map. For example, given a ring map φ : R → R′ , its image Im(φ) := φ(R) is a subring of R′ . An R-algebra is a ring R′ that comes equipped with a ring map φ : R → R′ , called the structure map. An R-algebra homomorphism, or R-algebra map, R′ → R′′ is a ring map between R-algebras compatible with their structure maps. (1.2) (Boolean rings). — The simplest nonzero ring has two elements, 0 and 1. It is unique, and denoted F2 . Given any ring R and any set X, let RX denote the set of functions f : X → R. Then RX is, clearly, a ring under valuewise addition and multiplication. For example, take R := F2 . Given f : X → R, put S := f −1 {1}. Then f (x) = 1 if x ∈ S, and f (x) = 0 if x ∈ / S; in other words, f is the characteristic function χS . Thus the characteristic functions form a ring, namely, FX 2 . Given T ⊂ X, clearly χS · χT = χs∩T . Further, χS + χT = χS△T , where S△T is September 3, 2012 11Nts.tex 2 1. Rings and Ideals the symmetric difference: S△T := (S ∪ T ) − (S ∩ T ) = (S − T ) ∪ (T − S), where S − T denotes, as usual, the set of elements of S not in T . Thus the subsets of X form a ring: sum is symmetric difference, and product is intersection. This ring is canonically isomorphic to FX 2 . A ring B is said to be Boolean if f 2 = f for all f ∈ B. Clearly, FX 2 is Boolean. Suppose X is a topological space, and give F2 the discrete topology; that is, every subset is both open and closed. Consider the continuous functions f : X → F2 . Clearly, they are just the χS where S is both open and closed. Clearly, they form a Boolean subring of FX 2 . Conversely, Stone’s Theorem (13.7) asserts that every Boolean ring is canonically isomorphic to the ring of continuous functions from a compact Hausdorff topological space to F2 , or equivalently, to the ring of open and closed subsets of that space. (1.3) (Polynomial rings). — Let R be a ring, P := R[X1 , . . . , Xn ] the polynomial ring in n variables (see [1, pp. 352–3] or [4, p. 268]). Recall that P has this Universal Mapping Property (UMP): given a ring map φ : R → R′ and given an element xi of R′ for each i, there is a unique ring map π : P → R′ with π|R = φ and π(Xi ) = xi . In fact, since π is a ring map, necessarily π is given by the formula: (∑ ) ∑ a(i1 ,...,in ) X1i1 · · · Xnin = π φ(a(i1 ,...,in ) )xi11 · · · xinn . In other words, P is the universal example of an R-algebra equipped with a list of n elements: P is one example, and it maps uniquely to any other [1, (3.4), p. 353]. Similarly, let P ′ := R[{Xλ }λ∈Λ ] be the polynomial ring in an arbitrary set of variables: its elements are the polynomials in any finitely many of the Xλ ; sum and product are defined as in P . Thus P ′ contains as a subring the polynomial ring in any finitely many Xλ , and P ′ is the union of these subrings. Clearly, P ′ has essentially the same UMP as P : given φ : R → R′ and given xλ ∈ R′ for each λ, there is a unique π : P ′ → R′ with π|R = φ and π(Xλ ) = xλ . (1.4) (Ideals). — Let R be a ring. Recall that a subset a is called an ideal if (1) 0 ∈ a (or a is nonempty), (2) whenever a, b ∈ a, also a + b ∈ a, and (3) whenever x ∈ R and a ∈ a, also xa ∈ a. Given elements aλ ∈ R for λ ∈ Λ, by the ideal ⟨aλ ⟩λ∈Λ they generate, we mean the smallest ideal containing them all. If Λ = ∅, then this ideal consists just of 0. ∑ Any ideal containing all the aλ contains any (finite) linear combination xλ aλ with xλ ∈ R and almost all 0. Form the set a of all such linear combinations; clearly, a is an ideal containing all aλ . Thus a is the ideal generated by the aλ . Given a single element a, we say that the ideal ⟨a⟩ is principal. By the preceding observation, ⟨a⟩ is equal to the set of all multiples xa with x ∈ R. Similarly, given ideals aλ of R, by the ideal∑ they generate, we mean the smallest ∑ ideal aλ that contains them all. Clearly, aλ is equal to the set of all finite ∑ linear combinations xλ aλ with xλ ∈ R and aλ ∈ aλ . September 3, 2012 11Nts.tex 1. Rings and Ideals 3 Given two ideals a and b, consider these three nested sets: a + b := {a + b | a ∈ a and b ∈ b}, a ∩ b := {a | a ∈ a and a ∈ b}, ∑ ab := { ai bi | ai ∈ a and bi ∈ b}. They are clearly ideals. They are known as the sum, intersection, and product of a and b. Further, for any ideal c, the distributive law holds: a(b + c) = ab + ac. Let a be an ideal. Then a = R if and only if 1 ∈ a. Indeed, if 1 ∈ a, then x = x · 1 ∈ a for every x ∈ R. It follows that a = R if and only if a contains a unit. Further, if ⟨x⟩ = R, then x is a unit, since then there is an element y such that xy = 1. If a ̸= R, then a is said to be proper. Let φ : R → R′ be a ring map. Let aR′ denote the ideal of R′ generated by φ(a); we call aR′ the extension of a. Let a′ be an ideal of R′ . Clearly, the preimage φ−1 (a′ ) is an ideal of R; we call φ−1 (a′ ) the contraction of a′ . (1.5) (Residue rings). — Let φ : R → R′ be a ring map. Recall its kernel Ker(φ) is defined to be the ideal φ−1 (0) of R. Recall Ker(φ) = 0 if and only if φ is injective. Conversely, let a be an ideal of R. Form the set of cosets of a: R/a := {x + a | x ∈ R}. Recall that R/a inherits a ring structure, and is called the residue ring (or quotient ring or factor ring) of R modulo a. Form the quotient map κ : R → R/a by κx := x + a. The element κx ∈ R/a is called the residue of x. Clearly, κ is surjective, κ is a ring map, and κ has kernel a. Thus every ideal is a kernel! Note that Ker(φ) ⊃ a if and only if φa = 0. Recall that, if Ker(φ) ⊃ a, then there is a ring map ψ : R/a → R′ with ψκ = φ; that is, the following diagram is commutative: κ // R/a R ❏ ❏❏ ❏❏ φ ❏❏❏ ψ  $$ R′ Conversely, if ψ exists, then Ker(φ) ⊃ a, or φa = 0, or aR′ = 0, since κa = 0. Further, if ψ exists, then ψ is unique as κ is surjective. Finally, as κ is surjective, if ψ exists, then ψ is surjective if and only if φ is so. In addition, then ψ is injective if and only if a = Ker(φ). Hence then ψ is an isomorphism if and only if φ is surjective and a = Ker(φ). In particular, always ∼ Im(φ). R/ Ker(φ) −→ (1.5.1) In practice, it is usually more convenient to view R/a not as a set of cosets, but simply as another ring R′ that comes equipped with a surjective ring map φ : R → R′ whose kernel is the given ideal a. Finally, R/a has, as we saw, this UMP: κ(a) = 0 and, given φ : R → R′ such that φ(a) = 0, there is a unique ring map ψ : R/a → R′ such that ψκ = φ. In other words, R/a is the universal example of an R-algebra R′ such that aR′ = 0. The UMP applies, first of all, to the underlying sets, providing a unique map ψ September 3, 2012 11Nts.tex 4 1. Rings and Ideals of sets. Now, φ and κ are ring maps, and ψκ = φ; so ( ) ψ κ(a) + κ(b) = ψκ(a + b) = ψκ(a) + ψκ(b), ( ) ψ κ(a)κ(b) = ψκ(ab) = ψκ(a) · ψκ(b), and ψ(1) = ψκ(1) = 1. But κ is surjective; so κ(a), κ(b) ∈ R/a are arbitrary. Thus ψ is a ring map. The UMP serves to determine R/a up to unique isomorphism. Indeed, say R′ , equipped with φ : R → R′ , has the UMP too. Then φ(a) = 0; so there is a unique ψ : R/a → R′ with ψκ = φ. And κ(a) = 0; so there is a unique ψ ′ : R′ → R/a with ψ ′ φ = κ. Then, as shown, (ψ ′ ψ)κ = κ, but 1 ◦ κ = κ where 1 R/a ♦77 ⑧ ♦ ♦ ⑧ ♦ κ ♦ ⑧ ♦♦♦ ⑧⑧ ψ ♦♦♦φ // ′ ⑧ 1 R ❖❖ R ❄ ❖❖❖ ′ ❖❖❖ ❄❄❄❄ψ ❖❖❖ ❄  κ '' R/a is the identity map of R/a; hence, ψ ′ ψ = 1 by uniqueness. Similarly, ψψ ′ = 1 where 1 now stands for the identity map of R′ . Thus ψ and ψ ′ are inverse isomorphisms. The preceding proof is completely formal, and so works widely. There are many more constructions to come, and each one has an associated UMP, which therefore serves to determine the construction up to unique isomorphism. Exercise (1.6). — Let R be a ring, a an ideal, and P := R[X1 , . . . , Xn ] the polynomial ring. Construct an isomorphism ψ from P/aP onto (R/a)[X1 , . . . , Xn ]. Proposition (1.7). — Let R be a ring, P := R[X] the polynomial ring in one variable, a ∈ R, and π : P → / R the R-algebra map defined by π(X) := a. Then ∼ R. Ker(π) = ⟨X − a⟩, and R[X] ⟨X − a⟩ −→ Proof: Given F (X) ∈ P , the Division Algorithm yields F (X) = G(X)(X−a)+b with G(X) ∈ P and/b ∈ R. Then π(F (X)) = b. Hence Ker(π) = ⟨X − a⟩. Finally, ∼ R. (1.5.1) yields R[X] ⟨X − a⟩ −→ □ (1.8) (Nested ideals). — Let R be a ring, a an ideal, and κ : R → R/a the quotient map. Given an ideal b ⊃ a, form the corresponding set of cosets of a: b/a := {b + a | b ∈ b} = κ(b). Clearly, b/a is an ideal of R/a. Also b/a = b(R/a). Clearly, the operations b 7→ b/a and b′ 7→ κ−1 (b′ ) are inverse to each other, and establish a bijective correspondence between the set of ideals b of R containing a and the set of all ideals b′ of R/a. Moreover, this correspondence preserves inclusions. Given an ideal b ⊃ a, form the composition of the quotient maps / φ : R → R/a → (R/a) (b/a). Clearly, φ is surjective, and Ker(φ) = b. Hence, owing to (1.5), φ factors through the canonical isomorphism ψ in this commutative diagram: R −−−−−→ R/b     ψ y≃ y / R/a − → (R/a) (b/a) September 3, 2012 11Nts.tex 1. Rings and Ideals 5 Exercise (1.9). — Let R be ring, and P := R[X1 , . . . , Xn ] the polynomial ring. Let m ≤ n and a1 , . . . , am ∈ R. Set p := ⟨X1 − a1 , . . . , Xm − am ⟩. Prove that P/p = R[Xm+1 , . . . , Xn ]. (1.10) (Idempotents). — Let R be a ring. Let e ∈ R be an idempotent; that is, e2 = e. Then Re is a ring with e as 1, because (xe)e = xe. But Re is not a subring of R unless e = 1, although Re is an ideal. Set e′ := 1 − e. Then e′ is idempotent, and e · e′ = 0. We call e and e′ complementary and orthogonal idempotents Example (1.11). — Let R := R′ × R′′ be a product of two rings: its operations are performed componentwise. The additive identity is (0, 0); the multiplicative identity is (1, 1). Set e := (1, 0) and e′ := (0, 1). Then e and e′ are complementary idempotents. The next proposition shows this example is the only one possible. Proposition (1.12). — Let R be a ring with complementary idempotents e and e′ . Set R′ := Re and R′′ := Re′ , and form the map φ : R → R′ × R′′ defined by φ(x) := (xe, xe′ ). Then φ is a ring isomorphism. Proof: Define a map φ′ : R → R′ by φ′ (x) := xe. Then φ′ is a ring map since xye = xye2 = (xe)(ye). Hence φ is a ring map. Further, φ is surjective, since (xe, x′ e′ ) = φ(xe + x′ e′ ). Also φ is injective, since if xe = 0 and xe′ = 0, then x = xe + xe′ = 0. Thus φ is an isomorphism. □ Exercise (1.13) (Chinese Remainder Theorem). — Let R be a ring. (1) Let a and b be comaximal ideals; that is, a + b = R. Prove (a) ab = a ∩ b and (b) R/ab = (R/a) × (R/b). (2) Let a be comaximal to both b and b′ . Prove a is also comaximal to bb′ . (3) Let a, b be comaximal, and m, n ≥ 1. Prove am and bn are comaximal. (4) Let a1 , . . . , an be pairwise comaximal. Prove (a) a1 and a2 · · · an are comaximal; (b) a1 ∩ · · · ∩ an = a1∏ · · · an ; ∼ (c) R/(a1 · · · an ) −→ (R/ai ). Exercise (1.14). — First, given a prime number p and a k ≥ 1, find the idempotents in Z/⟨pk ⟩. Second, find the idempotents in Z/⟨12⟩. Third, find the number ∏N of idempotents in Z/⟨n⟩ where n = i=1 pni i with pi distinct prime numbers. Exercise (1.15). — Let R := R′ × R′′ be a product of rings, a ⊂ R an ideal. Show a = a′ × a′′ with a′ ⊂ R′ and a′′ ⊂ R′′ ideals. Show R/a = (R′ /a′ ) × (R′′ /a′′ ). Exercise (1.16). — Let R be a ring, and e, e′ idempotents. (See (10.6) also.) (1) Set a := ⟨e⟩. Show a is idempotent; that is, a2 = a. (2) Let a be a principal idempotent ideal. Show a = ⟨f ⟩ with f idempotent. (3) Assume ⟨e⟩ = ⟨e′ ⟩. Show e = e′ . (4) Set e′′ := e + e′ − ee′ . Show ⟨e, e′ ⟩ = ⟨e′′ ⟩ and e′′ is idempotent. (5) Let e1 , . . . , er be idempotents. Show ⟨e1 , . . . , er ⟩ = ⟨f ⟩ with f idempotent. (6) Assume R is Boolean. Show every finitely generated ideal is principal. September 3, 2012 11Nts.tex 6 2. Prime Ideals 2. Prime Ideals Prime ideals are the key to the structure of commutative rings. So we review the basic theory. Specifically, we define prime ideals, and show their residue rings are domains. We show maximal ideals are prime, and discuss examples. Then we use Zorn’s Lemma to prove the existence of maximal ideals in every nonzero ring. Definition (2.1). — Let R be a ring. An element x is called a zerodivisor if there is a nonzero y with xy = 0; otherwise, x is called a nonzerodivisor. Denote the set of zerodivisors by z.div(R). A subset S is called multiplicative if 1 ∈ S and if x, y ∈ S implies xy ∈ S. An ideal p is called prime if its complement R − p is multiplicative, or equivalently, if 1 ∈ / p and if xy ∈ p implies x ∈ p or y ∈ p. Exercise (2.2). — Let a and b be ideals, and p a prime ideal. Prove that these conditions are equivalent: (1) a ⊂ p or b ⊂ p; and (2) a ∩ b ⊂ p; and (3) ab ⊂ p. (2.3) (Fields, Domains). — A ring is called a field if 1 ̸= 0 and if every nonzero element is a unit. Standard examples include the rational numbers Q, the real numbers R, and the complex numbers C. A ring is called an integral domain, or simply a domain, if ⟨0⟩ is prime, or equivalently, if R is nonzero and has no nonzero zerodivisors. Every domain R is a subring of its fraction field Frac(R), which consists of the fractions x/y with x, y ∈ R and y ̸= 0. Conversely, any subring R of a field K, including K itself, is a domain; indeed, any nonzero x ∈ R cannot be a zerodivisor, because, if xy = 0, then (1/x)(xy) = 0, so y = 0. Further, Frac(R) has this UMP: the inclusion of R into any field L extends uniquely to an inclusion of Frac(R) into L. For example, the ring of integers Z is a domain, and Frac(Z) = Q ⊂ R ⊂ C. Let R be a domain, and R[X] the polynomial ring in one variable. Then R[X] is a domain too; in fact, given any two nonzero polynomials f and g, not only is their product f g nonzero, but its leading coefficient is the product of the leading coefficients of f and g. By induction, the polynomial ring in n variables R[X1 , . . . , Xn ] is a domain, since R[X1 , . . . , Xn ] = R[X1 , . . . , Xn−1 ][Xn ]. Hence the polynomial ring in an arbitrary set of variables R[{Xλ }λ∈Λ ] is a domain, since any two elements lie in a polynomial subring in finitely many of the Xλ . A similar argument proves that R× is full group of units in the polynomial ring. This statement can fail if R is not a domain. For example, if a2 = 0 in R, then (1 + aX)(1 − aX) = 1 in R[X]. The fraction field Frac(R[{Xλ }λ∈Λ ]) is called the field of rational functions, and is also denoted by K({Xλ }λ∈Λ ) where K := Frac(R). Exercise (2.4). — Given a prime number p and an integer n ≥ 2, prove that the residue ring Z/⟨pn ⟩ does not contain a domain. Exercise (2.5). — Let R := R′ × R′′ be a product of two rings. Show that R is a domain if and only if either R′ or R′′ is a domain and the other is 0. September 3, 2012 11Nts.tex 2. Prime Ideals 7 (2.6) (Unique factorization). — Let R be a domain, p a nonzero nonunit. We call p prime if, whenever p | xy (that is, there exists z ∈ R such that pz = xy), either p | x or p | y. Clearly, p is prime if and only if the ideal ⟨p⟩ is prime. We call p irreducible if, whenever p = yz, either y or z is a unit. We call R a Unique Factorization Domain (UFD) if every nonzero element is a product of irreducible elements in a unique way up to order and units. In general, prime elements are irreducible; in a UFD, irreducible elements are prime. Standard examples of UFDs include any field, the integers Z, and a polynomial ring in n variables over a UFD; see [1, p. 398, p. 401], [4, Cor. 18.23, p. 297]. Lemma (2.7). — Let φ : R → R′ be a ring map, and T ⊂ R′ a subset. If T is multiplicative, then φ−1 T is multiplicative; the converse holds if φ is surjective. Proof: Both assertions are easy to check. □ ′ ′ Proposition (2.8). — Let φ : R → R be a ring map, and q ⊂ R an ideal. If q is prime, then φ−1 q is prime; the converse holds if φ is surjective. Proof: By (2.7), R − p is multiplicative if and only if R′ − q is. So the assertion results from Definitions (2.1). □ Proposition (2.9). — Let R be a ring, p an ideal. Then p is prime if and only if R/p is a domain. Proof: By (2.8), p is prime if and only if ⟨0⟩ ⊂ R/p is. So the assertion results from the definition of domain in (2.3). □ Exercise (2.10). — Let R be a ring, p a prime ideal, R[X] the polynomial ring. Show that pR[X] and pR[X] + ⟨X⟩ are prime ideals of R[X]. Exercise (2.11). — Let R be a domain, and R[X1 , . . . , Xn ] the polynomial ring in n variables. Let m ≤ n, and set p := ⟨X1 , . . . , Xm ⟩. Prove p is a prime ideal. Exercise (2.12). — Let R := R′ × R′′ be a product of rings. Show every prime ideal of R has the form p′ × R′′ with p′ ⊂ R′ prime or R′ × p′′ with p′′ ⊂ R′′ prime. Definition (2.13). — Let R be a ring. An ideal m is said to be maximal if m is proper and if there is no proper ideal a with m ⫋ a. Example (2.14). — Let R be a domain. In the polynomial ring R[X, Y ] in two variables, ⟨X⟩ is prime by (2.11). However, ⟨X⟩ is not maximal since ⟨X⟩ ⫋ ⟨X, Y ⟩. Proposition (2.15). — A ring R is a field if and only if ⟨0⟩ is a maximal ideal. Proof: Suppose R is a field. Let a be a nonzero ideal, and a a nonzero element of a. Since R is a field, a ∈ R× . So (1.4) yields a = R. Conversely, suppose ⟨0⟩ is maximal. Take x ̸= 0. Then ⟨x⟩ ̸= ⟨0⟩. So ⟨x⟩ = R. So x is a unit by (1.4). Thus R is a field. □ Exercise (2.16). — Let k be a field, R a nonzero ring, φ : k → R a ring map. Prove φ is injective. Proposition (2.17). — Let R be a ring, m an ideal. Then m is maximal if and only if R/m is a field. Proof: Clearly, m is maximal in R if and only if ⟨0⟩ is maximal in R/m by (1.8). Hence the assertion results from (2.15). □ September 3, 2012 11Nts.tex 8 2. Prime Ideals Exercise (2.18). — Let B be a Boolean ring. Show that every prime p is maximal, and B/p = F2 . Example (2.19). — Let k be a field, a1 , . . . , an ∈ k, and P := k[X1 , . . . , Xn ] the polynomial ring in n variables. Set m := ⟨X1 − a1 , . . . , Xn − an ⟩. Then P/m = k by (1.9); so m is maximal by (2.17). Exercise (2.20). — Prove the following statements or give a counterexample. (1) The complement of a multiplicative subset is a prime ideal. (2) Given two prime ideals, their intersection is prime. (3) Given two prime ideals, their sum is prime. (4) Given a ring map φ : R → R′ , the operation φ−1 carries maximal ideals of R′ to maximal ideals of R. (5) In (1.8), κ−1 takes maximal ideals of R/a to maximal ideals of R. Exercise (2.21). — Let k be a field, P := k[X1 , . . . , Xn ] the polynomial ring, f ∈ P nonzero. Let d be the highest power of any variable appearing in f . (1) Let S ⊂ k have at least dn + 1 elements. Proceeding by induction on n, find a1 , . . . , an ∈ S with f (a1 , . . . , an ) ̸= 0. (2) Using the algebraic closure K of k, find a maximal ideal m of P with f ∈ / m. Corollary (2.22). — In a ring, every maximal ideal is prime. Proof: A field is a domain by (2.3). So (2.9) and (2.17) yield the result. □ (2.23) (PIDs). — A domain R is called a Principal Ideal Domain (PID) if every ideal is principal. Examples include the polynomial ring k[X] in one variable over a field k, and the ring Z of integers. Every PID is a UFD by [1, (2.12), p. 396], [4, Thm. 18.11, p. 291]. Let R be a PID, and p ∈ R irreducible. Then ⟨p⟩ is maximal; indeed, if ⟨p⟩ ⫋ ⟨x⟩, then p = xy for some nonunit y, and so x must be a unit since p is irreducible. So (2.17) implies that R/⟨p⟩ is a field. Exercise (2.24). — Prove that, in a PID, elements x and y are relatively prime (share no prime factor) if and only if the ideals ⟨x⟩ and ⟨y⟩ are comaximal. Example (2.25). — Let R be a PID, and p ∈ R a prime. Set k := R/⟨p⟩. Let P := R[X] be the polynomial ring in one variable. Take g ∈ P , let g ′ be its image in k[X], and assume g ′ is irreducible. Set m := ⟨p, g⟩. Then m is maximal by (2.17); ∼ k[X]/⟨g ′ ⟩ by (1.5), and k[X]/⟨g ′ ⟩ is a field by (2.23). indeed, P/m −→ Theorem (2.26). — Let R be a PID. Let P := R[X] be the polynomial ring in one variable, and p a prime ideal of P . (1) Then p = ⟨0⟩, or p = ⟨f ⟩ with f prime, or p is maximal. (2) Assume p is maximal. Then either( p = ⟨f) ⟩ with f prime, or p = ⟨p, g⟩ with p ∈ R prime and g ∈ P with image g ′ ∈ R/⟨p⟩ [X] prime. Proof: Assume p ̸= ⟨0⟩. Take a nonzero f1 ∈ p. Since p is prime, p contains a prime factor f1′ of f1 . Replace f1 by f1′ . Assume p ̸= ⟨f1 ⟩. Then there is an prime f2 ∈ p − ⟨f1 ⟩. Set K := Frac(R). Gauss’s Lemma [1, p. 401], [4, Thm. 18.15, p. 295] implies that f1 and f2 are also prime in K[X]. So f1 and f2 are relatively prime in K[X]. So (2.23) and (2.24) yield g1 , g2 ∈ P and c ∈ R with (g1 /c)f1 + (g2 /c)f2 = 1. So c = g1 f1 + g2 f2 ∈ R ∩ p. Hence R ∩ p ̸= 0. But R ∩ p September 3, 2012 11Nts.tex 2. Prime Ideals 9 is prime, and R is a PID; so R ∩ p = ⟨p⟩ where p is prime by (2.6). Set k := R/⟨p⟩. Then k is a field by (2.23). Set q := p/⟨p⟩ ⊂ k[X]. Then k[X]/q = P/p by (1.6) and (1.8). But P/p is a domain as p is prime. Hence q = ⟨g ′ ⟩ where g ′ is prime in k[X] by (2.6). Then q is maximal by (2.23). So p is maximal by (1.7). Take g ∈ p with image g ′ . Then p = ⟨p, g⟩ as p/⟨p⟩ = ⟨g ′ ⟩. □ Exercise (2.27). — Preserve the setup of (2.26). Let f := a0 X n + · · · + an be a polynomial of positive degree n. Assume that R has infinitely many prime elements p, or simply that there is a p such that p ∤ a0 . Show that ⟨f ⟩ is not maximal. Theorem (2.28). — Every proper ideal a is contained in some maximal ideal. Proof: Set S := {ideals b | b ⊃ a and b ̸∋ 1}. Then a ∈ S, and S ∪ is partially ordered by inclusion. Given a totally ordered subset {bλ } of S, set b := bλ . Then b is clearly an ideal, and 1 ∈ / b; so b is an upper bound of {bλ } in S. Hence by Zorn’s Lemma [7, pp. 25, 26], [6, p. 880, p. 884], S has a maximal element, and it is the desired maximal ideal. □ Corollary (2.29). — Let R be a ring, x ∈ R. Then x is a unit if and only if x belongs to no maximal ideal. Proof: By (1.4), x is a unit if and only if ⟨x⟩ is not proper. So (2.28) yields the assertion. □ September 3, 2012 11Nts.tex 10 3. Radicals 3. Radicals Two radicals of a ring are commonly used in Commutative Algebra: the Jacobson radical, which is the intersection of all maximal ideals, and the nilradical, which is the set of all nilpotent elements. Closely related to the nilradical is the radical of a subset. We define these three radicals, and discuss examples. In particular, we study local rings; a local ring has only one maximal ideal, which is then its Jacobson radical. We prove two important general results: Prime Avoidance, which states that, if an ideal lies in a finite union of primes, then it lies in one of them, and the Scheinnullstellensatz, which states that the nilradical of an ideal is equal to the intersection of all the prime ideals containing it. Definition (3.1). — Let R be a ring. Its (Jacobson) radical rad(R) is defined to be the intersection of all its maximal ideals. Proposition (3.2). — Let R be a ring, x ∈ R. Then x ∈ rad(R) if and only if 1 − xy is a unit for all y ∈ R. Proof: Assume x ∈ rad(R). Let m be a maximal ideal. Suppose 1 − xy ∈ m. Since x ∈ m too, also 1 ∈ m, a contradiction. So 1 − xy is a unit by (2.29). Conversely, assume x ∈ / rad(R). Then there is a maximal ideal m with x ∈ / m. So ⟨x⟩ + m = R. Hence there exist y ∈ R and m ∈ m such that xy + m = 1. Then 1 − xy = m ∈ m. So 1 − xy is not a unit by (2.29), or directly by (1.4). □ Definition (3.3). — A ring A is called local if it has exactly one maximal ideal, and semilocal if it has at least one and at most finitely many. Lemma (3.4) (Nonunit Criterion). — Let A be a ring, n the set of nonunits. Then A is local if and only if n is an ideal; if so, then n is the maximal ideal. Proof: Every proper ideal a lies in n as a contains no unit. So, if n is an ideal, then it is a maximal ideal, and the only one. Thus A is local. Conversely, assume A is local with maximal ideal m. Then A − n = A − m by (2.29). So n = m. Thus n is an ideal. □ Example (3.5). — The product ring R′ × R′′ is not local by (3.4) if both R′ and R′′ are nonzero. Indeed, (1, 0) and (0, 1) are nonunits, but their sum is a unit. Exercise (3.6). — Let A be a ring, m a maximal ideal such that 1 + m is a unit for every m ∈ m. Prove A is local. Is this assertion still true if m is not maximal? Example (3.7). — Let R be a ring. A formal power series in the n variables ∑ X1 , . . . , Xn is a formal infinite sum of the form a(i) X1i1 · · · Xnin where a(i) ∈ R and where (i) = (i1 , . . . , in ) with each ij ≥ 0. Addition and multiplication are performed as for polynomials; with these operations, these series form a ring R[[X1 , . . . , Xn ]]. ∑ Set P := R[[X1 , . . . , Xn ]] and a := ⟨X1 , . . . , Xn ⟩. Then a(i) X1i1 · · · Xnin 7→ a(0) is a canonical surjective ring map P → R with kernel a; hence, P/a = R. Given an ideal m ⊂ R, set n := a + mP . Then (1.8) yields P/n = R/m. Suppose R is a local ring with maximal ideal m. Then any power series f ∈ / n is of the form f = a(1 − g) with a ∈ R× and g ∈ a. Set h := a−1 (1 + g + g 2 + · · · ); this sum makes sense as the component of degree d involves only the first d + 1 September 3, 2012 11Nts.tex 3. Radicals 11 summands. Clearly f · h = 1. Hence the nonunits constitute n. Thus P is local with maximal ideal n by (3.4). Example (3.8). — Let k be a ring, and A := k[[X]] the formal∑ power series ring ∞ in one variable. A Laurent series is a formal sum of the form i=−m ai X i with ai ∈ k and m ∈ Z. The Laurent series form a ring k{{X}}. Set K := k{{X}}. ∑∞ Set f := i=−m ai X i . If a−m ∈ k × , then f ∈ K × ; indeed, f = a−m X −m (1 − g) m 2 where g ∈ A, and f · a−1 −m X (1 + g + g + · · · ) = 1. Assume k is a field. If f ̸= 0, then f = X −m u where u ∈ A× . Let a ⊂ A be a nonzero ideal. Suppose f ∈ a. Then X −m ∈ a. Let n be the smallest integer such that X n ∈ a. Then −m ≥ n. Set b := X −m−n u. Then b ∈ A and f = bX n . Hence a = ⟨X n ⟩. Thus A is a PID. Further, K is a field. In fact, K = Frac(A) as any nonzero f ∈ K is of the form f = u/X m where u, X m ∈ A. Let A[Y ] be the polynomial ring in one variable, and ι : A ֒→ K the inclusion. Define φ : A[Y ] → K by φ|A = ι and φ(Y ) := X −1 . Then φ is surjective. Set m := Ker(φ). Then m is maximal by (2.17) and (1.5). So by (2.26), m has the form ⟨f ⟩ with f irreducible, or the form ⟨p, g⟩ with p ∈ A irreducible and g ∈ A[Y ]. But m ∩ A = 0 as ι is injective. So m = ⟨f ⟩. But XY − 1 belongs to m, and is clearly irreducible; hence, XY − 1 = f u with u a/unit. Thus ⟨XY − 1⟩ is maximal. In addition, / ⟨X, Y ⟩ is/maximal. Indeed, A[Y ] ⟨Y ⟩ = A by (1.7), and so (3.7) yields A[Y ] ⟨X, Y ⟩ = A ⟨X⟩ = k. However, ⟨X, Y ⟩ is not principal, as no nonunit of A[Y ] divides both X and Y . Thus A[Y ] has both principal and nonprincipal maximal ideals, the two types allowed by (2.26). Proposition (3.9). — Let R be a ring, S a multiplicative subset, and a an ideal with a ∩ S = ∅. Set S := {ideals b | b ⊃ a and b ∩ S = ∅}. Then S has a maximal element p, and every such p is prime. Proof: CLearly, a ∈ S, and S is∪partially ordered by inclusion. Given a totally ordered subset {bλ } of S, set b := bλ . Then b is an upper bound for {bλ } in S. So by Zorn’s Lemma, S has a maximal element p. Let’s show p is prime. Take x, y ∈ R − p. Then p + ⟨x⟩ and p + ⟨y⟩ are strictly larger than p. So there are p, q ∈ p and a, b ∈ R with p + ax ∈ S and q + by ∈ S. Since S is multiplicative, pq + pby + qax + abxy ∈ S. But pq + pby + qax ∈ p, so xy ∈ / p. Thus p is prime. □ Exercise (3.10). — Let φ : R → R′ be a ring map, p an ideal of R. Prove (1) there is an ideal q of R′ with φ−1 (q) = p if and only if φ−1 (pR′ ) = p; (2) if p is prime with φ−1 (pR′ ) = p, then there’s a prime q of R′ with φ−1 (q) = p. Exercise (3.11). — Use Zorn’s lemma to prove that any prime ideal p contains a prime ideal q that is minimal containing any given subset s ⊂ p. (3.12) (Saturated multiplicative subsets). — Let R be a ring, and S a multiplicative subset. We say S is saturated if, given x, y ∈ R with xy ∈ S, necessarily x, y ∈ S. For example, it’s easy to see the following. The group of units R× is a saturated multiplicative subset. Further, let φ : R → R′ be a ring map, T ⊂ R′ a subset. If T is saturated multiplicative, then so is φ−1 T . The converse holds if φ is surjective. Exercise (3.13). — Let R be a ring, S a subset. Show that S is saturated multiplicative if and only if R − S is a union of primes. September 3, 2012 11Nts.tex 12 3. Radicals Exercise (3.14). — Let R be a ring, and S a multiplicative subset. Define its saturation to be the subset S := { x ∈ R | there is y ∈ R with xy ∈ S }. (1) Show (a) that S ⊃ S, and (b) that S is saturated multiplicative, and (c) that any saturated multiplicative subset T containing S also contains S. (2) Show that R − S is the union U of all the primes p with p ∩ S = ∅. ∪ (3) Let a be an ideal; assume S = 1 + a; set W := p∈V(a) p. Show R − S = W . Lemma (3.15) (Prime Avoidance). — Let R be a ring, a a subset of R that is stable under addition and multiplication, and p1 , . . . , pn ideals such that p3 , . . . , pn are prime. If a ̸⊂ pj∪for all j, then there is an x ∈ a such that x ∈ / pj for all j; or n equivalently, if a ⊂ i=1 pi , then a ⊂ pi for some i. Proof: Proceed by induction on n. If n = 1, the assertion is trivial. Assume that n ≥ 2 and by induction that, for every i, there is an xi ∈ a such that xi ∈ / pj for all j ̸= i. We may assume xi ∈ pi for every i, else we’re done. If n = 2, then clearly x1 + x2 ∈ / pj for j = 1, 2. If n ≥ 3, then (x1 · · · xn−1 ) + xn ∈ / pj for all j as, if j = n, then xn ∈ pn and pn is prime, and if j < n, then xn ∈ / pj and xj ∈ pj . □ Exercise (3.16). — Let k be an infinite field. ∪ (1) Let V be a vector space, W1 , . . . , Wr proper subspaces. Show Wi ̸= V . ∪ (2) In (1), let W ⊂ Wi be a subspace. Show W ⊂ Wi for some i. ∪ (3) Let R a k-algebra, a, a1 , . . . , ar ideals. If a ⊂ ai , show a ⊂ ai for some i. Exercise (3.17). — Let k be a field, R := k[X, Y ] the polynomial ring in two variables, m := ⟨X, Y ⟩. Show m is a union of smaller primes. (3.18) be a ring, a a subset. Then the radical of a is the √ (Nilradical ). — Let R √ by the formula a := {x ∈ R | xn ∈ a for some n = n(x) ≥ 1}. set a defined √√ √ √ Notice √ a = a. Also, if a is an intersection of prime ideals, then a = a. We call ⟨0⟩ the nilradical, and sometimes denote it by nil(R). We call an √ element x ∈ R nilpotent if x belongs to ⟨0⟩, that is, if xn = 0 for some n ≥ 1. We call R reduced if nil(R) = ⟨0⟩, that is, if R has no nonzero nilpotents. Exercise (3.19). — Find the nilpotents in Z/⟨n⟩. In particular, take n = 12. Exercise (3.20). — Let φ : R → R′ be a ring map, b ⊂ R′ a subset. Prove √ √ φ−1 b = φ−1 b. √ Exercise (3.21). — Let R be a ring, a ⊂ ⟨0⟩ an ideal, and P := R[Y ] the polynomial ring in one variable. Let u ∈ R be a unit, and x ∈ R a nilpotent. (1) Prove (a) that u + x is a unit in R and (b) that u + xY is a unit in P . (2) Suppose w ∈ R maps to a unit of R/a. Prove that w is a unit in R. Theorem (3.22) (Scheinnullstellensatz). — Let R be a ring, a an ideal. Then ∩ √ a = p⊃a p where p runs through all the prime ideals containing a. (By convention, the empty intersection is equal to R.) September 3, 2012 11Nts.tex 3. Radicals 13 √ and Proof: Take x ∈ / a. Set S := {1, x, x2 , . . .}. Then S is multiplicative, ∩ a ∩ S = ∅. By (3.9), there is a p ⊃ a, but x ∈ / p. So x ∈ / p⊃a p. Thus ∩ √ a ⊃ p⊃a p. ∩ √ √ Conversely, take x ∈ a. Say xn ∈ a ⊂ p. Then x ∈ p. Thus a = p⊃a p. □ Exercise (3.23). — Let B be a Boolean ring. Show that rad(B) = nil(B) = ⟨0⟩. √ Proposition (3.24). — Let R be a ring, a an ideal. Then a is an ideal. √ Proof: Take x, y ∈ a; say xn ∈ a and y m ∈ a. Then ( ) i j ∑ xy . (x + y)n+m−1 = i+j=m+n−1 n+m−1 j This sum belongs to a as, in each summand, either xi√or y j does, since, √ if i ≤ n − 1 a is an ideal. and j ≤ m − 1, then i + j ≤ m + n − 2. Thus x + y ∈ a. So clearly ∩ Alternatively, given any collection of ideals a , note that a is also an ideal. λ λ √ So a is an ideal owing to (3.22). □ √ Exercise (3.25). (√—)nLet R be a ring, and a an ideal. Assume a is finitely generated. Show a ⊂ a for all large n. Exercise (3.26). — Let R be a ring, q an ideal, p a finitely generated prime. √ Prove that p = q if and only if there is n ≥ 1 such that p ⊃ q ⊃ pn . Proposition (3.27). — Let R be a ring. Assume R is reduced with only one minimal prime q. Then R is a domain. √ Proof: Since R is reduced, ⟨0⟩ = ⟨0⟩ by (3.18). Hence ⟨0⟩ is equal to the intersection of all the prime ideals p by (3.22). By (3.11), every p contains q. So ⟨0⟩ = q. Thus R is a domain. □ Exercise (3.28). — Let R be a ring. Assume R ∏is reduced and has finitely many minimal prime ideals p1 , . . . , pn . Prove φ : R → (R/pi ) is injective, and for each i, there is some (x1 , . . . , xn ) ∈ Im(φ) with xi ̸= 0 but xj = 0 for j ̸= i. September 3, 2012 11Nts.tex 14 4. Modules 4. Modules In Commutative Algebra, it has proven advantageous to expand the study of rings to include modules. Thus we obtain a richer theory, which is more flexible and more useful. We begin the expansion here by discussing residue modules, kernels, and images. In particular, we identify the universal property of the residue module, and use it to construct the Noether isomorphisms. We also construct free modules, direct sums, and direct products, and we describe their universal properties. (4.1) (Modules). — Let R be a ring. Recall that an R-module M is an abelian group, written additively, with a scalar multiplication, R × M → M , written (x, m) 7→ xm, which is (1) distributive, x(m + n) = xm + xn and (x + y)m = xm + xn, (2) associative, x(ym) = (xy)m, and (3) unitary, 1 · m = m. For example, if R is a field, then an R-module is a vector space. Moreover, a Z-module is just an abelian group; multiplication is repeated addition. As in (1.1), for any x ∈ R and m ∈ M , we have x · 0 = 0 and 0 · m = 0. A submodule N of M is a subgroup that is closed under multiplication; that is, xn ∈ N for all x ∈ R and n ∈ N . For example, the ring R is itself an R-module, and the submodules are just the ideals. Given an ideal a, let aN denote the smallest submodule containing all products an with ∑ a ∈ a and n ∈ N . Similar to (1.4), clearly aN is equal to the set of finite sums ai ni with ai ∈ a and ni ∈ N . Given m ∈ M , we call the set of x ∈ R with xm = 0 the annihilator of m, and denote it Ann(m). We call the set of x ∈ R with xm = 0 for all m ∈ M the annihilator of M , and denote it Ann(M ). Clearly, Ann(m) and Ann(M ) are ideals. (4.2) (Homomorphisms). — Let R be a ring, M and N modules. Recall that a homomorphism, or module map is a map α : M → N that is R-linear: α(xm + yn) = x(αm) + y(αn). Associated to a homomorphism α : M → N are its kernel and its image Ker(α) := α−1 (0) ⊂ M and Im(α) := α(M ) ⊂ N. They are defined as subsets, but are obviously submodules. A homomorphism α is called an isomorphism if it is bijective. If so, then we ∼ N . Then the set-theoretic inverse α−1 : N → M is a homomorwrite α : M −→ phism too. So α is an isomorphism if and only if there is a set map β : N → M such that βα = 1M and αβ = 1N , and then β = α−1 . If there is an unspecified isomorphism between M and N , then we write M = N when it is canonical (that is, it does not depend on any artificial choices), and we write M ≃ N otherwise. The set of homomorphisms α is denoted by HomR (M, N ) or simply Hom(M, N ). It is an R-module with addition and scalar multiplication defined by (α + β)m := αm + βm and (xα)m := x(αm) = α(xm). September 3, 2012 11Nts.tex 4. Modules 15 Homomorphisms α : L → M and β : N → P induce, via composition, a map Hom(α, β) : Hom(M, N ) → Hom(L, P ), which is obviously a homomorphism. When α is the identity map 1M , we write Hom(M, β) for Hom(1M , β); similarly, we write Hom(α, N ) for Hom(α, 1N ). Exercise (4.3). — Let R be a ring, M a module. Consider the set map θ : Hom(R, M ) → M defined by θ(ρ) := ρ(1). Show that θ is an isomorphism, and describe its inverse. (4.4) (Endomorphisms). — Let R be a ring, M a module. An endomorphism of M is a homomorphism α : M → M . The module of endomorphisms Hom(M, M ) is also denoted EndR (M ).It is a ring, usually noncommutative, with multiplication given by composition. Further, EndR (M ) is a subring of EndZ (M ). Given x ∈ R, let µx : M → M denote the map of multiplication by x, defined by µx (m) := xm. It is an endomorphism. Further, x 7→ µx is a ring map µR : R → EndR (M ) ⊂ EndZ (M ). (Thus we may view µR as representing R as a ring of operators on the abelian group M .) Note that Ker(µR ) = Ann(M ). Conversely, given an abelian group N and a ring map ν : R → EndZ (N ), we obtain a module structure on N by setting xn := (νx)(n). Then µR = ν. We call M faithful if µR : R → EndR (M ) is injective, or Ann(M ) = 0. For example, R is a faithful R-module, as x · 1 = 0 implies x = 0. (4.5) (Algebras). — Fix two rings R and R′ . Suppose R′ is an R-algebra with structure map φ. Let M ′ be an R′ -module. Then M ′ is also an R-module by restriction of scalars: xm := φ(x)m. In other words, the R-module structure on M ′ corresponds to the composition φ µ ′ R R− → R′ −− → EndZ (M ′ ). In particular, R′ it is an R-module; further, for all x ∈ R and y, z ∈ R′ , (xy)z = x(yz). ′ ′ Indeed, R is an R -module, so an R-module by restriction of scalars; further, (xy)z = x(yz) since (φ(x)y)z = φ(x)(yz) by associativity in R′ . Conversely, suppose R′ is an R-module such that (xy)z = x(yz). Then R′ has an R-algebra structure that is compatible with the given R-module structure. Indeed, define φ : R → R′ by φ(x) := x · 1. Then φ(x)z = xz as (x · 1)z = x(1 · z). So the composition µR′ φ : R → R′ → EndZ (R′ ) is equal to µR . Hence φ is a ring map, because µR is one and µR′ is injective by (4.4). Thus R′ is an R-algebra, and restriction of scalars recovers its given R-module structure. Suppose that R′ = R/a for some ideal a. Then an R-module M has a compatible ′ R -module structure if and only if aM = 0; if so, then the R′ -structure is unique. Indeed, the ring map µR : R → EndZ (M ) factors through R′ if and only if µR (a) = 0 by (1.5), so if and only if aM = 0; as EndZ (M ) may be noncommutative, we must apply (1.5) to µR (R), which is commutative. Again suppose R′ is an arbitrary R-algebra with structure map φ. A subalgebra ′′ R of R′ is a subring such that φ maps into R′′ . The subalgebra generated by September 3, 2012 11Nts.tex 16 4. Modules x1 , . . . , xn ∈ R′ is the smallest R-subalgebra that contains them. We denote it by R[x1 , . . . , xn ]. It clearly contains all polynomial combinations f (x1 , . . . , xn ) with coefficients in R. In fact, the set R′′ of these polynomial combinations is itself clearly an R-subalgebra; hence, R′′ = R[x1 , . . . , xn ]. We say R′ is a finitely generated R-algebra or is algebra finite over R if there exist x1 , . . . , xn ∈ R′ such that R′ = R[x1 , . . . , xn ]. (4.6) (Residue modules). — Let R be a ring, M a module, M ′ ⊂ M a submodule. Form the set of cosets M/M ′ := {m + M ′ | m ∈ M }. Recall that M/M ′ inherits a module structure, and is called the residue module or quotient of M modulo M ′ . Form the quotient map κ : M → M/M ′ by κ(m) := m + M ′ . Clearly κ is surjective, κ is linear, and κ has kernel M ′ . Let α : M → N be linear. Note that Ker(α) ⊃ M ′ if and only if α(M ′ ) = 0. Recall that, if Ker(α) ⊃ M ′ , then there exists a homomorphism β : M/M ′ → N such that βκ = α; that is, the following diagram is commutative: κ // M/M ′ M ❏ ❏❏ ❏❏ ❏ β α ❏❏  $$ N Conversely, if β exists, then Ker(α) ⊃ M ′ , or α(M ′ ) = 0, as κ(M ′ ) = 0. Further, if β exists, then β is unique as κ is surjective. Finally, since κ is surjective, if β exists, then β is surjective if and only if α is so. In addition, then β is injective if and only if M ′ = Ker(α). Hence β is an isomorphism if and only if α is surjective and M ′ = Ker(α). In particular, always ∼ Im(α). M/ Ker(α) −→ (4.6.1) In practice, it is usually more convenient to view M/M ′ not as a set of cosets, but simply another module M ′′ that comes equipped with a surjective homomorphism α : M → M ′′ whose kernel is the given submodule M ′ . Finally, as we have seen, M/M ′ has the following UMP: κ(M ′ ) = 0, and given α : M → N such that α(M ′ ) = 0, there is a unique homomorphism β : M/M ′ → N such that βκ = α. Formally, the UMP determines M/M ′ up to unique isomorphism. (4.7) (Cyclic modules). — Let R be a ring. A module M is said to be cyclic if there exists m ∈ M such that M = Rm. If so, form α : R → M by x 7→ xm; then ∼ M as Ker(α) = Ann(m); see (4.6.1). α induces an isomorphism R/ Ann(m) −→ Note that Ann(m) = Ann(M ). Conversely, given any ideal a, the R-module R/a is cyclic, generated by the coset of 1, and Ann(R/a) = a. (4.8) (Noether Isomorphisms). — Let R be a ring, N a module, and L and M submodules. First, assume L ⊂ M ⊂ N . Form the following composition of quotient maps: / α : N → N/L → (N/L) (M/L). Clearly α is surjective, and Ker(α) = M . Hence owing to (4.6), α factors through September 3, 2012 11Nts.tex 4. Modules 17 the isomorphism β in this commutative diagram: N −−−−−→ N/M     β y≃ y / N/L − → (N/L) (M/L) (4.8.1) Second, let L + M denote the set of all sums ℓ + m with ℓ ∈ L and m ∈ M . Clearly L + M is a submodule of N . It is called the sum of L and M . Form the composition α′ of the inclusion map L → L + M and the quotient map L + M → (L + M )/M . Clearly α′ is surjective and Ker(α′ ) = L ∩ M . Hence owing to (4.6), α′ factors through the isomorphism β ′ in this commutative diagram: L −−−→ L/(L ∩ M )     β ′ y≃ y (4.8.2) L+M − → (L + M )/M The isomorphisms of (4.6.1) and (4.8.1) and (4.8.2) are called Noether’s First, Second, and Third Isomorphisms. (4.9) (Cokernels, coimages). — Let R be a ring, α : M → N a linear map. Associated to α are its cokernel and its coimage, Coker(α) := N/ Im(α) and Coim(α) := M/ Ker(α); they are quotient modules, and their quotient maps are both denoted by κ. Note (4.6) yields the UMP of the cokernel: κα = 0, and given a map β : N → P with βα = 0, there is a unique map γ : Coker(α) → P with γκ = β as shown below α // N κ // Coker(α) M ❏ ❏❏ tt ❏❏ β tt ❏❏ ❏$$  zzttt γ P ∼ Im(α). Further, (4.6.1) becomes Coim(α) −→ (4.10) (Free modules). — Let R be a ring, Λ a set, M a module. Given elements mλ ∈ M for λ ∈ Λ, by the submodule they generate, we mean the smallest submodule that contains them all. Clearly,∑any submodule that contains them all contains any (finite) linear combination xλ mλ with xλ ∈ R. On the other hand, consider the set N of all such linear combinations; clearly, N is a submodule containing the mλ . Thus N is the submodule generated by the mλ . ∑ The mλ are said to be free or linearly independent if, whenever xλ mλ = 0, also xλ = 0 for all λ. Finally, the mλ are said to form a free basis of M if they are free and generate M ; if so, then we say M is free on the mλ . We say M is finitely generated if it has a finite set of generators. We say M is free if it has a free basis. If so, then by (10.5) below, any two free bases have the same number ℓ of elements, and we say M is free of rank ℓ. For example, form the set of restricted vectors R⊕Λ := {(xλ ) | xλ ∈ R with xλ = 0 for almost all λ}. It is a module under componentwise addition and scalar multiplication. It has a standard basis, which consists of the vectors eµ whose λth component is the value September 3, 2012 11Nts.tex 18 4. Modules of the Kronecker delta function; that is, eµ := (δµλ ) where δµλ := { 1, 0, if λ = µ; if λ = ̸ µ. Clearly the standard basis is free. If Λ has a finite number ℓ of elements, then R⊕Λ is often written Rℓ and called the direct sum of ℓ copies of R. The free module R⊕Λ has the following UMP: given a module M and elements mλ ∈ M for λ ∈ Λ, there is a unique homomorphism α : R⊕Λ → M with α(eλ ) = mλ for each λ ∈ Λ; ) ∑ (∑ xλ mλ . Note the following obvious statements: namely, α (xλ ) = α xλ eλ = (1) α is surjective if and only if the mλ generate M . (2) α is injective if and only if the mλ are linearly independent. (3) α is an isomorphism if and only if the mλ form a free basis. Thus M is free of rank ℓ if and only if M ≃ Rℓ . ( ) Example (4.11). — Take R := Z and M := Q. Then any two x, y in M are not free; indeed, if x = a/b and y = −c/d, then bcx + ady = 0. So M is not free. Also M is not finitely generated. Indeed, given any m1 /n1 , . . . , mr /nr ∈ M , let d be a common multiple of n1 , . . . , nr . Then (1/d)Z contains every linear combination x1 (m1 /n1 ) + · · · + xℓ (mℓ /nℓ ), but (1/d)Z ̸= M . Exercise (4.12). — Let R be a domain, and x ∈ R nonzero. Let M be the submodule of Frac(R) generated by 1, x−1 , x−2 , . . . . Suppose that M is finitely generated. Prove that x−1 ∈ R, and conclude that M = R. (4.13) (Direct Products, Direct Sums). — Let R be a ring, Λ a set, Mλ a module for λ ∈ Λ. The direct product of the Mλ is the set of arbitrary vectors: ∏ Mλ := {(mλ ) | mλ ∈ Mλ }. ∏ Clearly, Mλ is a module under componentwise addition and scalar multiplication. The direct sum of the Mλ is the subset of restricted vectors: ⊕ ∏ Mλ := {(mλ ) | mλ = 0 for almost all λ} ⊂ Mλ . ⊕ ∏ ⊕ ∏ Clearly, Mλ is a submodule of Mλ . Clearly, Mλ = Mλ if Λ is finite. If ⊕ Λ = {λ1 , . . . , λn }, then Mλ is also denoted by Mλ1 ⊕ · · · ⊕ Mλn . The direct product comes equipped with projections ( ) ∏ πκ : Mλ → Mκ given by πκ (mλ ) := mκ . ∏ It is easy to see that Mλ has this UMP: ∏ given homomorphisms ακ : N → Mκ , there is a unique( homomorphism α : N → Mλ satisfying πκ α = ακ for all κ ∈ Λ; ) namely, α(n) = αλ (n) . Often, α is denoted (αλ ). In other words, the πλ induce a bijection of sets, ( ∏ ) ∏ ∼ Hom(N, Mλ ). (4.13.1) Hom N, Mλ −→ Clearly, this bijection is an isomorphism of modules. Similarly, the direct sum comes equipped with injections ι κ : Mκ → ⊕ Mλ given by { m, ικ (m) := (mλ ) where mλ := 0, if λ = κ; if λ = ̸ κ. It is easy to see that it has this UMP: given homomorphisms βκ : Mκ → N , there is September 3, 2012 11Nts.tex 4. Modules 19 ⊕ a (unique) homomorphism β: Mλ → N satisfying βικ = βκ for all κ ∈ Λ; namely, ∑ ∑ β (mλ ) = βλ (mλ ). Often, β is denoted βλ ; often, (βλ ). In other words, the ικ induce this bijection of sets: (⊕ ) ∏ ∼ Hom Mλ , N −→ Hom(Mλ , N ). (4.13.2) Clearly, this bijection is an isomorphism of⊕modules. For example, if Mλ = R for all λ, then ∏Mλ = R⊕Λ by construction. Further, if Nλ := N for all λ, then Hom(R⊕Λ , N ) = Nλ by (4.13.2) and (4.3). ∏ Exercise (4.14). — Let Λ be an infinite set, Rλ a ring for λ ∈ Λ. Endow Rλ ⊕ ∏ and Rλ with componentwise addition and multiplication. Show that R has λ ⊕ a multiplicative identity (so is a ring), but that Rλ does not (so is not a ring). Exercise (4.15). — Let L, M , and N be modules. Consider a diagram α β → → L− M− N ← − ← − ρ σ where α, β, ρ, and σ are homomorphisms. Prove that M =L⊕N and α = ι L , β = πN , σ = ι N , ρ = πL if and only if the following relations hold: βα = 0, βσ = 1, ρσ = 0, ρα = 1, and αρ + σβ = 1. Exercise (4.16). — Let N be a module,⊕ Λ a nonempty set, Mλ a module for λ ∈ Λ. Prove that the injections ικ : Mκ → Mλ induce an injection ⊕ ⊕ Hom(N, Mλ ) ֒→ Hom(N, Mλ ), and that it is an isomorphism if N is finitely generated. Exercise set, Mλ a module for λ ∈ Λ. ) — (⊕(4.17). ⊕ Let a be an ideal, ∏ Λ a nonempty ∏ aMλ . Prove a( Mλ ) = aMλ if a is finitely generated. Prove a Mλ = September 3, 2012 11Nts.tex 20 5. Exact Sequences 5. Exact Sequences In the study of modules, the exact sequence plays a central role. We relate it to the kernel and image, the direct sum and direct product. We introduce diagram chasing, and prove the Snake Lemma, which is a fundamental result in homological algebra. We define projective modules, and characterize them in four ways. Finally, we prove Schanuel’s Lemma, which relates two arbitrary presentations of a module. Definition (5.1). — A (finite or infinite) sequence of module homomorphisms αi−1 α i · · · → Mi−1 −−−→ Mi −→ Mi+1 → · · · is said to be exact at Mi if Ker(αi ) = Im(αi−1 ). The sequence is said to be exact if it is exact at every Mi , except an initial source or final target. α Example (5.2). — (1) A sequence 0 → L − → M is exact if and only if α is injective. If so, then we often identify L with its image α(L). Dually — that is, in the analogous situation with all arrows reversed — a seβ quence M − → N → 0 is exact if and only if β is surjective. α β (2) A sequence 0 → L − →M − → N is exact if and only if L = Ker(β), where ‘=’ α β means “canonically isomorphic.” Dually, a sequence L − →M − → N → 0 is exact if and only if N = Coker(α) owing to (1) and (4.6.1). α β (5.3) (Short exact sequences). — A sequence 0 → L − →M − → N → 0 is exact if and only if α is injective and N = Coker(α), or dually, if and only if β is surjective and L = Ker(β). If so, then the sequence is called short exact, and often we regard L as a submodule of M , and N as the quotient M/L. For example, the following sequence is clearly short exact: ι π L 0 → L −→ L ⊕ N −−N → N → 0. Often, we identify L with ιL L and N with ιN N . Proposition (5.4). — For λ ∈ Λ, let Mλ′ → Mλ → Mλ′′ be a sequence of module homomorphisms. If every sequence is exact, then so are the two induced sequences ∏ ∏ ⊕ ′ ⊕ ⊕ ′′ ∏ ′ Mλ → Mλ → Mλ and Mλ → Mλ → Mλ′′ . Conversely, if either induced sequence is exact then so is every original one. Proof: The assertions are immediate from (5.1) and (4.13). □ Exercise (5.5). — Let M ′ and M ′′ be modules, N ⊂ M ′ a submodule. Set M := M ′ ⊕ M ′′ . Using (5.2)(1) and (5.3) and (5.4), prove M/N = M ′ /N ⊕ M ′′ . Exercise (5.6). — Let 0 → M ′ → M → M ′′ → 0 be a short exact sequence. Prove that, if M ′ and M ′′ are finitely generated, then so is M . α β Lemma (5.7). — Let 0 → M ′ − →M − → M ′′ → 0 be a short exact sequence, and ′ −1 N ⊂ M a submodule. Set N := α (N ) and N ′′ := β(N ). Then the induced sequence 0 → N ′ → N → N ′′ → 0 is short exact. Proof: It is simple and straightforward to verify the asserted exactness. September 3, 2012 11Nts.tex □ 5. Exact Sequences 21 Definition (5.8). — We say that a short exact sequence β α 0 → M′ − →M − → M ′′ → 0 ′ ∼ −→ (5.8.1) ′′ splits if there is an isomorphism φ : M M ⊕M with φα = ιM ′ and β = πM ′′ φ. We call a homomorphism ρ : M → M ′ a retraction of α if ρα = 1M ′ . Dually, we call a homomorphism σ : M ′′ → M a section of β if βσ = 1M ′′ . β α Proposition (5.9). — Let 0 → M ′ − →M − → M ′′ → 0 be a short exact sequence. Then the following conditions are equivalent: (1) The sequence splits. (2) There exists a retraction ρ : M → M ′ of α. (3) There exists a section σ : M ′′ → M of β. ∼ M ′ ⊕ M ′′ such that φα = ι ′ Proof: Assume (1). Then there exists φ : M −→ M −1 and β = πM ′′ φ. Set ρ := πM ′ φ and σ := φ ιM ′′ . Then clearly (2) and (3) hold. Assume (2). Set σ ′ := 1M − αρ. Then σ ′ α = α − αρα = 0. So there exists σ : M ′′ → M with σβ = σ ′ by (5.2)(2) and the UMP of (4.9). So 1M = αρ + σβ. Since βσβ = β and β is surjective, βσ = 1M ′′ . Hence αρσ = 0. Since α is injective, ρσ = 0. Thus (4.15) yields (1) and also (3). Assume (3). Then similarly (1) and (2) hold. □ Exercise (5.10). — Let M ′ , M ′′ be modules, and set M := M ′ ⊕ M ′′ . Let N be a submodule of M containing M ′ , and set N ′′ := N ∩ M ′′ . Prove N = M ′ ⊕ N ′′ . Exercise (5.11). — Criticize the following misstatement of (5.9): given a short α β →M − → M ′′ → 0, there is an isomorphism M ≃ M ′ ⊕ M ′′ exact sequence 0 → M ′ − if and only if there is a section σ : M ′′ → M of β. Lemma (5.12) (Snake). — Consider this commutative diagram with exact rows: α β α′ β′ ′ ′′ M →M →0  −  −→ M  −    ′′ ′ γ γ y γ y y 0− → N ′ −→ N −→ N ′′ It yields the following exact sequence: φ ψ ∂ φ′ ψ′ Ker(γ ′ ) − → Ker(γ) − → Ker(γ ′′ ) − → Coker(γ ′ ) −→ Coker(γ) −→ Coker(γ ′′ ). (5.12.1) Moreover, if α is injective, then so is φ; dually, if β ′ is surjective, then so is ψ ′ . Proof: Clearly α) yields a unique compatible homomorphism Ker(γ ′ ) → Ker(γ) ( ′ because γα Ker(γ ) = 0. By the UMP discussed in (4.9), α′ yields a unique compatible homomorphism φ′ because M ′ goes to 0 in Coker(γ). Similarly, β and β ′ induce corresponding homomorphisms ψ and ψ ′ . Thus all the homomorphisms in (5.12.1) are defined except for ∂. To define ∂, chase an m′′ ∈ Ker(γ ′′ ) through the diagram. Since β is surjective, there is m ∈ M such that β(m) = m′′ . By commutativity, γ ′′ β(m) = β ′ γ(m). So β ′ γ(m) = 0. By exactness of the bottom row, there is a unique n′ ∈ N ′ such that α′ (n′ ) = γ(m). Define ∂(m′′ ) to be the image of n′ in Coker(γ ′ ). To see ∂ is well defined, choose another m1 ∈ M with β(m1 ) = m′′ . Let n′1 ∈ N ′ be the unique element with α′ (n′1 ) = γ(m1 ) as above. Since β(m − m1 ) = 0, there is an m′ ∈ M ′ with α(m′ ) = m − m1 . But α′ γ ′ = γα. So α′ γ ′ (m′ ) = α′ (n′ − n′1 ). Hence γ ′ (m′ ) = n′ − n′1 since α′ is injective. So n′ and n′1 have the same image in September 3, 2012 11Nts.tex 22 5. Exact Sequences Coker(γ ′ ). Thus ∂ is well defined. Let’s show that (5.12.1) is exact at Ker(γ ′′ ). Take m′′ ∈ Ker(γ ′′ ). As in the construction of ∂, take m ∈ M such that β(m) = m′′ and take n′ ∈ N ′ such that α′ (n′ ) = γ(m). Suppose m′′ ∈ Ker(∂). Then the image of n′ in Coker(γ ′ ) is equal ′ ′ ′ ′ to 0; so there is m′ ∈ M ′ such that γ ′ (m′ ) = n′ . Clearly γα(m ( ) = α γ′ (m ) ). So ′ ′ ′ ′ γα(m ) = α (n ) = γ(m). Hence m − α(m ) ∈ Ker(γ). Since β m − α(m ) = m′′ , clearly m′′ = ψ(m − α(m′ )); so m′′ ∈ Im(ψ). Hence Ker(∂) ⊂ Im(ψ). Conversely, suppose m′′ ∈ Im(ψ). We may assume m ∈ Ker(γ). So γ(m) = 0 and ′ ′ α (n ) = 0. Since α′ is injective, n′ = 0. Thus ∂(m′′ ) = 0, and so Im(ψ) ⊂ Ker(∂). Thus Ker(∂) is equal to Im(ψ); that is, (5.12.1) is exact at Ker(γ ′′ ). The other verifications of exactness are similar or easier. The last two assertions are clearly true. □ Exercise (5.13). — Referring to (4.8), give an alternative proof that β is an isomorphism by applying the Snake Lemma to the diagram 0 −−→ M −−−→ N −−−−−→ N/M −−−−→ 0       κy βy y / λ 0− → M/L − → N/L − → (N/L) (M/L) − →0 Exercise (5.14) (Five Lemma). — Consider this commutative diagram: α α α α β4 β3 β2 β1 4 3 2 1 M 4 −−→ M 3 −−→ M 2 −−→ M 1 −−→ M 0      γ4 y γ3 y γ2 y γ1 y γ0 y N4 −−→ N3 −−→ N2 −−→ N1 −−→ N0 Assume it has exact rows. Via a chase, prove these two statements: (1) If γ3 and γ1 are surjective and if γ0 is injective, then γ2 is surjective. (2) If γ3 and γ1 are injective and if γ4 is surjective, then γ2 is injective. Exercise (5.15) (Nine Lemma). — Consider this commutative diagram: 0 0 0       y y y ′ ′′ 0 −→ L  −−→ L −→ 0  −−→ L    y y y ′ ′′ 0− →M →M →M →0  −  −  −    y y y ′ 0− →N  −→  y N  −→  y (5.15.1) N′′ − →0  y 0 0 0 Assume all the columns are exact and the middle row is exact. Applying the Snake Lemma, prove that the first row is exact if and only if the third is. Exercise (5.16). — Consider this commutative diagram with exact rows: β γ β′ γ′ ′′ ′ →M →M M  −  −     αy α′ y α′′ y N ′ −→ N −→ N ′′ September 3, 2012 11Nts.tex 5. Exact Sequences 23 Assume α′ and γ are surjective. Given n ∈ N and m′′ ∈ M ′′ with α′′ (m′′ ) = γ ′ (n), show that there is m ∈ M such that α(m) = n and γ(m) = m′′ . Theorem (5.17) (Left exactness of Hom). — (1) Let M ′ → M → M ′′ → 0 be a sequence of module homomorphisms. Then it is exact if and only if, for all modules N , the following induced sequence is exact: 0 → Hom(M ′′ , N ) → Hom(M, N ) → Hom(M ′ , N ). (5.17.1) (2) Let 0 → N ′ → N → N ′′ be a sequence of module homomorphisms. Then it is exact if and only if, for all modules M , the following induced sequence is exact: 0 → Hom(M, N ′ ) → Hom(M, N ) → Hom(M, N ′′ ). α β Proof: By (5.2)(2), the exactness of M ′ − → M − → M ′′ → 0 means simply ′′ that M = Coker(α). On the other hand, the exactness of (5.17.1) means that a φ ∈ Hom(M, N ) maps to 0, or equivalently φα = 0, if and only if there is a unique γ : M ′′ → N such that γβ = φ. So (5.17.1) is exact if and only if M ′′ has the UMP of Coker(α), discussed in (4.9); that is, M ′′ = Coker(α). Thus (1) holds. The proof of (2) is similar. □ Definition (5.18). — A (free) presentation of a module M is an exact sequence G→F →M →0 with G and F free. If G and F are free of finite rank, then the presentation is called finite. If M has a finite presentation, then M is said to be finitely presented. Proposition (5.19). — Given a module M and a set of generators {mλ }λ∈Λ , α there is an exact sequence 0 → K → R⊕Λ − →M → 0 with α(eλ ) = mλ , where {eλ } α →M → 0. is the standard basis; further, there is a presentation R⊕Σ → R⊕Λ − Proof: By (4.10)(1), there is a surjection α : R⊕Λ → → M with α(eλ ) = mλ . Set K := Ker(α). Then 0 → K → R⊕Λ → M → 0 is exact by (5.3). Take a set of generators {kσ }σ∈Σ of K, and repeat the process to obtain a surjection R⊕Σ → → K. Then R⊕Σ → R⊕Λ → M → 0 is a presentation. □ Definition (5.20). — A module P is called projective if, given any surjective homomorphism β : M → → N , every homomorphism α : P → N lifts to a homomorphism γ : P → M ; that is, α = βγ. Exercise (5.21). — Show that a free module R⊕Λ is projective. Theorem (5.22). — The following conditions on a module P are equivalent: (1) (2) (3) (4) The module P is projective. Every short exact sequence 0 → K → M → P → 0 splits. There is a module K such that K ⊕ P is free. Every exact sequence N ′ → N → N ′′ induces an exact sequence Hom(P, N ′ ) → Hom(P, N ) → Hom(P, N ′′ ). (5.22.1) (5) Every surjective homomorphism β : M → → N induces a surjection Hom(P, β) : Hom(P, M ) → Hom(P, N ). September 3, 2012 11Nts.tex 24 5. Exact Sequences Proof: Assume (1). In (2), the surjection M → → P and the identity P → P yield a section P → M . So the sequence splits by (5.9). Thus (2) holds. Assume (2). By (5.19), there is an exact sequence 0 → K → R⊕Λ → P → 0. Then (2) implies K ⊕ P ≃ R⊕Λ . Thus (3) holds. Assume (3); say K ⊕ P ≃ R⊕Λ . For each λ ∈ Λ, take a copy Nλ′ → Nλ → Nλ′ of the exact sequence N ′ → N → N ′′ of (4). Then the induced sequence ∏ ′ ∏ ∏ Nλ → Nλ → Nλ′′ . is exact by (5.4). But by the end of (4.13), that sequence is equal to this one: Hom(R⊕Λ , N ′ ) → Hom(R⊕Λ , N ) → Hom(R⊕Λ , N ′′ ). But K ⊕ P ≃ R⊕Λ . So owing to (4.13.2), the latter sequence is also equal to Hom(K, N ′ ) ⊕ Hom(P, N ′ ) → Hom(K, N ) ⊕ Hom(P, N ) → Hom(K, N ′′ ) ⊕ Hom(P, N ′′ ). Hence (5.22.1) is exact by (5.4). Thus (4) holds. β Assume (4). Then every exact sequence M − → N → 0 induces an exact sequence Hom(P, M ) → Hom(P, N ) → 0. In other words, (5) holds. Assume (5). By definition, Hom(P, β)(γ) = βγ. Therefore, (1) holds. □ Lemma (5.23) (Schanuel). — Given two short exact sequences i α 0→L→ − P − →M →0 and i′ α′ 0 → L′ − → P ′ −→ M → 0 with P and P ′ projective, there is an isomorphism of exact sequences: i⊕1 ′ (α 0) 1 ⊕i′ (0 α′ ) P ′ ′ 0− → L⊕ →0  −  P −−−−→ P ⊕  P −−−−→ M    ∼ ∼ γ =y1M = yβ =y 0− → P ⊕ L′ −−P−−→ P ⊕ P ′ −−−−→ M − →0 Proof: First, let’s construct an intermediate isomorphism of exact sequences: i⊕1 ′ (α 0) P ′ ′ 0− → L⊕ →0 x P −−−−→ P ⊕ x P −−−−→ M x −    ∼ ∼ =1M = λ = θ (α α′ ) 0 −−−→ K −−−−−−→ P ⊕ P ′ −−−−→ M − →0 Take K := Ker(α α′ ). To form θ, recall that P ′ is projective ) α is surjective. So ( and there is a map π : P ′ → P such that α′ = απ. Take θ := 10 π1 . ) ( Then θ has 01 −π1 as inverse. Further, the right-hand square is commutative: ( ) (α 0)θ = (α 0) 10 π1 = (α απ) = (α α′ ). ∼ L ⊕ P ′. So θ induces the desired isomorphism λ : K −→ ′ Symmetrically, form an automorphism θ of P ⊕P ′ , which induces an isomorphism ′ ∼ P ⊕ L′ . Finally, take γ := θ ′ θ −1 and β := λ′ λ−1 . λ : K −→ □ Exercise (5.24). — Let R be a ring, and 0 → L → Rn → M → 0 an exact sequence. Prove M is finitely presented if and only if L is finitely generated. Exercise (5.25). — Let R be a ring, X1 , X2 , . . . infinitely many variables. Set P := R[X1 , X2 , . . . ] and M := P/⟨X1 , X2 , . . . ⟩. Is M finitely presented? Explain. September 3, 2012 11Nts.tex 5. Exact Sequences 25 β α Proposition (5.26). — Let 0 → L − →M − → N → 0 be a short exact sequence with L finitely generated and M finitely presented. Then N is finitely presented. Proof: Let R be the ground ring, µ : Rm → M any surjection. Set ν := βµ, set K := Ker ν, and set λ := µ|K. Then the following diagram is commutative: 0− →K →  −  λy α ν Rm − →N →0  −  1  µy Ny β 0 −→ L −→ M −→ N − →0 ∼ Ker µ. But Ker µ is The Snake Lemma (5.12) yields an isomorphism Ker λ −→ finitely generated by (5.24). So Ker λ is finitely generated. Also, the Snake Lemma λ implies Coker λ = 0 as Coker µ = 0; so 0 → Ker λ → K − → L → 0 is exact. Hence K is finitely generated by (5.6). Thus N is finitely presented by (5.24). □ β α Exercise (5.27). — Let 0 → L − →M − → N → 0 be a short exact sequence with M finitely generated and N finitely presented. Prove L is finitely generated. α β Proposition (5.28). — Let 0 → L − →M − → N → 0 be a short exact sequence with L and N finitely presented. Prove M is finitely presented too. Proof: Let R be the ground ring, λ : Rℓ → L and ν : Rn → → N any surjections. Define γ : Rℓ → M by γ := αλ. Note Rn is projective by (5.21), and define δ : Rn → M by lifting ν along β. Define µ : Rℓ ⊕ Rn → M by µ := γ + δ. Then the following diagram is, plainly, commutative, where ι := ιRℓ and π := πRn : ι π ℓ n 0− →R → Rℓ ⊕ → Rn − →0 R −  −    µ ν λy y y α β 0 −→ L −−−−→ M −−−−→ N −→ 0 Since λ and ν are surjective, the Snake Lemma (5.12) yields an exact sequence 0 → Ker λ → Ker µ → Ker ν → 0, and implies Coker µ = 0. Also, Ker λ and Ker ν are finitely generated by (5.24). So Ker µ is finitely generated by (5.6). Thus M is finitely presented by (5.24). □ September 3, 2012 11Nts.tex 26 6. Direct Limits 6. Direct Limits Category theory provides the right abstract setting for certain common concepts, constructions, and proofs. Here we treat adjoints and direct limits. We elaborate on two key special cases of direct limits: coproducts (direct sums) and coequalizers (cokernels). Then we construct arbitrary direct limits of sets and of modules. Further, we prove direct limits are preserved by left adjoints; whence, direct limits commute with each other, and in particular, with coproducts and coequalizers. Although this section is the most abstract of the entire book, all the material here is elementary, and none of it is very deep. In fact, many statements are just concise restatements in more expressive language; they can be understood through a simple translation of terms. Experience shows that it pays to learn this more abstract language, but that doing so requires determined, yet modest effort. (6.1) (Categories). — A category C is a collection of elements, called objects. Each pair of objects A, B is equipped with a set HomC (A, B) of elements, called α maps or morphisms. We write α : A → B or A − → B to mean α ∈ HomC (A, B). Further, given objects A, B, C, there is a composition law HomC (A, B) × HomC (B, C) → HomC (A, C), written (α, β) 7→ βα, and there is a distinguished map 1B ∈ HomC (B, B), called the identity such that (1) composition is associative, or γ(βα) = (γβ)α for γ : C → D, and (2) 1B is unitary, or 1B α = α and β1B = β. We say α is an isomorphism with inverse β : B → A if αβ = 1B and βα = 1A . For example, four common categories are those of sets ((Sets)), of rings ((Rings)), of R-modules ((R-mod)), and of R-algebras ((R-alg)); the corresponding maps are the set maps, and the ring, R-module, and R-algebra homomorphisms. Given categories C and C′ , their product C × C′ is the category whose objects are the pairs (A, A′ ) with A an object of C and A′ an object of C′ and whose maps are the pairs (α, α′ ) of maps α in C and α′ in C′ . (6.2) (Functors). — A map of categories is known as a functor. Namely, given categories C and C′ , a (covariant) functor F : C → C′ is a rule that assigns to each object A of C an object F (A) of C′ and to each map α : A → B of C a map F (α) : F (A) → F (B) of C′ preserving composition and identity; that is, (1) F (βα) = F (β)F (α) for maps α : A → B and β : B → C of C, and (2) F (1A ) = 1F (A) for any object A of C. We also denote a functor F by F (•), by A 7→ F (A), or by A 7→ FA . Note that a functor F preserves isomorphisms. Indeed, if αβ = 1B and βα = 1A , then F (α)F (β) = 1F (B) and F (β)F (α) = F (1A ). For example, let R be a ring, M a module. Then clearly HomR (M, •) is a functor from ((R-mod)) to ((R-mod)). A second example is the forgetful functor from ((R-mod)) to ((Sets)); it sends a module to its underlying set and a homomorphism to its underlying set map. A map of functors is known as a natural transformation. Namely, given two functors F, F ′ : C ⇒ C′ , a natural transformation θ : F → F ′ is a collection of maps θ(A) : F (A) → F ′ (A), one for each object A of C, such that θ(B)F (α) = F ′ (α)θ(A) September 3, 2012 11Nts.tex 6. Direct Limits 27 for every map α : A → B of C; that is, the following diagram is commutative: F (α) F (A) −−−−→ F (B)     θ(B)y θ(A)y F ′ (α) F ′ (A) −−−−→ F ′ (B) For example, the identity maps 1F (A) trivially form a natural transformation 1F from any functor F to itself. We call F and F ′ isomorphic if there are natural transformations θ : F → F ′ and θ′ : F ′ → F with θ′ θ = 1F and θθ′ = 1F ′ . A contravariant functor G from C to C′ is a rule similar to F , but G reverses the direction of maps; that is, G(α) carries G(B) to G(A), and G satisfies the analogues of (1) and (2). For example, fix a module N ; then Hom(•, N ) is a contravariant functor from ((R-mod)) to ((R-mod)). Exercise (6.3). — (1) Show that the condition (6.2)(1) is equivalent to the commutativity of the corresponding diagram: ( ) HomC (B, C) − → HomC′ F (B), F (C)     y y ( ) HomC (A, C) − → HomC′ F (A), F (C) (2) Given γ : C → D, show (6.2)(1) yields the commutativity of this diagram: ( ) HomC (B, C) − → HomC′ F (B), F (C)     y y ( ) HomC (A, D) − → HomC′ F (A), F (D) (6.4) (Adjoints). — Let C and C′ be categories, F : C → C′ and F ′ : C′ → C functors. We call (F, F ′ ) an adjoint pair, F the left adjoint of F ′ , and F ′ the right adjoint of F if, for each object A ∈ C and object A′ ∈ C′ , there is a natural bijection HomC′ (F (A), A′ ) ≃ HomC (A, F ′ (A′ )). (6.4.1) Here natural means that maps B → A and A′ → B ′ induce a commutative diagram: HomC′ (F (A), A′ ) ≃ HomC (A, F ′ (A′ ))     y y HomC′ (F (B), B ′ ) ≃ HomC (B, F ′ (B ′ )) Naturality serves to determine an adjoint up to canonical isomorphism. Indeed, let F and G be two left adjoints of F ′ . Given A ∈ C, define θ(A) : G(A) → F (A) to be the image of 1F (A) under the adjoint bijections HomC′ (F (A), F (A)) ≃ HomC (A, F ′ F (A)) ≃ HomC′ (G(A), F (A)). To see that θ(A) is natural in A, take a map α : A → B. It induces the following September 3, 2012 11Nts.tex 28 6. Direct Limits diagram, which is commutative owing to the naturality of the adjoint bijections: HomC′ (F (A), F (A)) ≃ HomC (A, F ′ F (A)) ≃ HomC′ (G(A), F (A))       y y y HomC′ (F (A), F (B)) ≃ HomC (A, F ′ F (B)) ≃ HomC′ (G(A), F (B)) x x x       HomC′ (F (B), F (B)) ≃ HomC (B, F ′ F (B)) ≃ HomC′ (G(B), F (B)) Chase after 1F (A) and 1F (B) . Both map to F (α) ∈ HomC′ (F (A), F (B)). So both map to the same image in HomC′ (G(A), F (B)). But clockwise, 1F (A) maps to F (α)θ(A); counterclockwise, 1F (B) maps to θ(B)G(α). So θ(B)G(α) = F (α)θ(A). Thus the θ(A) form a natural transformation θ : G → F . Similarly, there is a natural transformation θ′ : F → G. It remains to show ′ θ θ = 1G and θθ′ = 1F . However, by naturality, this diagram is commutative: HomC′ (F (A), F (A)) ≃ HomC (A, F ′ F (A)) ≃ HomC (G(A), F (A))       y y y HomC′ (F (A), G(A)) ≃ HomC (A, F ′ G(A)) ≃ HomC (G(A), G(A)) Chase after 1F (A) . Clockwise, its image is θ′ (A)θ(A) in the lower right corner. Counterclockwise, its image is 1G(A) , owing to the definition of θ′ . Thus θ′ θ = 1G . Similarly, θθ′ = 1F , as required. For example, the “free module” functor is the left adjoint of the forgetful functor from ((R-mod)) to ((Sets)), since by (4.10), Hom((R-mod)) (R⊕Λ , M ) = Hom((Sets)) (Λ, M ). Similarly, the “polynomial ring” functor is the left adjoint of the forgetful functor from ((R-alg)) to ((Sets)), since by (1.3), ( ) ( ) Hom((R-alg)) R[X1 , . . . , Xn ], R′ = Hom((Sets)) {X1 , . . . , Xn }, R′ . Exercise (6.5). — Let C and C′ be categories, F : C → C′ and F ′ : C′ → C an ∼ Hom (A, F ′ A′ ) denote the natural adjoint pair. Let φA,A′ : HomC′ (F A, A′ ) −→ C bijection, and set ηA := φA,F A (1F A ). Do the following: (1) Prove ηA is natural in A; that is, given g : A → B, the induced square ηA ′ A FA  −−→ F    gy yF ′ F g ηB B −−→ F ′ F B is commutative. We call the natural transformation A 7→ ηA the unit of (F, F ′ ). (2) Given f ′ : F A → A′ , prove φA,A′ (f ′ ) = F ′ f ′ ◦ ηA . (3) Prove the natural map ηA : A → F ′ F A is universal from A to F ′ ; that is, given f : A → F ′ A′ , there is a unique map f ′ : F A → A′ with F ′ f ′ ◦ ηA = f . (4) Conversely, instead of assuming (F, F ′ ) is an adjoint pair, assume given a natural transformation η : 1C → F ′ F satisfying (1) and (3). Prove the equation in (2) defines a natural bijection making (F, F ′ ) an adjoint pair, whose unit is η. (5) Identify the units in the two examples in (6.4): the “free module” functor and the “polynomial ring” functor. (Dually, we can define a counit ε : F F ′ → 1C′ , and prove similar statements.) September 3, 2012 11Nts.tex 6. Direct Limits 29 (6.6) (Direct limits). — Let Λ, C be categories. Assume Λ is small; that is, its objects form a set. Given a functor λ 7→ Mλ from Λ to C, its direct limit or colimit, denoted lim Mλ or limλ∈Λ Mλ , is defined as the universal example of −→ −→ an object P of C equipped with maps βµ : Mµ → P , called insertions, that are compatible with the transition maps αµκ : Mκ → Mµ , which are the images of the maps of Λ. In other words, there is a unique map β such that all these diagrams commute: κ αµ Mκ −−→  β y κ 1 αµ Mµ −−→ lim Mλ −→  β β y µ y 1 P P −−− → P −−−P−→ P To indicate this context, the functor λ 7→ Mλ is often called a direct system. As usual, universality implies that, once equipped with its insertions αµ , the limit lim Mλ is determined up to unique isomorphism, assuming it exists. In practice, −→ there is usually a canonical choice for lim Mλ , given by a construction. In any case, −→ let us use lim Mλ to denote a particular choice. −→ We say that C has direct limits indexed by Λ if, for every functor λ 7→ Mλ from Λ to C, the direct limit lim Mλ exists. We say that C has direct limits −→ if it has direct limits indexed by every small category Λ. We say that a functor F : C → C′ preserves direct limits if, given any direct limit lim Mλ in C, the −→ direct limit lim F (Mλ ) exists, and is equal to F (lim Mλ ); more precisely, the maps −→ −→ F (αµ ) : F (Mµ ) → F (lim Mλ ) induce a canonical map −→ ϕ : lim F (Mλ ) → F (lim Mλ ), −→ −→ and ϕ is an isomorphism. Sometimes, we construct lim F (Mλ ) by showing that −→ F (lim Mλ ) has the requisite UMP. −→ Assume C has direct limits indexed by Λ. Then, given a natural transformation from λ 7→ Mλ to λ 7→ Nλ , universality yields unique commutative diagrams Mµ − → lim Mλ −→    y y Nµ −→ lim Nλ −→ To put it in another way, form the functor category CΛ : its objects are the functors λ 7→ Mλ from Λ to C; its maps are the natural transformations (they form a set as Λ is one). Then taking direct limits yields a functor lim from CΛ to C. −→ In fact, it is just a restatement of the definitions that the “direct limit” functor lim is the left adjoint of the diagonal functor −→ ∆ : C → CΛ . By definition, ∆ sends each object M to the constant functor ∆M , which has the same value M at every λ ∈ Λ and has the same value 1M at every map of Λ; further, ∆ carries a map γ : M → N to the natural transformation ∆γ : ∆M → ∆N , which has the same value γ at every λ ∈ Λ. (6.7) (Coproducts). — Let C ⨿be a category, Λ a⨿set, and Mλ an object of C for each λ ∈ Λ. The coproduct λ∈Λ Mλ , or simply Mλ , is defined as the universal example of an object ⨿ P equipped with a map βµ : Mµ → P for each µ ∈ Λ. The maps ιµ : Mµ → Mλ are called the inclusions. Thus, given an example P , there September 3, 2012 11Nts.tex 30 6. Direct Limits ⨿ exists a unique map β : Mλ → P with βιµ = βµ for all µ ∈ Λ. If Λ = ∅, then the coproduct is an object B with a unique map β to every other object P . There are no µ in Λ, so no inclusions ιµ : Mµ → B, so no equations βιµ = βµ to restrict β. Such a B is called an initial object. For instance, suppose C⨿= ((R-mod)). Then the zero ⊕ module is an initial object. For any Λ, the coproduct Mλ is just the direct sum Mλ (a convention if Λ = ∅). Further, suppose C = ((Sets)). Then the empty set is an initial object. For any Λ, ⨿ ⊔ the coproduct Mλ is the disjoint union Mλ (a convention if Λ = ∅). Note that the coproduct is a special case of the direct limit. Indeed, regard Λ as a discrete category: its objects ⨿ are the λ ∈ Λ, and it has just the required maps, namely, the 1λ . Then lim Mλ = Mλ with the insertions equal to the inclusions. −→ (6.8) (Coequalizers). — Let α, α′ : M → N be two maps in a category C. Their coequalizer is defined as the universal example of an object P equipped with a map η : N → P such that ηα = ηα′ . For instance, if C = ((R-mod)), then the coequalizer is Coker(α − α′ ). In particular, the coequalizer of α and 0 is just Coker(α). Suppose C = ((Sets)). Take the smallest equivalence relation ∼ on N with α(m) ∼ α′ (m) for all m ∈ M ; explicitly, n ∼ n′ if there are elements m1 , . . . , mr with α(m1 ) = n, with α′ (mr ) = n′ , and with α(mi ) = α′ (mi+1 ) for 1 ≤ i < r. Clearly, the coequalizer is the quotient N/∼ equipped with the quotient map. Note that the coequalizer is a special case of the direct limit. Indeed, let Λ be the category consisting of two objects κ, µ and two nontrivial maps φ, φ′ : κ → µ. Define λ 7→ Mλ in the obvious way: set Mκ := M and Mµ := N ; send φ to α and φ′ to α′ . Then the coequalizer is lim Mλ . −→ Exercise (6.9). — Let α : L → M and β : L → N be two maps. Their pushout is defined as the universal example of an object P equipped with a pair of maps γ : M → P and δ : N → P such that γα = δβ. Express the pushout as a direct limit. Show that, in ((Sets)), the pushout is the disjoint union M ⊔ N modulo the smallest equivalence relation ∼ with m ∼ n if there is ℓ ∈ L with α(ℓ) = m and β(ℓ) = n. Show that, in ((R-mod)), the pushout is equal to the direct sum M ⊕ N modulo the image of L under the map (α, −β). Lemma (6.10). — A category C has direct limits if and only if C has coproducts and coequalizers. If a category C has direct limits, then a functor F : C → C′ preserves them if and only if F preserves coproducts and coequalizers. Proof: If C has direct limits, then C has coproducts and coequalizers because they are special cases by (6.7) and (6.8). By the same token, if F : C → C′ preserves direct limits, then F preserves coproducts and coequalizers. Conversely, assume that C has coproducts and coequalizers. Let Λ be a small category, and λ 7→ Mλ a functor from Λ to C. Let Σ be the set of transition maps ⨿ λ αµλ : M⨿ λ → Mµ . For each σ := αµ ∈ Σ, set Mσ := Mλ . Set M := σ∈Σ Mσ and N := λ∈Λ Mλ . For each σ, there are two maps Mσ := Mλ → N : the inclusion ιλ and the composition ιµ αµλ . Correspondingly, there are two maps α, α′ : M → N . Let C be their coequalizer, and η : N → C the insertion. Given maps βλ : Mλ → P with βµ αµλ = βλ , there is a unique map β : N → P with βιλ = βλ by the UMP of the coproduct. Clearly βα = βα′ ; so β factors uniquely through C by the UMP of the coequalizer. Thus C = lim MΛ , as desired. −→ September 3, 2012 11Nts.tex 6. Direct Limits 31 Finally, if F : C → C′ preserves coproducts and coequalizers, then F preserves arbitrary direct limits as F preserves the above construction. □ Theorem (6.11). — The categories ((R-mod)) and ((Sets)) have direct limits. Proof: The assertion follows from (6.10) because ((R-mod)) and ((Sets)) have coproducts by (6.7) and have coequalizers by (6.8). □ Theorem (6.12). — Every left adjoint F : C → C′ preserves direct limits. Proof: Let Λ be a small category, λ 7→ Mλ a functor from Λ to C such that lim Mλ exists. Given an object P ′ of C′ , consider all possible commutative diagrams −→ F (ακ µ) F (αµ ) F (Mκ ) −−−−→ F (Mµ ) −−−−→ F (lim Mλ ) −→     ′ β ′ β ′ yβ y κ y µ 1 (6.12.1) 1 P ′ −−−−−−−−→ P ′ −−−−−−−−−→ P ′ where αµκ is any transition map and αµ is the corresponding insertion. Given the βκ′ , we must show there is a unique β ′ . Say F is the left adjoint of F ′ : C′ → C. Then giving (6.12.1) is equivalent to giving this corresponding commutative diagram: ακ µ Mκ −−−→  β y κ 1 αµ Mµ −−→ lim Mλ −→  β β y µ y 1 F ′ (P ′ ) − → F ′ (P ′ ) − → F ′ (P ′ ) However, given the βκ , there is a unique β by the UMP of lim Mλ . −→ □ Proposition (6.13). — Let C be a category, Λ and Σ small categories. Assume C has direct limits indexed by Σ. Then the functor category CΛ does too. Proof: Let σ 7→ (λ 7→ Mσλ ) be a functor from Σ to CΛ . Then a map σ → τ in Σ yields a natural transformation from λ → Mσλ to λ 7→ Mτ λ . So a map λ → 7 µ in Λ yields a commutative square Mσλ − → Mσµ     y y (6.13.1) Mτ λ − → Mτ µ in a manner compatible with composition in Σ. Hence, with λ fixed, the rule σ 7→ Mσλ is a functor from Σ to C. By hypothesis, limσ∈Σ Mσλ exists. So λ 7→ limσ∈Σ Mσλ is a functor from Λ to −→ −→ C. Further, as τ ∈ Σ varies, there are compatible natural transformations from the λ 7→ Mτ λ to λ 7→ limσ∈Σ Mσλ . Finally, the latter is the direct limit of the functor −→ τ 7→ (λ 7→ Mτ λ ) from Σ to CΛ , because, given any functor λ 7→ Pλ from Λ to C equipped with, for τ ∈ Σ, compatible natural transformations from the λ 7→ Mτ λ to λ 7→ Pλ , there are, for λ ∈ Λ, compatible unique maps limσ∈Σ Mσλ → Pλ . □ −→ September 3, 2012 11Nts.tex 32 6. Direct Limits Theorem (6.14) (Direct limits commute). — Let C be a category with direct limits indexed by small categories Σ and Λ. Let σ 7→ (λ 7→ Mσλ ) be a functor from Σ to CΛ . Then limσ∈Σ limλ∈Λ Mσ,λ = limλ∈Λ limσ∈Σ Mσ,λ . −→ −→ −→ −→ Proof: By (6.6), the functor limλ∈Λ : CΛ → C is a left adjoint. By (6.13), the −→ category CΛ has direct limits indexed by Σ. So (6.12) yields the assertion. □ Corollary (6.15). — Let Λ be a small category, R a ring, and C either ((Sets)) or ((R-mod)). Then functor lim : CΛ → C preserves coproducts and coequalizers. −→ Proof: By (6.7) and (6.8), both coproducts and coequalizers are special cases of direct limits, and C has them. So (6.14) yields the assertion. □ Exercise (6.16). — Let C be a category, Σ and Λ small categories. (1) Prove CΣ×Λ = (CΛ )Σ with (σ, λ) 7→ Mσ,λ corresponding to σ 7→ (λ 7→ Mσ,λ ). (2) Assume C has direct limits indexed by Σ and by Λ. Prove that C has direct limits indexed by Σ × Λ and that limλ∈Λ limσ∈Σ = lim(σ,λ)∈Σ×Λ . −→ −→ −→ Exercise (6.17). — Let λ 7→ Mλ and λ 7→ Nλ be two functors from a small category Λ to ((R-mod)), and {θλ : Mλ → Nλ } a natural transformation. Show lim Coker(θλ ) = Coker(lim Mλ → lim Nλ ). −→ −→ −→ Show that the analogous statement for kernels can be false by constructing a counterexample using the following commutative diagram with exact rows: µ2 Z −−→  µ 2 y µ2 Z− → Z/⟨2⟩ − →0   µ2 µ 2 y y Z −−→ Z − → Z/⟨2⟩ − →0 September 3, 2012 11Nts.tex 7. Filtered Direct Limits Filtered direct limits are direct limits indexed by a filtered category, which is a more traditional sort of index set. We give an alternative construction of these limits for modules. We conclude that forming them preserves exact sequences, and so commutes with forming the module of homomorphisms out of a fixed finitely presented source. We end by proving that every module is a filtered direct limit of finitely presented modules. (7.1) (Filtered categories). — We call a small category Λ filtered if (1) given objects κ and λ, for some µ there are maps κ → µ and λ → µ, (2) given two maps σ, τ : η ⇒ κ with the same source and the same target, for some µ there is a map φ : κ → µ such that φσ = φτ . Given a category C, we say a functor λ 7→ Mλ from Λ to C is filtered if Λ is filtered. If so, then we say the direct limit lim Mλ is filtered if it exists. −→ For example, let Λ be a partially ordered set. Suppose Λ is directed; that is, given κ, λ ∈ Λ, there is a µ with κ ≤ µ and λ ≤ µ. Regard Λ as a category whose objects are its elements and whose sets Hom(κ, λ) consist of a single element if κ ≤ λ, and are empty if not; morphisms can be composed as the ordering is transitive. Clearly, the category Λ is filtered. Exercise (7.2). — ∪ Let R be a ring, M a module, Λ a set, Mλ a submodule for each λ ∈ Λ. Assume Mλ = M . Assume, given λ, µ ∈ Λ, there is ν ∈ Λ such that Mλ , Mµ ⊂ Mν . Order Λ by inclusion: λ ≤ µ if Mλ ⊂ Mµ . Prove that M = lim Mλ . −→ Exercise (7.3). — Show that every module M is the filtered direct limit of its finitely generated submodules. Exercise (7.4). — Show that every direct sum of modules is the filtered direct limit of its finite direct subsums. Example (7.5). — Let Λ be the set ∪ of all positive integers, and for each n ∈ Λ, set Mn := {r/n | r ∈ Z} ⊂ Q. Then Mn = Q and Mm , Mn ⊂ Mmn . Then (7.2) yields Q = lim Mn where Λ is ordered by inclusion of the Mn . −→ However, Mm ⊂ Mn if and only if 1/m = s/n for some s, if and only if m | n. Thus we may view Λ as ordered by divisibility of the n ∈ Λ. For each n ∈ Λ, set Rn := Z, and define βn : Rn → Mn by βn (r) := r/n. Clearly, βn is a Z-module isomorphism. And if n = ms, then this diagram is commutative: µs Rm −−→ Rn   βn y ≃ β m y≃ ιm n (7.5.1) Mm ֒−→ Mn R where the transition maps are the where ιm n is the inclusion. Hence Q = lim −→ n multiplication maps µs . Exercise (7.6). — Keep the setup of (7.5). For each n ∈ Λ, set Nn := Z/⟨n⟩; if n = ms, define αnm : Nm → Nn by αnm (x) := xs (mod n). Show lim Nn = Q/Z. −→ September 3, 2012 11Nts.tex 34 7. Filtered Direct Limits Proposition (7.7). — Let Λ be a filtered category, R a ring, and C either ((Sets)) or ((R-mod)) or ((R-alg)). Let⊔λ 7→ Mλ be a functor from Λ to C. Define a relation ∼ on the disjoint union Mλ as follows: m1 ∼ m2 for mi ∈ Mλi if there are transition maps αµλi : Mλi → Mµ such that αµλ1 m1 = αµλ2 m2 . Then ∼ is an (⊔ )/ equivalence relation. Set M := Mλ ∼. Then M = lim Mλ , and for each µ, the −→ canonical map αµ : Mµ → M is equal to the insertion map Mµ → lim Mλ . −→ Proof: Clearly ∼ is reflexive and symmetric. Let’s show it is transitive. Given mi ∈ Mλi for i = 1, 2, 3 with m1 ∼ m2 and m2 ∼ m3 , there are αµλi for i = 1, 2 and ανλi for i = 2, 3 with αµλ1 m1 = αµλ2 m2 and ανλ2 m2 = ανλ3 m3 . Then (7.1)(1) yields αρµ and αρν . Possibly, αρµ αµλ2 ̸= αρν ανλ2 , but in any case, (7.1)(2) yields ασρ with ασρ (αρµ αµλ2 ) = ασρ (αρν ανλ2 ). Hence, (ασρ αρµ )αµλ1 m1 = (ασρ αρν )ανλ3 m3 . Thus m1 ∼ m3 . If C = ((R-mod)), define addition in M as follows. Given mi ∈ Mλi for i = 1, 2, there are αµλi by (7.1)(1). Set αλ1 m1 + αλ2 m2 := αµ (αµλ1 m1 + αµλ2 m2 ). We must check that this addition is well defined. First, consider µ. Suppose there are ανλi too. Then (7.1)(1) yields αρµ and αρν . Possibly, αρµ αµλi ̸= αρν ανλi , but (7.1)(2) yields ασρ with ασρ (αρµ αµλ1 ) = ασρ (αρν ανλ1 ) and then ατσ with ατσ (ασρ αρµ αµλ2 ) = ατσ (ασρ αρν ανλ2 ). Therefore, (ατσ ασρ αρµ )(αµλ1 m1 + αµλ2 m2 ) = (ατσ ασρ αρν )(ανλ1 m1 + ανλ2 m2 ). Thus both µ and ν yield the same value for αλ1 m1 + αλ2 m2 . Second, suppose m1 ∼ m′1 ∈ Mλ′1 . Then a similar, but easier, argument yields αλ1 m1 + αλ2 m2 = αλ′1 m′1 + αλ2 m2 . Thus addition is well defined on M . Define scalar multiplication on M similarly. Then clearly M is an R-module. If C = ((R-alg)), then we can see similarly that M is canonically an R-algebra. κ κ βλ induce Finally, ⊔ let βλ : Mλ → N be maps with βλ αλ = βκ for all αλ . λThe a map Mλ → N . Suppose m1 ∼ m2 for mi ∈ Mλi ; that is, αµ1 m1 = αµλ2 m2 for some αµλi . Then βλ1 m1 = βλ2 m2 as βµ αµλi = βλi . So there is a unique map β : M → N with βαλ = βλ for all λ. Further, if C = ((R-mod)) or C = ((R-alg)), then clearly β is a homomorphism. The proof is now complete. □ Corollary (7.8). — Preserve the conditions of (7.7). (1) Given m ∈ lim Mλ , for some λ, there is mλ ∈ Mλ such that m = αλ mλ . −→ (2) Given mi ∈ Mλi for i = 1, 2 such that αλ1 m1 = αλ2 m2 , there are αµλi such that αµλ1 m1 = αµλ2 m2 . (3) Suppose C = ((R−mod)) or C = ((R−alg)). Then given mλ ∈ Mλ such that αλ mλ = 0, there is αµλ such that αµλ mλ = 0. Proof: The assertions follow immediately from (7.7). □ Exercise (7.9). — Let R be a filtered direct limit of rings Rλ . Show R = 0 if and only if Rλ = 0 for some λ. Show R is a domain if Rλ is a domain for every λ. Theorem (7.10) (Exactness of filtered direct limits). — Let R be a ring, Λ a filtered category. Let C be the category of 3-term exact sequences of R-modules: its September 3, 2012 11Nts.tex 7. Filtered Direct Limits 35 objects are the 3-term exact sequences, and its maps are the commutative diagrams L  −→  y M  −→  y N   y L′ − → M′ − → N′ βλ γλ Then, for any functor λ 7→ (Lλ −−→ Mλ −→ Nλ ) from Λ to C, the induced sequence β γ → lim Nλ is exact. → lim Mλ − lim Lλ − −→ −→ −→ Proof: Abusing notation, in all three cases, denote by αλκ the transition maps and by αλ the insertions. Then given ℓ ∈ lim Lλ , there is ℓλ ∈ Lλ with αλ ℓλ = ℓ −→ by (7.8)(1). By hypothesis, γλ βλ ℓλ = 0; so γβℓ = 0. Thus Im(β) ⊂ Ker(γ). For the opposite inclusion, take m ∈ lim Mλ with γm = 0. By (7.8)(1), there is −→ mλ ∈ Mλ with αλ mλ = m. Now, αλ γλ mλ = 0 by commutativity. So by (7.8)(3), there is αµλ with αµλ γλ mλ = 0. So γµ αµλ mλ = 0 by commutativity. Hence there is ℓµ ∈ Lµ with βµ ℓµ = αµλ mλ by exactness. Apply αµ to get βαµ ℓµ = αµ βµ ℓµ = αµ αµλ mλ = m. Thus Ker(γ) ⊂ Im(β). So Ker(γ) = Im(β) as asserted. □ Exercise (7.11). — Let M := lim Mλ be a filtered direct limit of modules, and −→ N ⊂ M a submodule. For each λ, let αλ : Mλ → M be the insertion, and set −1 Nλ := αλ N ⊂ Mλ . Prove that N = lim Nλ . −→ Proposition (7.12). — Let Λ a filtered category, R a ring, λ 7→ Mλ a functor from Λ to ((R-mod)), and N an R-module. Consider the canonical homomorphism θ(N ) : lim Hom(N, Mλ ) → Hom(N, lim Mλ ), −→ −→ which is induced by the insertions Mλ → lim Mλ . Then θ(N ) is injective if N is −→ finitely generated; further, θ(N ) is bijective if N is finitely presented. Proof: If N := R, then θ(N ) is bijective by (4.3). Assume N is finitely generated, and take a presentation R⊕Σ → Rn → N → 0 with Σ finite if N is finitely presented. It induces the following commutative diagram: 0− → lim Hom(N, Mλ ) − → lim Hom(Rn , Mλ ) − → lim Hom(R⊕Σ , Mλ ) −→ −→ −→       θ(Rn )y≃ θ(N )y θ(R⊕Σ )y 0− → Hom(N, lim Mλ ) − → Hom(Rn , lim Mλ ) − → Hom(R⊕Σ , lim Mλ ) −→ −→ −→ The rows are exact owing to (5.17), the left exactness of Hom, and to (7.10), the exactness of filtered direct limits. Now, Hom preserves finite direct sums by (4.13), and direct limit does so by (6.15) and (6.7); hence, θ(Rn ) is bijective, and θ(R⊕Σ ) is bijective if Σ is finite. A diagram chase yields the assertion. □ Exercise (7.13). — Let Λ and Λ′ be small categories, C : Λ′ → Λ a functor. Assume Λ′ is filtered. Assume C is cofinal; that is, (1) given λ ∈ Λ, there is a map λ → Cλ′ for some λ′ ∈ Λ′ , and (2) given ψ, φ : λ ⇒ Cλ′ , there is χ : λ′ → λ′1 with (Cχ)ψ = (Cχ)φ. September 3, 2012 11Nts.tex 36 7. Filtered Direct Limits Let λ 7→ Mλ be a functor from Λ to C whose direct limit exists. Show that limλ′ ∈Λ′ MCλ′ = limλ∈Λ Mλ ; −→ −→ more precisely, show that the right side has the UMP characterizing the left. Exercise (7.14). — Show that every R-module M is the filtered direct limit over a directed set of finitely presented modules. September 3, 2012 11Nts.tex 8. Tensor Products Given two modules, their tensor product is the target of the universal bilinear map. We construct the product, and establish various properties: bifunctoriality, commutativity, associativity, cancellation, and most importantly, adjoint associativity; the latter relates the product to the module of homomorphisms. With one factor fixed, the product becomes a linear functor. We prove Watt’s Theorem; it characterizes “tensor-product” functors as those linear functors that commute with direct sums and cokernels. Lastly, we discuss the tensor product of algebras. (8.1) (Bilinear maps). — Let R a ring, and M , N , P modules. We call a map α: M × N → P bilinear if it is linear in each variable; that is, given m ∈ M and n ∈ N , the maps m′ 7→ α(m′ , n) and n′ 7→ α(m, n′ ) are R-linear. Denote the set of all these maps by BilR (M, N ; P ). It is clearly an R-module, with sum and scalar multiplication performed valuewise. (8.2) (Tensor product). — Let R be a ring, and M , N modules. Their tensor product, denoted M ⊗R N or simply M ⊗ N , is constructed as the quotient of the free module R⊕(M ×N ) modulo the submodule generated by the following elements, where (m, n) stands for the standard basis element e(m,n) : (m + m′ , n) − (m, n) − (m′ , n) and (xm, n) − x(m, n) and (m, n + n′ ) − (m, n) − (m, n′ ), (m, xn) − x(m, n) (8.2.1) for all m, m′ ∈ M and n, n′ ∈ N and x ∈ R. Note that M ⊗ N is the target of the canonical map with source M × N β : M × N → M ⊗ N, which sends each (m, n) to its residue class m ⊗ n. By construction, β is bilinear. Theorem (8.3) (UMP of tensor product). — Let R be a ring, M , N modules. Then β : M × N → M ⊗ N is the universal example of a bilinear map with source M × N ; in fact, β induces, not simply a bijection, but a module isomorphism, ∼ Bil (M, N ; P ). θ : HomR (M ⊗R N, P ) −→ R (8.3.1) Proof: Note that, if we follow any bilinear map with any linear map, then the result is bilinear; hence, θ is well defined. Clearly, θ is a module homomorphism. Further, θ is injective since M ⊗R N is generated by the image of β. Finally, given any bilinear map α : M ×N → P , by (4.10) it extends to a map α′ : R⊕(M ×N ) → P , and α′ carries all the elements in (8.2.1) to 0; hence, α′ factors through β. Thus θ is also surjective, so an isomorphism, as asserted. □ (8.4) (Bifunctoriality). — Let R be a ring, α : M → M ′ and α′ : N → N ′ module homomorphisms. Then there is a canonical commutative diagram: α×α′ M × N −−−→ M ′  × N′  ′ β yβ y α⊗α′ M ⊗ N −−−→ M ′ ⊗ N ′ September 3, 2012 11Nts.tex 38 8. Tensor Products Indeed, β ′ ◦ (α × α′ ) is clearly bilinear; so the UMP (8.3) yields α ⊗ α′ . Thus • ⊗ N and M ⊗ • are commuting linear functors, that is, linear on maps (see (9.2)). Proposition (8.5). — Let R be a ring, M and N modules. (1) Then the switch map M × N → N × M induces an isomorphism M ⊗R N = N ⊗R M. (commutative law) (2) Then multiplication of R on M induces an isomorphism R ⊗R M = M. (unitary law) ∼ R⊕(N ×M ) , and −→ Proof: The switch map induces an isomorphism R it preserves the elements of (8.2.1). Thus (1) holds. Define β : R × M → M by β(x, m) := xm. Clearly β is bilinear. Let’s check β has the requisite UMP. Given a bilinear map α : R × M → P , define γ : M → P by γ(m) := α(1, m). Then γ is linear as α is bilinear. Also, α = γβ as ⊕(M ×N ) α(x, m) = xα(1, m) = α(1, xm) = γ(xm) = γβ(x, m). Further, γ is unique as β is surjective. Thus the UMP holds, so (2) does too. □ Exercise (8.6). — Let R be a domain, a a nonzero ideal. Set K := Frac(R). Show that a ⊗R K = K. (8.7) (Bimodules). — Let R and R′ be rings. An abelian group N is an (R, R′ )bimodule if it is both an R-module and an R′ -module and if x(x′ n) = x′ (xn) for all x ∈ R, all x′ ∈ R′ , and all n ∈ N . At times, we think of N as a left Rmodule, with multiplication xn, and as a right R′ -module, with multiplication nx′ . Then the compatibility condition becomes the associative law: x(nx′ ) = (xn)x′ . A (R, R′ )-homomorphism of bimodules is a map that is both R-linear and R′ -linear. Let M be an R-module, and let N be an (R, R′ )-bimodule. Then M ⊗R N is an (R, R′ )-bimodule with R-structure as usual and with R′ -structure defined by x′ (m ⊗ n) := m ⊗ (x′ n) for all x′ ∈ R′ , all m ∈ M , and all n ∈ N . The latter multiplication is well defined and the two multiplications commute because of bifunctoriality (8.4) with α := µx and α′ := µx′ . For instance, suppose R′ is an R-algebra. Then R′ is an (R, R′ )-bimodule. So M ⊗R R′ is an R′ -module. It is said to be obtained by extension of scalars. Exercise (8.8). — Let R be a ring, R′ an R-algebra, M, N two R′ -modules. Show there is a canonical R-linear map τ : M ⊗R N → M ⊗R′ N . Let K ⊂ M ⊗R N denote the R-submodule generated by all the differences (x′ m) ⊗ n − m ⊗ (x′ n) for x′ ∈ R′ and m ∈ M and n ∈ N . Show K = Ker(τ ). Show τ is surjective, and is an isomorphism if R′ is a quotient of R. Theorem (8.9). — Let R and R′ be rings, M an R-module, P an R′ -module, N an (R, R′ )-bimodule, . Then there are two canonical (R, R′ )-isomorphisms: M ⊗R (N ⊗R′ P ) = (M ⊗R N ) ⊗R′ P, ( ) HomR′ (M ⊗R N, P ) = HomR M, HomR′ (N, P ) . (associative law) (adjoint associativity) Proof: Note that M ⊗R (N ⊗R′ P ) and (M ⊗R N ) ⊗R′ P are (R, R′ )-bimodules. For each (R, R′ )-bimodule Q, call a map τ : M × N × P → Q trilinear if it is R-bilinear in M × N and R′ -bilinear in N × P . Denote the set of all these τ by Tril(M, N, P ; Q). It is, clearly, an (R, R′ )-bimodule. September 3, 2012 11Nts.tex 8. Tensor Products 39 A trilinear map τ yields an R-bilinear map M × (N ⊗R′ P ) → Q, whence a map M ⊗R (N ⊗R′ P ) → Q, which is both R-linear and R′ -linear, and vice versa. Thus ( ) Tril(R,R′ ) (M, N, P ; Q) = Hom M ⊗R (N ⊗R′ P ), Q . Similarly, there is a canonical isomorphism of (R, R′ )-bimodules ( ) Tril(R,R′ ) (M, N, P ; Q) = Hom (M ⊗R N ) ⊗R′ P, Q . Hence both M ⊗R (N ⊗R′ P ) and (M ⊗R N ) ⊗R′ P are universal examples of a target of a trilinear map with source M × N × P . Thus they are equal, as asserted. To establish the isomorphism of adjoint associativity, define a map ( ) α : HomR′ (M ⊗R N, P ) → HomR M, HomR′ (N, P ) by ( ) α(γ)(m) (n) := γ(m ⊗ n). Let’s check α is well defined. First, α(γ)(m) is R′ -linear, because given x′ ∈ R′ , γ(m ⊗ (x′ n)) = γ(x′ (m ⊗ n)) = x′ γ(m ⊗ n) since γ is R′ -linear. Further, α(γ) is R-linear, because given x ∈ R, ( ) ( ) (xm) ⊗ n = m ⊗ (xn) and so α(γ)(xm) (n) = α(γ)(m) (xn). ( ) Thus α(γ) ∈ HomR M, HomR′ (N, P ) . Clearly, α is an (R, R′ )-homomorphism. ( ) To obtain an inverse to α, given η ∈ HomR M, HomR′ (N, P ) , define a map ζ : M × N → P by ζ(m, n) := (η(m))(n). Clearly, ζ is Z-bilinear, so ζ induces a Z-linear map δ : M ⊗Z N → P . Given x ∈ R, clearly (η(xm))(n) = (η(m))(xn); so δ((xm) ⊗ n) = δ(m ⊗ (xn)). Hence, δ induces a Z-linear map β(η) : M ⊗R N → P owing to (8.8) with Z for R and with R for R′ . Clearly, β(η) is R′ -linear as η(m) is so. Finally, it is easy to verify that α(β(η)) = η and β(α(γ)) = γ, as desired. □ Corollary (8.10). — Let R and R′ be rings, M an R-module, P an R′ -module. If R′ is an R-algebra, then there are two canonical (R, R′ )-isomorphisms: (M ⊗R R′ ) ⊗R′ P = M ⊗R P, (cancellation law) ′ HomR′ (M ⊗R R , P ) = HomR (M, P ). ′ (left adjoint) ′ Instead, if R is an R -algebra, then there is another canonical (R, R )-isomorphism: HomR′ (M, P ) = HomR (M, HomR′ (R, P )). (right adjoint) ′ In other words, • ⊗R R is the left adjoint of restriction of scalars from R′ to R, and HomR′ (R, •) is the right adjoint of restriction of scalars from R to R′ . Proof: The cancellation law results from the associative and unitary laws; the adjoint isomorphisms, from adjoint associativity, (4.3) and the unitary law. □ Corollary (8.11). — Let R, R′ be rings, N a bimodule. Then the functor •⊗R N preserves direct limits, or equivalently, direct sums and cokernels. Proof: By adjoint associativity, •⊗R N is the left adjoint of HomR′ (N, •). Thus the assertion results from (6.12) and from (6.7) and (6.8). □ Example (8.12). — Tensor product does not preserve kernels, nor even injections. Indeed, consider the injection µ2 : Z → Z. Tensor it with N := Z/⟨2⟩, obtaining µ2 : N → N . This map is zero, but not injective as N ̸= 0. September 3, 2012 11Nts.tex 40 8. Tensor Products Exercise (8.13). — Let R be a ring, a and b ideals, and M a module. (1) Use (8.11) to show that (R/a) ⊗ M = M/aM . (2) Use (1) to show that (R/a) ⊗ (R/b) = R/(a + b). Exercise (8.14). — Let k be a field, M and N nonzero vector spaces. Prove that M ⊗ N ̸= 0. Theorem (8.15) (Watts). — Let F : ((R-mod)) → ((R-mod)) be a linear functor. Then there is a natural transformation θ(•) : • ⊗F (R) → F (•) with θ(R) = 1, and θ(•) is an isomorphism if and only if F preserves direct sums and cokernels. Proof: As F is a linear functor, there is, by definition, a natural R-linear map θ(M ) : Hom(R, M ) → Hom(F (R), F (M )). But Hom(R, M ) = M by (4.3). Set N := F (R). Then, with P := F (M ), adjoint associativity yields the desired map ( ) θ(M ) ∈ Hom M, Hom(N, F (M )) = Hom(M ⊗ N, F (M )). Explicitly, θ(M )(m ⊗ n) = F (ρ)(n) where ρ : R → M is defined by ρ(1) = m. Alternatively, this formula can be used to construct θ(M ), as (m, n) 7→ F (ρ)(n) is clearly bilinear. Either way, it is not hard to see that θ(M ) is natural in M . If θ(•) is an isomorphism, then F preserves direct sums and cokernels by (8.11). β α To prove the converse, take a presentation R⊕Λ − → R⊕Σ − → M → 0; one exists by (5.19). Applying θ, we get this commutative diagram: R⊕Λ⊗ N − → R⊕Σ⊗ N − →M ⊗N − →0  ⊕Λ  ⊕Σ θ(M ) yθ(R ) yθ(R ) y F (R ⊕Λ ) −−→ F (R ⊕Σ (8.15.1) ) −−→ F (M ) −→ 0 By construction, θ(R) = 1N . If F preserves direct sums, then θ(R⊕Λ ) = 1N ⊕Λ and θ(R⊕Σ ) = 1N ⊕Σ ; in fact, given any natural transformation θ : T → U , let’s show that, if T and U preserve direct sums, then so does θ. ⊕ Given a collection of modules Mλ , each inclusion ιλ : Mλ → Mλ yields, because of naturality, the following commutative diagram: T (ιλ ) ⊕ T (Mλ ) −−−−→ T (Mλ )   θ(⊕ M ) θ(M ) λ y y λ U (ιλ ) ⊕ U (Mλ ) −−−−→ U (Mλ ) ⊕ ⊕ Hence θ( Mλ )T (ιλ )⊕ = θ(Mλ )T (ιλ ). But the UMP of direct sum says that, given any N , a map T⊕ (Mλ ) → N ⊕ is determined by its compositions with the inclusions T (ιλ ). Thus θ( Mλ ) = θ(Mλ ), as desired. Suppose F preserves cokernels. Since • ⊗ N does too, the rows of (8.15.1) are exact by (5.2). Therefore, θ(M ) is an isomorphism. □ Exercise (8.16). — Let F : ((R-mod)) → ((R-mod)) be a linear functor. Show that F always preserves finite direct sums. Show that θ(M ) : M ⊗ F (R) → F (M ) is surjective if F preserves surjections and M is finitely generated, and that θ(M ) is an isomorphism if F preserves cokernels and M is finitely presented. (8.17) (Additive functors). — Let R be a ring, M a module, and form the diagram δ σ M M −− → M ⊕ M −−M →M where δM := (1M , 1M ) and σM := 1M + 1M . September 3, 2012 11Nts.tex 8. Tensor Products 41 Let α, β : M → N be two maps of modules. Then σN (α ⊕ β)δM = α + β, (8.17.1) because, for any m ∈ M , we have (σN (α ⊕ β)δM )(m) = σN (α ⊕ β)(m, m) = σN (α(m), β(m)) = α(m) + β(m). Let F : ((R-mod)) → ((R-mod)) be a functor that preserves finite direct sums. Then F (α ⊕ β) = F (α) ⊕ F (β). Also, F (δM ) = δF (M ) and F (σM ) = σF (M ) as F (1M ) = 1F (M ) . Hence F (α + β) = F (α) + F (β) by (8.17.1). Thus F is additive, that is, Z-linear. Conversely, every additive functor preserves finite direct sums owing to (8.16). However, not every additive functor is R-linear. For example, take R := C. Define F (M ) to be M , but with the scalar product of x ∈ C and m ∈ M to be xm where x is the conjugate. Define F (α) to be α. Then F is additive, but not linear. Lemma (8.18) (Equational Criterion for Vanishing). — Let R be a ring, M and N modules, and {nλ }λ∈Λ a set∑of generators of N . Then any element ∑ of M ⊗ N can be written as a finite sum mλ ⊗ nλ with mλ ∈ M . Further, mλ ⊗ nλ = 0 if and only if there are mσ ∈ M and xλσ ∈ R for σ ∈ Σ for some Σ such that ∑ ∑ σ xλσ mσ = mλ for all λ and λ xλσ nλ = 0 for all σ. Proof: By (8.2), M ⊗ N ∑ is generated by elements of the form ∑ m ⊗ n with m ∈ M and n ∈ N , and if n = xλ nλ with xλ ∈ R,∑then m ⊗ n = (xλ m) ⊗ nλ . Thus any element of M ⊗ N has the asserted form mλ ⊗ nλ . Assume the mσ and the xλσ exist. Then ) ) ∑ ∑ (∑ ∑ ( ∑ mλ ⊗ nλ = λ σ xλσ mσ ⊗ nλ = σ mσ ⊗ λ xλσ nλ = 0. β α → N → 0 with Conversely, by (5.19), there is a presentation R⊕Σ − → R⊕Λ − α(eλ ) = nλ for all λ where {eλ } is the standard basis of R⊕Λ . Then by (8.11) the following sequence is exact: 1⊗β 1⊗α M ⊗ R⊕Σ −−−→ M ⊗ R⊕Λ −−−→ M ⊗ N → 0. (∑ ) Further, (1 ⊗ α) mλ ⊗ eλ = 0. So the exactness implies there is an element ∑ s ∈ M ⊗ R⊕Σ such that∑(1 ⊗ β)(s) = mλ ⊗ eλ . Let {eσ } be the ∑standard basis of R⊕Σ , and write s =∑ mσ ⊗ eσ with mσ ∈ M . Write β(eσ ) = λ xλσ eλ . Then clearly 0 = αβ(eσ ) = λ xλσ nλ , and (∑ ) ∑ ( ) ∑ ∑ ∑ 0 = λ mλ ⊗ e λ − σ mσ ⊗ λ xλσ eλ = λ mλ − σ xλσ mσ ⊗ eλ . ∑ Since the eλ are independent, mλ = σ xλσ mσ , as asserted. □ (8.19) (Algebras). — Let R be a ring, S and T algebras with structure maps σ : R → S and τ : R → T . Set U := S ⊗R T ; it is an R-module. Now, define S × T × S × T → U by (s, t, s′ , t′ ) 7→ ss′ ⊗ tt′ . This map is clearly linear in each factor. So it induces a bilinear map µ: U × U → U with µ(s ⊗ t, s′ ⊗ t′ ) = (ss′ ⊗ tt′ ). It is easy to check that U is a ring with µ as product. In fact, U is an R-algebra with structure map ω given by ω(r) := σ(r) ⊗ 1 = 1 ⊗ τ (r), called the tensor product of S and T over R. September 3, 2012 11Nts.tex 42 8. Tensor Products Define ιS : S → S ⊗R T by ιS (s) := s ⊗ 1. Clearly ιS is an R-algebra homomorphism. Define ιT : T → S ⊗ T similarly. Given an R-algebra V , define a map γ : Hom((R-alg)) (S ⊗R T, V ) → Hom((R-alg)) (S, V ) × Hom((R-alg)) (T, V ). by γ(ψ) := (ψιS , ψιT ). Conversely, given R-algebra homomorphisms θ : S → V and ζ : T → V , define η : S × T → V by η(s, t) := θ(s) · ζ(t). Then η is clearly bilinear, so it defines a linear map ψ : S ⊗R T → V . It is easy to see that the map (θ, ζ) 7→ ψ is an inverse to γ. Thus γ is bijective. In other words, S ⊗R T is the coproduct of S and T in ((R-alg)): ❏❏❚❏❚❚❚❚ :: S ❚ ❏ ❚❚❚θ❚❚ ttt t t ιS ❏❏❏ ❚❚❚❚ σ $$ ttt ψ ❚❚**// S ⊗ T 44 V R ❏❏ R :: ❥❥❥❥ ❥ ❏❏❏τ t ❥ ❥ t ι ❥ T t ❏❏❏ t ❥❥❥ $$ tt❥t❥❥❥❥ ζ ❥ T Example (8.20). — Let R be a ring, S an algebra, and X1 , . . . , Xn variables. Then there is a canonical S-algebra isomorphism S ⊗R R[X1 , . . . , Xn ] = S[X1 , . . . , Xn ]. Indeed, given an S-algebra homomorphism S → T and elements x1 , . . . , xn of T , there is an R-algebra homomorphism R[X1 , . . . , Xn ] → T by (1.3). So by (8.19), there is a unique S-algebra homomorphism S ⊗R R[X1 , . . . , Xn ] → T . Thus both S ⊗R R[X1 , . . . , Xn ] → T and S[X1 , . . . Xn ] possess the same UMP. In particular, for variables Y1 , . . . , Ym , we obtain R[X1 , . . . , Xn ] ⊗R R[Y1 , . . . , Ym ] = R[X1 , . . . , Xn , Y1 , . . . , Ym ]. √ Exercise (8.21). — Let X be a variable, ω a complex cubic root of 1, and 3 2 √ the real cube root of 2. Set k := Q(ω) and K := k[ 3 2]. Show K = k[X]/⟨X 3 − 2⟩ and then K ⊗k K = K × K × K. September 3, 2012 11Nts.tex 9. Flatness A module is called flat if tensor product with it is an exact functor. First, we study exact functors in general. Then we prove various properties of flat modules. Notably, we prove Lazard’s Theorem, which characterizes the flat modules as the filtered direct limits of free modules of finite rank. Lazard’s Theorem yields the Ideal Criterion for Flatness, which characterizes the flat modules as those whose tensor product with any finitely generated ideal is equal to the ordinary product. Lemma (9.1). — Let R be a ring, α : M → N a homomorphism of modules. Then there is a diagram with two short exact sequences involving N ′ 0 // M ′ α // M // N ❏❏❏α′ α′′ tt:: ❏❏ $$ ttt ′ / / // 0 0 N // P // 0 (9.1.1) if and only if M ′ = Ker(α) and N ′ = Im(α) and P = Coker(α). ∼ Im(α) by (4.9). Proof: The equations yield the diagram since Coim(α) −→ ′ Conversely, given the diagram, note that Ker(α) = Ker(α ) since α′′ is injective. So M ′ = Ker(α). So N ′ = Coim(α) since α′ is surjective. Hence N ′ = Im(α). Therefore, P = Coker(α). Thus the equations hold. □ (9.2) (Exact Functors). — Let R be a ring, R′ an algebra, F a functor from ((R-mod)) to ((R′ -mod)). Assume F is R-linear; that is, the associated map HomR (M, N ) → HomR′ (F M, F N ) is R-linear. Then, if a map α : M → N is 0, so is F α : F M → F N . But M = 0 if and only if 1M = 0. Further, F (1M ) = 1F M . Thus if M = 0, then F M = 0. We call F exact if it preserves exact sequences. For example, Hom(P, •) is exact if and only if P is projective by (5.22). We call F left exact if it preserves kernels. When F is contravariant, we call F left exact if it takes cokernels to kernels. For example, Hom(N, •) and Hom(•, N ) are left exact covariant and contravariant functors. We call F right exact if it preserves cokernels. For example, M ⊗ • is right exact. Proposition (9.3). — Let R be a ring, R′ an algebra, F an R-linear functor. Then the following conditions are equivalent: (1) F preserves exact sequences; that is, F is exact. (2) F preserves short exact sequences. (3) F preserves kernels and surjections. (4) F preserves cokernels and injections. (5) F preserves kernels and images. Proof: Trivially, (1) implies (2). In view of (5.2), clearly (1) yields (3) and (4). Assume (3). Let 0 → M ′ → M → M ′′ → 0 be a short exact sequence. Since F preserves kernels, 0 → F M ′ → F M → F M ′′ is exact; since F preserves surjections, F M → F M ′′ → 0 is also exact. Thus (2) holds. Similarly, (4) implies (2). Assume (2). Given α : M → N , form the diagram (9.1.1). Applying F to it and September 3, 2012 11Nts.tex 44 9. Flatness using (2), we obtain a similar diagram for F (α). Hence (9.1) yields (5). α β Finally, assume (5). Let M ′ − →M − → M ′′ be exact; that is, Ker(β) = Im(α). Now, (5) yields Ker(F (β)) = F (Ker(β)) and Im(F (α)) = F (Im(α)). Therefore, Ker(F (β)) = Im(F (α)). Thus (1) holds. □ (9.4) (Flatness). — An R-module M is said to be flat over R or R-flat if the functor M ⊗R • preserves injections. It is equivalent by (9.3) that M ⊗R • be exact since it is right exact. An R-algebra R′ and its structure map are said to be flat if R′ is flat as an R-module. ⊕ Lemma (9.5). — A direct sum Mλ is flat if and only if each summand is flat. Proof: Let β : N ′ → N be an injective map. Then (8.11) yields ) (⊕ ⊕ Mλ ⊗ β = (Mλ ⊗ β); see the end of the proof of (8.15), taking T (M ) := M ⊗ N ′ and U (M ) := M ⊗ N . Now, the map on the right is injective if and only if each summand Mλ ⊗ β is injective by (5.4). The assertion follows. □ Proposition (9.6). — A free module is flat; in fact, a projective module is flat. Proof: The unitary law implies that R is flat over R. Hence a free module is flat by (9.5). Finally, a projective module is a direct summand of a free module by (5.22), and therefore flat by (9.5). □ Exercise (9.7). — Let R be a ring, R′ a flat algebra, and P a flat R′ -module. Show that P is a flat R-module. Exercise (9.8). — Let R be a ring, M a flat module, and R′ an algebra. Show that M ⊗R R′ is a flat R′ -module. Exercise (9.9). — Let R be a ring, a an ideal. Assume that R/a is R-flat. Show that a = a2 . Exercise (9.10). — Let R be a ring, R′ a flat algebra with structure map φ. Then R′ is said to be faithfully flat if for every R-module M , the map M → M ⊗ R′ given by x 7→ x ⊗ 1 is injective. Show that the following conditions are equivalent: (1) (2) (3) (4) (5) R′ is faithfully flat. φ−1 (aR′ ) = a for every ideal a of R. Spec(R′ ) → Spec(R) is surjective. For every maximal ideal m of R, the ideal mR′ ̸= R′ . For any nonzero R-module M , the module M ⊗R R′ ̸= 0. Exercise (9.11). — Let A and B be local rings, m and n their maximal ideals. Let φ : A → B be a local homomorphism; that is, φ(m) ⊂ n. Assume φ is flat. Show that φ is faithfully flat. Proposition (9.12). — Let R be a ring, 0 → M ′ → M → M ′′ → 0 an exact sequence of modules. Assume M ′′ is flat. (1) Then 0 → M ′ ⊗ N → M ⊗ N → M ′′ ⊗ N → 0 is exact for any module N . (2) Then M is flat if and only if M ′ is flat. September 3, 2012 11Nts.tex 9. Flatness 45 Proof: By (5.19), there is an exact sequence 0 → K → R⊕Λ → N → 0. Tensor it with the given sequence to obtain the following commutative diagram: 0   y M′ ⊗ K −−−→ M  ⊗ K −−−→ M ′′⊗ K − →0 α   y y y β 0− → M ′ ⊗R⊕Λ − → M ⊗R⊕Λ − → M ′′ ⊗ R⊕Λ   y y γ M ′ ⊗ N −−−→ M  ⊗N   y y 0 0 Here α and β are injective by Definition (9.4), as M ′′ and R⊕Λ are flat by hypothesis and by (9.6). So the rows and columns are exact, as tensor product is right exact. Finally, the Snake Lemma, (5.12), implies γ is injective. Thus (1) holds. To prove (2), take an injection β : N ′ → N , and form this commutative diagram: ′ 0− → M′  ⊗ N′ − →M⊗ → M ′′  ⊗ N′ − →0 N −    ′′ ′ αy α y αy 0 −→ M ′ ⊗ N −→ M ⊗ N −→ M ′′ ⊗ N −→ 0 Its rows are exact by (1). Assume M is flat. Then α is injective. Hence α′ is too. Thus M ′ is flat. Conversely, assume M ′ is flat. Then α′ is injective. But α′′ is injective as M ′′ is flat. Hence α is injective by the Snake lemma. Thus M is flat. Thus (2) holds. □ Proposition (9.13). — A filtered direct limit of flat modules lim Mλ is flat. −→ Proof: Let β : N ′ → N be injective. Then Mλ ⊗ β is injective for each λ since Mλ is flat. So lim(Mλ ⊗ β) is injective by the exactness of filtered direct limits, −→ (7.10). So (lim Mλ ) ⊗ β is injective by (8.11). Thus lim Mλ is flat. □ −→ −→ Proposition (9.14). — Let R and R′ be rings, M an R-module, N an (R, R′ )bimodule, and P an R′ -module. Then there is a canonical homomorphism θ : HomR (M, N ) ⊗R′ P → HomR (M, N ⊗R′ P ). (9.14.1) Assume P is flat. If M is finitely generated, then θ is injective; if M is finitely presented, then θ is an isomorphism. Proof: The map θ exists by Watts’s Theorem, (8.15), with R′ for R, applied to HomR (M, N ⊗R′ •). Explicitly, θ(φ ⊗ p)(m) = φ(m) ⊗ p. Alternatively, this formula can be used to construct θ, as (φ, n) 7→ ψ, where ψ(m) := φ(m) ⊗ p, is clearly bilinear. Clearly, θ is bijective if M = R. So θ is bijective if M = Rn for any n, as HomR (•, Q) preserves finite direct sums for any Q by (4.13). Assume that M is finitely generated. Then from (5.19), we obtain a presentation R⊕Λ → Rn → M → 0, with Λ finite if P is finitely presented. Since θ is natural, it September 3, 2012 11Nts.tex 46 9. Flatness yields this commutative diagram: 0 − → HomR (M, N ) ⊗R′ P − → HomR (Rn , N ) ⊗R′ P −→ HomR (R⊕Λ , N ) ⊗R′ P       ≃y θy y 0 − → HomR (M, N ⊗R′ P ) − → HomR (Rn , N ⊗R′ P ) − → HomR (R⊕Λ , N ⊗R′ P ) Its rows are exact owing to the left exactness of Hom and to the flatness of P . The right-hand vertical map is bijective if Λ is finite. The assertion follows. □ Exercise (9.15). — Let R be a ring, R′ an algebra, M and N modules. Show that there is a canonical map σ : HomR (M, N ) ⊗R R′ → HomR′ (M ⊗R R′ , N ⊗R R′ ). Assume R′ is flat over R. Show that if M is finitely generated, then σ is injective, and that if M is finitely presented, then σ is an isomorphism. Definition (9.16). — Let R be a ring, M a module. Let ΛM be the category whose objects are the pairs (Rm , α) where α : Rm → M is a homomorphism, and whose maps (Rm , α) → (Rn , β) are the homomorphisms φ : Rm → Rn with βφ = α. Proposition (9.17). — Let R be a ring, M a module, and (Rm , α) 7→ Rm the forgetful functor from ΛM to ((R-mod)). Then M = lim(Rm ,α)∈Λ Rm . −→ M Proof: By the UMP, the α : Rm → M induce a map ζ : lim Rm → M . Let’s −→ show ζ is bijective. First, ζ is surjective, because each x ∈ M is in the image of (R, αx ) where αx (r) := rx. ⊕ For injectivity, let y ∈ Ker(ζ). By construction, (Rm ,α) Rm → lim Rm is surjec−⊕ → tive; see the proof of (6.10). So y is in the image of some finite sum (Rmi , αi ) Rmi . ∑ ⊕ mi ∑ Set m := mi . Then R = Rm . Set α := αi . Then y is the image of some ′ m m m y ∈ R under the insertion ιm : R → lim R . But y ∈ Ker(ζ). So α(y ′ ) = 0. −→ Let θ, φ : R ⇒ Rm be the homomorphisms with θ(1) := y ′ and φ(1) := 0. They yield maps in ΛM . So, by definition of direct limit, they have the same compositions with the insertion ιm . Hence y = ιm (y ′ ) = 0. Thus ζ is injective, so bijective. □ Theorem (9.18) (Lazard). — Let R be a ring, M a module. Then the following conditions are equivalent: (1) M is flat. (2) Given a finitely presented module P , this version of (9.14.1) is surjective: HomR (P, R) ⊗R M → HomR (P, M ). (3) Given a finitely presented module P and a map β : P → M , there exists a γ α → M; factorization β : P − → Rn − m (4) Given an α : R → M and a k ∈ Ker(α), there exists a factorization φ → Rn → M such that φ(k) = 0. α : Rm − (5) Given an α : Rm → M and k1 , . . . , kr ∈ Ker(α) there exists a factorization φ → Rn → M such that φ(ki ) = 0 for i = 1, . . . , r. α : Rm − ρ α → M such that αρ = 0, there exists a factorization (6) Given Rr − → Rm − φ α : Rm − → Rn → M such that φρ = 0. (7) ΛM is filtered. (8) M is a filtered direct limit of free modules of finite rank. September 3, 2012 11Nts.tex 9. Flatness 47 Proof: Assume (1). Then (9.14) yields (2). Assume (2). Consider (3). There are γ1 , . . . , γn ∈ Hom(P, R) and x1 , . . . , xn ∈ M ∑ such that β(p) = γi (p)xi .∑Let γ : P → Rn be (γ1 , . . . , γn ), and let α : Rn → M be given by α(r1 , . . . , rn ) = ri xi . Then β = αγ, as (3) requires. Assume (3), and consider (4). Set P := Rm /Rk, and let κ : Rm → P denote the quotient map. Then P is finitely presented, and there is β : P → M such that γ βκ = α. By (3), there is a factorization β : P − → Rn → M . Set φ := γκ. Then φ → Rn → M is a factorization of β and φ(k) = 0. β : Rm − Assume (4), and consider (5). Set m0 := m and α0 = α. Inductively, (4) yields φi α i M αi−1 : Rmi−1 −→ Rmi −→ for i = 1, . . . , r such that φi · · · φ1 (ki ) = 0. Set φ := φr · · · φ1 and n := mr . Then (5) holds. Assume (5), and consider (6). Let e1 , . . . , er be the standard basis of Rr , and set φ → Rn → M such ki := ρ(ei ). Then α(ki ) = 0. So (5) yields a factorization α : Rm − that φ(ki ) = 0. Then φρ = 0, as required by (6). Assume (6). Given (Rm1 , α1 ) and (Rm2 , α2 ) in ΛM , set m := m1 + m2 and α := α1 + α2 . Then the inclusions Rmi → Rm induce maps in ΛM . Thus the first condition of (7.1) is satisfied. Given σ, τ : (Rr , ω) ⇒ (Rm , α) in ΛM , set ρ := σ − τ . Then αρ = 0. So (6) φ yields a factorization α : Rm − → Rn → M with φρ = 0. Then φ is a map of ΛM , and φσ = φτ . Hence the second condition of (7.1) is satisfied. Thus (7) holds. If (7) holds, then (8) does too, since M = lim(Rm ,α)∈Λ Rm by (9.17). −→ M Assume (8). Say M = lim Mλ with the Mλ free. Each Mλ is flat by (9.4), and −→ a filtered direct limit of flat modules is flat by (9.13). Thus M is flat □ Exercise (9.19) (Equational Criterion for Flatness). — Prove that ∑ the Condition (9.18)(4) can be reformulated as follows: For every relation i xi yi = 0 with xi ∈ R and yi ∈ M , there are xij ∈ R and yj′ ∈ M such that ∑ ∑ ′ (9.19.1) j xij yj = yi for all i and i xij xi = 0 for all j. Lemma (9.20) (Ideal Criterion for Flatness). — A module N is flat if and only if, for every finitely generated ideal a, the natural map is an isomorphism: ∼ aN. a ⊗ N −→ ∼ N with a⊗ x 7→ ax. If N is flat, Proof: In any case, (8.5)(2) implies R ⊗N −→ ∼ aN . then the inclusion a ֒→ R yields an injection a ⊗ N ֒→ R ⊗ N , and so a ⊗ N −→ ∑n To prove the converse, let’s check the criterion (9.19). Given ∑ i=1 xi yi = 0 with ∼ aN , then xi ∈ R and yi ∈ N , set a := ⟨x1 , . . . , xn ⟩. If a ⊗ N −→ i xi ⊗ yi = 0; so the Equational Criterion for Vanishing (8.18) yields (9.19.1). Thus N is flat. □ Example (9.21). — Let R be a domain, and set K := Frac(R). Then K is flat, but K is not projective unless R = K. Indeed, (8.6) says a ⊗R K = K, with a ⊗ x = ax, for any ideal a of R. So K is flat by (9.20). Suppose K is projective. Then K ֒→ RΛ for some Λ by (5.22). So there is a nonzero map α : K → R. So there is an x ∈ K with α(x) ̸= 0. Set a := α(x). Take any nonzero b ∈ R. Then ab · α(x/ab) = α(x) = a. Since R is a domain, b · α(x/ab) = 1. Hence b ∈ R× . Thus R is a field. So (2.3) yields R = K. September 3, 2012 11Nts.tex 48 9. Flatness Exercise (9.22). — Let R be a domain, M a module. Prove that, if M is flat, then M is torsion free; that is, µx : M → M is injective for all nonzero x ∈ R. Prove that, conversely, if R is a PID and M is torsion free, then M is flat. September 3, 2012 11Nts.tex 10. Cayley–Hamilton Theorem The Cayley–Hamilton Theorem says that a matrix satisfies its own characteristic polynomial. We prove an equivalent form, known as the “Determinant Trick.” Using the Trick, we obtain various results, including the uniqueness of the rank of a finitely generated free module. We also obtain Nakayama’s Lemma, and use it to study finitely generated modules further. Then we turn to the important notions of integrality and module finiteness for an algebra. Using the Trick, we relate these notions to each other, and study their properties. We end with a discussion of integral extensions and normal rings. (10.1) (Cayley–Hamilton Theorem). — Let R be a ring, and M := (aij ) an n × n matrix with aij ∈ R. Let In be the n × n identity matrix, and T a variable. The characteristic polynomial of M is the following polynomial: pM (T ) := T n + a1 T n−1 + · · · + an := det(T In − M). Let a be an ideal. If aij ∈ a for all i, j, then clearly ak ∈ ak for all k. The Cayley–Hamilton Theorem asserts that, in the ring of matrices, pM (M) = 0. It is a special case of (10.2) below; indeed, take M := Rn , take m1 , . . . , mn to be the standard basis, and take φ to be the endomorphism defined by M. Conversely, given the setup of (10.2), form the surjection α : Rn → → M taking the ith standard basis element to mi , and form the map Φ : Rn → Rn associated to the matrix M. Then φα = αΦ. Hence, given any polynomial p(T ), we have p(φ)α = αp(Φ). Hence, if p(Φ) = 0, then p(φ) = 0 as α is surjective. Thus the Cayley–Hamilton Theorem and the Determinant Trick (10.2) are equivalent. Theorem (10.2) (Determinant Trick). — Let M be an R-module ∑n generated by m1 , . . . , mn , and φ : M → M an endomorphism. Say φ(mi ) =: j=1 aij mj with aij ∈ R, and form the matrix M := (aij ). Then pM (φ) = 0 in End(M ). Proof: Let δij be the Kronecker delta function, µaij the multiplication map. Let ∆ stand for the matrix (δij φ − µaij ) with entries in End(M ), and X for the column vector (mj ). Then clearly ∆X = 0. Multiply on the left by the matrix of cofactors Γ of ∆: the (i, j)th entry of Γ is (−1)i+j times the determinant of the matrix obtained by deleting the jth row and the ith column of ∆. Then Γ∆X = 0. Now, Γ∆ = det(∆)In . Hence det(∆)mj = 0 for all j. Thus pM (φ) = 0. □ Proposition (10.3). — Let M be a finitely generated module, a an ideal. Then M = aM if and only if there exists a ∈ a such that (1 + a)M = 0. ∑n Proof: Assume M = aM . Say m1 , . . . , mn generate M , and mi = j=1 aij mj with aij ∈ a. Set M := (aij ). Say pM (T ) = T n + a1 T n−1 + · · · + an . Set a := a1 + · · · + an ∈ a. Then (1 + a)M = 0 by (10.2) with φ := 1M . Conversely, if there exists a ∈ a such that (1 + a)M = 0, then m = −am for all m ∈ M . So M ⊂ aM ⊂ M . Thus M = aM . □ Corollary (10.4). — Let R be a ring, M a finitely generated module, and φ an endomorphism of M . If φ is surjective, then φ is an isomorphism. September 3, 2012 11Nts.tex 50 10. Cayley–Hamilton Theorem Proof: Let P := R[X] be the polynomial ring in one variable. By the UMP of P , there is an R-algebra homomorphism µ : P → End(M ) with µ(X) = φ. So M is an P -module such that p(X)M = p(φ)M for any p(X) ∈ P by (4.4). Set a := ⟨X⟩. Since φ is surjective, M = aM . By (10.3), there is a ∈ a with (1 + a)M = 0. Say a = Xr for some polynomial r. Then 1M + φr(φ) = 0. Thus φ is invertible. □ Corollary (10.5). — Let R be a nonzero ring, m and n positive integers. (1) Then any n generators v1 , . . . , vn of the free module Rn form a free basis. (2) If Rm ≃ Rn , then m = n. Proof: Form the surjection α : Rn → → Rn taking the ith standard basis element to vi . Then φ is an isomorphism by (10.4). So the vi form a free basis by (4.10)(3). To prove (2), say m ≤ n. Then Rn has m generators. Add to them n − m zeros. The result is a free basis by (1), so can contain no zeros. Thus n − m = 0. □ Exercise (10.6). — Let R be a ring, a an ideal. Assume a is finitely generated and idempotent (or a = a2 ). Prove there is a unique idempotent e with ⟨e⟩ = a. Proposition (10.7). — Let R be a ring, a an ideal. Then these conditions are equivalent: (1) (2) (3) (4) (5) R/a is projective over R. R/a is flat over R, and a is finitely generated. a is finitely generated and idempotent. a is generated by an idempotent. a is a direct summand of R. Proof: Suppose (1) holds. Then R/a is flat by (9.6). Further, the sequence 0 → a → R → R/a → 0 splits by (5.22), and so a is principal. Thus (2) holds. If (2) holds, then (3) holds by (9.9). If (3) holds, then (4) holds by (10.6). If (4) holds, then (5) holds by (1.12). If (5) holds, then (1) holds by by (5.22). □ Exercise (10.8). — Prove the following conditions on a ring R are equivalent: (1) (2) (3) (4) R is absolutely flat; that is, every module is flat. Every finitely generated ideal is a direct summand of R. Every finitely generated ideal is idempotent. Every principal ideal is idempotent. Exercise (10.9). — Let R be a ring. (1) (2) (3) (4) Assume Assume Assume Assume R R R R is is is is Boolean. Prove R is absolutely flat. absolutely flat. Prove any quotient ring R′ is absolutely flat. absolutely flat. Prove every nonunit x is a zerodivisor. absolutely flat and local. Prove R is a field. Lemma (10.10) (Nakayama). — Let R be a ring, m ⊂ rad(R) an ideal, M a finitely generated module. Assume M = mM . Then M = 0. Proof: By (10.3), there is a ∈ m with (1 + a)M = 0. By (3.2), 1 + a is a unit. Thus M = (1 + a)−1 (1 + a)M = 0. Alternatively, suppose M ̸= 0. Say m1 , . . . , mn generate M with n minimal. Then m1 = a1 m1 +· · ·+an mn with ai ∈ m. By (3.2), we may set xi := (1−a1 )−1 ai . Then m1 = x2 m2 + · · · + xn mn , contradicting minimality of n. Thus M = 0. □ September 3, 2012 11Nts.tex 10. Cayley–Hamilton Theorem 51 Proposition (10.11). — Let R be a ring, m ⊂ rad(R) an ideal, N ⊂ M modules. (1) If M/N is finitely generated and if N + mM = M , then N = M . (2) Assume M is finitely generated. Then elements m1 , . . . , mn generate M if and only if their images m′1 , . . . , m′n generate M ′ := M/mM . Proof: In (1), the second hypothesis holds if and only if m(M/N ) = M/N . Hence (1) holds by (10.10) applied with M/N for M . In (2), let N be the submodule generated by m1 , . . . , mn . Since M is finitely generated, so is M/N . Hence N = M if the m′i generate M/mM by (1). The converse is obvious. □ Exercise (10.12). — Let R be a ring, m ⊂ rad(R) an ideal. Let α, β : M → N be two maps of finitely generated modules. Assume α is surjective and β(M ) ⊂ mN . Set γ := α + β. Show that γ is an isomorphism. Exercise (10.13). — Let A be a local ring, m the maximal ideal, M a finitely generated A-module, and m1 , . . . , mn ∈ M . Set k := A/m and M ′ := M/mM , and write m′i for the image of mi in M ′ . Prove that m′1 , . . . , m′n ∈ M ′ form a basis of the k-vector space M ′ if and only if m1 , . . . , mn form a minimal generating set of M (that is, no proper subset generates M ), and prove that every minimal generating set of M has the same number of elements. Exercise (10.14). — Let A be a local ring, k its residue field, M and N finitely generated modules. (1) Show that M = 0 if and only if M ⊗A k = 0. (2) Show that M ⊗A N ̸= 0 if M ̸= 0 and N ̸= 0. Proposition (10.15). — Consider these conditions on an R-module P : (1) P is free and of finite rank; (2) P is projective and finitely generated; (3) P is flat and finitely presented. Then (1) implies (2), and (2) implies (3); all three are equivalent if R is local. Proof: A free module is always projective by (5.21), and a projective module is always flat by (9.6). Further, each of the three conditions requires P to be finitely generated; so assume it is. Thus (1) implies (2). Let p1 , . . . , pn ∈ P generate, and let 0 → L → Rn → P → 0 be the short exact sequence defined by sending the ith standard basis element to pi . Set F := Rn . Assume P is projective. Then the sequence splits by (5.22). So (5.9) yields a surjection ρ : F → L. Hence L is finitely generated. Thus (2) implies (3). Assume P is flat and R is local. Denote the residue field of R by k. Then, by (9.12)(1), the sequence 0 → L ⊗ k → F ⊗ k → P ⊗ k → 0 is exact. Now, F ⊗ k = (R ⊗ k)n = k n by (8.11) and the unitary law; so dimk F ⊗ k = n. Finally, rechoose the pi so that n is minimal. Then dimk P ⊗ k = n, because the pi ⊗ 1 form a basis by (10.13). Therefore, dimk L ⊗ k = 0; so L ⊗ k = 0. Assume P is finitely presented. Then L is finitely generated by (5.24). Hence L = 0 by (10.14)(1). So F = P . Thus (3) implies (1). □ Definition (10.16). — Let R be a ring, R′ an R-algebra. Then R′ is said to be module finite over R if R′ is a finitely generated R-module. An element x ∈ R′ is said to be integral over R or integrally dependent on September 3, 2012 11Nts.tex 52 10. Cayley–Hamilton Theorem R if there exist a positive integer n and elements ai ∈ R such that xn + a1 xn−1 + · · · + an = 0. (10.16.1) Such an equation is called an equation of integral dependence of degree n. If every x ∈ R′ is integral over R, then R′ is said to be integral over R. Exercise (10.17). — Let G be a finite group acting on a domain R, and R′ the ring of invariants. Show every x ∈ R is integral over R′ , in fact, over the subring R′′ generated by the elementary symmetric functions in the conjugates gx for g ∈ G. Proposition (10.18). — Let R be a ring, R′ an R-algebra, n a positive integer, and x ∈ R′ . Then the following conditions are equivalent: (1) x satisfies an equation of integral dependence of degree n; (2) R[x] is generated as an R-module by 1, x, . . . , xn−1 ; (3) x lies in a subalgebra R′′ generated as an R-module by n elements; (4) there is a faithful R[x]-module M generated over R by n elements. Proof: Assume (1) holds. Say p(X) is a monic polynomial of degree n with p(x) = 0. For any m, let Mm ⊂ R[x] be the R-submodule generated by 1, . . . , xm . For m ≥ n, clearly xm − xm−n p(x) is in Mm−1 . But p(x) = 0. So also xm ∈ Mm−1 . So by induction, Mm = Mn−1 . Hence Mn−1 = R[x]. Thus (2) holds. If (2) holds, then trivially (3) holds with R′′ := R[x]. If (3) holds, then (4) holds with M := R′′ , as xM = 0 implies x = x · 1 = 0. Assume (4) holds. In (10.2), take φ := µx . We obtain a monic polynomial p of degree n with p(x)M = 0. Since M is faithful, p(x) = 0. Thus (1) holds. □ Exercise (10.19). — Let k be a field, P := k[X] the polynomial ring in one variable, f ∈ P . Set R := k[X 2 ] ⊂ P . Using the free basis 1, X of P over R, find an explicit equation of integral dependence of degree 2 on R for f . Corollary (10.20). — Let R be a ring, P the polynomial ring in one variable, and a an ideal of P . Set R′ := P/a, and let x be the image of X in R′ . Let n be a positive integer. Then the following conditions are equivalent: (1) a = ⟨p⟩ where p is a monic polynomial of degree n; (2) 1, x, . . . , xn−1 form a free basis of R′ over R; (3) R′ is a free R-module of rank n. Proof: Assume (1) holds. Then p(x) = 0 is an equation of integral dependence of degree n. So 1, x, . . . , xn−1 generate R′ by (1)⇒(2) of (10.18). Suppose b1 xn−1 + · · · + bn = 0 with the bi ∈ R. Set q(X) := b1 X n−1 + · · · + bn . Then q(x) = 0. So q ∈ a. Hence q = f p for some f ∈ P . But p is monic of degree n. Hence q = 0. Thus (2) holds. Trivially, (2) implies (3). Finally, assume (3) holds. Then (3)⇒(1) of (10.18) yields a monic polynomial p ∈ a of degree n. Form the induced homomorphism ψ : P/⟨p⟩ → R′ . It is obviously surjective. Since (1) implies (3), the quotient P/⟨p⟩ is free of rank n. So ψ is an isomorphism by (10.4). Hence ⟨p⟩ = a. Thus (1) holds. □ Lemma (10.21). — Let R be a ring, R′ a module-finite R-algebra, and M a finitely generated R′ -module. Then M is a finitely generated R-module. September 3, 2012 11Nts.tex 10. Cayley–Hamilton Theorem 53 Proof: Say elements xi generate R′ as a module over R, and say elements mj generate M over R′ . Then clearly the products xi mj generate M over R. □ Theorem (10.22) (Tower Law for Integrality). — Let R be a ring, R′ an algebra, and R′′ an R′ -algebra. If x ∈ R′′ is integral over R′ and if R′ is integral over R, then x is integral over R. Proof: Say xn + a1 xn−1 + · · · + an = 0 with ai ∈ R′ . For m = 1, . . . , n, set Rm := R[a1 , . . . , am ] ⊂ R′′ . Then Rm is module finite over Rm−1 by (1)⇒(2) of (10.18). So Rm is module finite over R by (10.21) and induction on m. Moreover, x is integral over Rn . So Rn [x] is module finite over Rn by (1)⇒(2) of (10.18). Hence Rn [x] is module finite over R by (10.21). So x is integral over R by (3)⇒(1) of (10.18), as desired. □ Theorem (10.23). — Let R be a ring, R′ an R-algebra. Then the following conditions are equivalent: (1) R′ is finitely generated as an R-algebra and is integral over R; (2) R′ = R[x1 , . . . , xn ] with all xi integral over R; (3) R′ is module-finite over R. Proof: Trivially, (1) implies (2). Assume (2) holds. To prove (3), set R′′ := R[x1 ] ⊂ R′ . Then R′′ is module finite over R by (1)⇒(2) of (10.18). We may assume R′ is module finite over R′′ by induction on n. So (10.21) yields (3). If (3) holds, then R′ is integral over R by (3)⇒(1) of (10.18); so (1) holds. □ Exercise (10.24). — Let R1 , . . . , Rn be R-algebras, integral over R. Show that ∏ their product Ri is a integral over R. Definition (10.25). — Let R be a ring, R′ an algebra. The integral closure or normalization of R in R′ is the subset R of elements that are integral over R. If R ⊂ R′ and R = R, then R is said to be integrally closed in R′ . If R is a domain, then its integral closure R in its fraction field Frac(R) is called simply its normalization, and R is said to be normal if R = R. Exercise (10.26). i ≤∏r, let Ri be a ring, Ri′ an extension of Ri , and ∏— For 1 ≤ ′ ′ xi ∈ Ri . Set R := Ri , set R := Ri′ , and set x := (x1 , . . . , xr ). Prove (1) x is integral over R if and only if xi is integral over Ri for each i; (2) R is integrally closed in R′ if and only if each Ri is integrally closed in Ri′ . Corollary (10.27). — Let R be a ring, R′ an R-algebra, R the integral closure of R in R′ . Then R is an R-algebra, and is integrally closed in R′ . Proof: Take a ∈ R and x, y ∈ R. Then the ring R[x, y] is integral over R by (2)⇒(1) of (10.23). So ax and x + y and xy are integral over R. Thus R is an R-algebra. Finally, R is integrally closed in R′ owing to (10.22). □ Theorem (10.28) (Gauss). — A UFD is normal. Proof: Let R be the UFD. Given x ∈ Frac(R), say x = r/s with r, s ∈ R relatively prime. Suppose x satisfies (10.16.1). Then rn = −(a1 rn−1 + · · · + an sn−1 )s. So any prime element dividing s also divides r. Hence s is a unit. Thus x ∈ R. September 3, 2012 11Nts.tex □ 54 10. Cayley–Hamilton Theorem Example (10.29). — (1) A polynomial ring in n variables over a field is a UFD, so normal by (10.28). √ (2) The ring R := Z[ 5] is not a UFD, since √ √ (1 + 5)(1 − 5) = −4 = −2 · 2, √ √ and 1 + 5, √ and 1 − 5 and 2 are irreducible, but not associates. However, set τ := (1 + 5)/2, the “golden ratio.” The ring Z[τ ] is known to be a PID; see [8, p. 292]. Hence, Z[τ ] is a UFD, so normal by (10.28); hence, Z[τ ] contains the normalization R of R. On the other hand, τ 2 − τ − 1 = 0; hence, Z[τ ] ⊂ R. Thus Z[τ ] = R. √ (3) Let d ∈ Z be square-free. In the field K := Q( d), form R := Z + Zδ where { √ (1 + d)/2, if d ≡ 1 (mod 4); δ := √ d, if not. Then R is the normalization Z of Z in K; see [1, pp. 412–3]. (4) Let k be a field, k[t] the polynomial ring in one variable. Set R := k[t2 , t3 ]. Then Frac(R) = k(t). Further, t is integral over R as t satisfies X 2 − t2 = 0; hence, k[t] ⊂ R. However, k[t] is normal by (1); hence, k[t] ⊃ R. Thus k[t] = R. Let k[X, Y ] be the polynomial ring in two variables, and φ : k[X, Y ] → R the k-algebra homomorphism defined by φ(X) := t2 and φ(Y ) := t3 . Clearly φ is surjective. Set p := Ker φ. Since R is a domain, but not a field, p is prime by (2.9), but not maximal by (2.17). Clearly p ⊃ ⟨Y 2 − X 3 ⟩. Since Y 2 − X 3 is ∼ R, irreducible, (2.26) implies that p = ⟨Y 2 − X 3 ⟩. So k[X, Y ]/⟨Y 2 − X 3 ⟩ −→ which provides us with another description of R. Exercise (10.30). — Let k be a field, X and Y variables. Set R := k[X, Y ]/⟨Y 2 − X 2 − X 3 ⟩, and let x, y ∈ R be the residues of X, Y . Prove that R is a domain, but not a field. Set t := y/x ∈ Frac(R). Prove that k[t] is the integral closure of R in Frac(R). September 3, 2012 11Nts.tex 11. Localization of Rings Localization generalizes construction of the fraction field of a domain. We localize an arbitrary ring using as denominators the elements of any given multiplicative subset. The result is the universal example of an algebra where all these elements become units. When the multiplicative subset is the complement of a prime ideal, we obtain a local ring. We relate the ideals in the original ring to those in the localized ring. We end by localizing algebras and then varying the set of denominators. (11.1) (Localization). — Let R be a ring, and S a multiplicative subset. Define a relation on R × S by (x, s) ∼ (y, t) if there is u ∈ S such that xtu = ysu. This relation is an equivalence relation. Indeed, it is reflexive as 1 ∈ S and is trivially symmetric. As to transitivity, let (y, t) ∼ (z, r). Say yrv = ztv with v ∈ S. Then xturv = ysurv = ztvsu. Thus (x, s) ∼ (z, r). Denote by S −1 R the set of equivalence classes, and by x/s the class of (x, s). Define x/s · y/t := xy/st. This product is well defined. Indeed, say y/t = z/r. Then there is v ∈ S such that yrv = ztv. So xsyrv = xsztv. Thus xy/st = xz/sr. Define x/s + y/t := (tx + sy)/(st). Then, similarly, this sum is well defined. It is easy to check S −1 R is a ring, with 0/1 for 0 and 1/1 for 1. It is called the ring of fractions with respect to S or the localization at S. Let φS : R → S −1 R be the map given by φS (x) := x/1. Then φS is a ring map, and it carries elements of S to units in S −1 R as s/1 · 1/s = 1. Exercise (11.2). — Let R be a ring, S a multiplicative subset. Prove S −1 R = 0 if and only if S contains a nilpotent element. Exercise (11.3). — Let R be a ring, S a multiplicative subset, S its saturation. Set T := (S −1 R)× . Show T = { x/s | x ∈ S and s ∈ S }. Show φ−1 S T = S. (11.4) (Total quotient ring). — Let R be a ring, S the set of all nonzerodivisors of R. Clearly S is a multiplicative subset. The map φS : R → S −1 R is injective, because if φS x = 0, then sx = 0 for some s ∈ S, and so x = 0. We call S −1 R the total quotient ring of R, and view R as a subring. Let T ⊂ S be a multiplicative subset. Clearly, R ⊂ T −1 R ⊂ S −1 R. Suppose R is a domain. Then S = R − {0}; so the total quotient ring is just the fraction field Frac(R), and φS is just the natural inclusion of R into Frac(R). Further, T −1 R is a domain by (2.3) as T −1 R ⊂ S −1 R = Frac(R). Exercise (11.5). — Find all intermediate rings Z ⊂ R ⊂ Q, and describe each R as a localization of Z. As a starter, prove Z[2/3] = S −1 Z where S = {3i | i ≥ 0}. Theorem (11.6) (UMP). — Let R be a ring, S a multiplicative subset. Then S −1 R is the universal example of an R-algebra in which all the elements of S become units. In fact, given a ring map ψ : R → R′ , then ψ(S) ⊂ R′× if and only if there is a ring map ρ : S −1 R → R′ with ρφS = ψ; that is, this diagram commutes: φS // S −1 R R ❏ ❏❏ ❏❏ ρ ❏ ψ ❏❏$$  R′ September 3, 2012 11Nts.tex 56 11. Localization of Rings Further, there is at most one ρ. Moreover, R′ may be noncommutative. Proof: First, suppose that ρ exists. Let s ∈ S. Then ψ(s) = ρ(s/1). Hence ψ(s)ρ(1/s) = ρ(s/1 · 1/s) = 1. Thus ψ(S) ⊂ R′× . Next, note that ρ is determined by ψ as follows: ρ(x/s) = ρ(x/1)ρ(1/s) = ψ(x)ψ(s)−1 . Conversely, suppose ψ(S) ⊂ R′× . Set ρ(x/s) := ψ(s)−1 ψ(x). Let’s check that ρ is well defined. Say x/s = y/t. Then there is u ∈ S such that xtu = ysu. Hence ψ(x)ψ(t)ψ(u) = ψ(y)ψ(s)ψ(u). Since ψ(u) is a unit, ψ(x)ψ(t) = ψ(y)ψ(s). Now, st = ts, so ψ(t)−1 ψ(s)−1 = ψ(s)−1 ψ(t)−1 . Hence ψ(x)ψ(s)−1 = ψ(y)ψ(t)−1 . Thus ρ is well defined. Clearly, ρ is a ring map. Clearly, ψ = ρφS . □ Corollary (11.7). — Let R be a ring, and S a multiplicative subset. Then the canonical map φS : R → S −1 R is an isomorphism if and only if S consists of units. Proof: If φS is an isomorphism, then S consists of units, because φS (S) does so. Conversely, if S consists of units, then the identity map R → R has the UMP that characterizes φS ; whence, φS is an isomorphism. □ Exercise (11.8). — Let R′ and R′′ be rings. Consider R := R′ × R′′ and set S := { (1, 1), (1, 0) }. Prove R′ = S −1 R. Exercise (11.9). — Take R and S as in (11.8). On R × S, impose this relation: (x, s) ∼ (y, t) if xt = ys. Prove that it is not an equivalence relation. Definition (11.10). — Let R be a ring, f ∈ R. Set S := {f n | n ≥ 0}. We call the ring S −1 R the localization of R at f , and set Rf := S −1 R and φf := φS . Proposition (11.11). — Let R be a ring, f ∈ R, and X a variable. Then / Rf = R[X] ⟨1 − f X⟩. / Proof: Set R′ := R[X] ⟨1 − f X⟩, and let φ : R → R′ be the canonical map. Let’s show that R′ has the UMP characterizing localization (11.6). First, let x ∈ R′ be the residue of X. Then 1 − xφ(f ) = 0. So φ(f ) is a unit. So φ(f n ) is a unit for n ≥ 0. Second, let ψ : R → R′′ be a homomorphism carrying f to a unit. Define θ : R[X] → R′′ by θ|R = ψ and θX = ψ(f )−1 . Then θ(1 − f X) = 0. So θ factors via a homomorphism ρ : R′ → R′′ , and ψ = ρφ. Further, ρ is unique, since every element of R′ is a polynomial in x and since ρx = ψ(f )−1 as 1 − (ρx)(ρφf ) = 0. □ Proposition (11.12). — Let R be a ring, S a multiplicative subset, a an ideal. (1) Then aS −1 R = {a/s ∈ S −1 R | a ∈ a and s ∈ S}. −1 R) = R. (2) Then a∩S ̸= ∅ if and only if aS −1 R = S −1 R if and only if φ−1 S (aS September 3, 2012 11Nts.tex 11. Localization of Rings 57 Proof: Let a, b ∈ a and x/s, y/t ∈ S −1 R. Then ax/s + by/t = (axt + bys)/st; further, axt + bys ∈ a and st ∈ S. So aS −1 R ⊂ {a/s | a ∈ a and s ∈ S}. But the opposite inclusion is trivial. Thus (1) holds. As to (2), if a ∩ S ∋ s, then aS −1 R ∋ s/s = 1, so aS −1 R = S −1 R; whence, −1 −1 φS (aS −1 R) = R. Conversely, suppose φ−1 R) = R. Then aS −1 R ∋ 1. So S (aS (1) yields a ∈ a and s ∈ S such that a/s = 1. So there exists a t ∈ S such that at = st. But at ∈ a and st ∈ S. So a ∩ S ̸= ∅. Thus (2) holds. □ Definition (11.13). — Let R be a ring, S a multiplicative subset, a a subset of R. The saturation of a with respect to S is the set denoted by aS and defined by aS := {a ∈ R | there is s ∈ S with as ∈ a}. If a = aS , then we say a is saturated. Proposition (11.14). — Let R be a ring, S a multiplicative subset, a an ideal. (1) Then Ker(φS ) = ⟨0⟩S . (2) Then a ⊂ aS . (3) Then aS is an ideal. Proof: Clearly, (1) holds, for a/1 = 0 if and only if there is s ∈ S with as = 0. Clearly, (2) holds as 1 ∈ S. Clearly, (3) holds, for if as, bt ∈ a, then (a + b)st ∈ a, and if x ∈ R, then xas ∈ a. □ Exercise (11.15). — Let R be a ring, S a multiplicative subset, a and b ideals. Show (1) if a ⊂ b, then aS ⊂ bS ; (2) (aS )S = aS ; and (3) (aS bS )S = (ab)S . Exercise (11.16). — Let R be a ring, S a multiplicative subset. Prove that nil(R)(S −1 R) = nil(S −1 R). Proposition (11.17). — Let R be a ring, S a multiplicative subset. (1) Let b be an ideal of S −1 R. Then −1 S (a) φ−1 S b = (φS b) and −1 R). (b) b = (φ−1 S b)(S −1 R) = aS . (2) Let a be an ideal of R. Then φ−1 S (aS (3) Let p be a prime ideal of R, and assume p ∩ S = ∅. Then (a) p = pS and (b) pS −1 R is prime. Proof: To prove (1)(a), take a ∈ R and s ∈ S with as ∈ φ−1 S b. Then as/1 ∈ b; −1 S so a/1 ∈ b because 1/s ∈ S −1 R. Hence a ∈ φ−1 b. Therefore, (φ−1 S S b) ⊂ φS b. The opposite inclusion holds as 1 ∈ S. Thus (1)(a) holds. To prove (1)(b), take a/s ∈ b. Then a/1 ∈ b. So a ∈ φ−1 S b. Hence a/1 · 1/s is in −1 −1 −1 −1 (φS b)(S R). Thus b ⊂ (φS b)(S R). Now, take a ∈ φ−1 S b. Then a/1 ∈ b. So −1 R). Thus (1)(b) holds too. b ⊃ (φ−1 S b)(S To prove (2), take a ∈ aS . Then there is s ∈ S with as ∈ a. But a/1 = as/1 · 1/s. −1 −1 R). Then R) ⊃ aS . Now, take x ∈ φ−1 So a/1 ∈ aS −1 R. Thus φ−1 S (aS S (aS x/1 = a/s with a ∈ a and s ∈ S by (11.14)(1). Hence there is t ∈ S such that −1 R) ⊂ aS . Thus (2) holds. xst = at ∈ a. So x ∈ aS . Thus φ−1 S (aS S To prove (3), note p ⊂ p as 1 ∈ S. Conversely, if sa ∈ p with s ∈ S ⊂ R − p, then a ∈ p as p is prime. Thus (a) holds. −1 R), and the latter is As for (b), say a/s · b/t ∈ pS −1 R. Then ab ∈ φ−1 S (pS S equal to p by (2), so to p by (a). Hence ab ∈ p, so either a ∈ p or b ∈ p. So either a/s ∈ pS −1 R or b/t ∈ pS −1 R. Thus pS −1 R is prime. Thus (3) holds. □ September 3, 2012 11Nts.tex 58 11. Localization of Rings Corollary (11.18). — Let R be a ring, S a multiplicative subset. (1) Then a 7→ aS −1 R is an inclusion-preserving bijection from the set of all ideals a of R with a = aS to the set of all ideals b of S −1 R. The inverse is b 7→ φ−1 S b. (2) Then p 7→ pS −1 R is an inclusion-preserving bijection from the set of all primes of R with p ∩ S = ∅ to the set of all primes q of S −1 R. The inverse is q 7→ φ−1 S q. Proof: In (1), the maps are inverses by (11.17)(1), (2); clearly, they preserve inclusions. Further, (1) implies (2) by (11.17)(3), by (2.8), and by (11.12)(2). □ Definition (11.19). — Let R be a ring, p a prime ideal. Set S := R − p. We call the ring S −1 R the localization of R at p, and set Rp := S −1 R and φp := φS . Proposition (11.20). — Let R be a ring, p a prime ideal. Then Rp is local with maximal ideal pRp . Proof: Let b be a proper ideal of Rp . Then φ−1 p b ⊂ p owing to (11.12)(2). Hence (11.18)(1) yields b ⊂ pRp . Thus pRp is a maximal ideal, and the only one. Alternatively, let x/s ∈ Rp . Suppose x/s is a unit. Then there is a y/t with xy/st = 1. So there is a u ∈ / p with xyu = stu. But stu ∈ / p. Hence x ∈ / p. Conversely, let x ∈ / p. Then s/x ∈ Rp . So x/s is a unit in Rp if and only if x ∈ / p, so if and only if x/s ∈ / pRp . Thus by (11.12)(1), the nonunits of Rp form pRp , which is an ideal. Hence (3.4) yields the assertion. □ ∩ Proposition (11.21). — Let R be a domain, a an ideal. Then a = m aRm where m runs through the maximal ideals and the intersection is taken in Frac(R). ∩ Proof: Set I := m aRm . Clearly, a ⊂ I. For the opposite inclusion, given x ∈ I, set b := {y ∈ R | yx ∈ a}. Suppose x ∈ / a. Then 1 ∈ / b, so there is a maximal ideal n ⊃ b by (2.27). But x ∈ aRn . So there is s ∈ R − n with sx ∈ a by (11.12). Then s ∈ b ⊂ n, a contradiction. Thus x ∈ a, as desired. □ (11.22) (Algebras). — Let R be a ring, S a multiplicative subset, R′ an R-algebra. It is easy to generalize (11.1) as follows. Define a relation on R′ ×S by (x, s) ∼ (y, t) if there is u ∈ S with xtu = ysu. It is easy to check, as in (11.1), that this relation is an equivalence relation. Denote by S −1 R′ the set of equivalence classes, and by x/s the class of (x, s). Clearly, S −1 R′ is an S −1 R-algebra with addition and multiplication given by x/s + y/t := (xt + ys)/(st) and x/s · y/t := xy/st. We call S −1 R′ the localization of R′ with respect to S. Let φ′S : R′ → S −1 R′ be the map given by φ′S (x) := x/1. Then φ′S makes S −1 R′ into an R′ -algebra, so also into an R-algebra, and φ′S is an R-algebra map. Note that elements of S become units in S −1 R′ . Moreover, it is easy to check, as in (11.6), that S −1 R′ has the following UMP: φ′S is an algebra map, and elements of S become units in S −1 R′ ; further, given an algebra map ψ : R′ → R′′ such that elements of S become units in R′′ , there is a unique R-algebra map ρ : S −1 R′ → R′′ such that ρφ′S = ψ; that is, the following diagram is commutative: φ′S // S −1 R′ R′ ❏ ❏❏ ❏❏ ρ ❏❏  ψ $$ R′′ September 3, 2012 11Nts.tex 11. Localization of Rings 59 In other words, S −1 R′ is the universal example of an R′ -algebra in which the elements of S become units. Let τ : R′ → R′′ be an R-algebra map. Then there is a commutative diagram of R-algebra maps τ ′ ′′ R  −−−−−−−→ R   φS y yφ′S S −1 τ S −1 R′ −−−→ S −1 R′′ Further, S −1 τ is an S −1 R-algebra map. Let T ⊂ R′ be the image of S ⊂ R. Then T is multiplicative. Further, S −1 R′ = T −1 R′ , even though R′ ×S and R′ ×T are rarely equal, because the two UMPs are essentially the same; indeed, any ring map R′ → R′′ may be viewed as an R-algebra map, and trivially the elements of S become units in R′′ if and only if the elements of T do. Exercise (11.23). — Let R′ /R be a integral extension of rings, S a multiplicative subset of R. Show that S −1 R′ is integral over S −1 R. Exercise (11.24). — Let R be a domain, K its fraction field, L a finite extension field, and R the integral closure of R in L. Show L is the fraction field of R. Show every element of L can, in fact, be expressed as a fraction b/a with b ∈ R and a ∈ R. Exercise (11.25). — Let R ⊂ R′ be domains, K and L their fraction fields. Assume that R′ is a finitely generated R-algebra, and that L is a finite dimensional K-vector space. Find an f ∈ R such that Rf′ is module finite over Rf . Proposition (11.26). — Let R be a ring, S a multiplicative subset. Let T ′ be a ′ multiplicative subset of S −1 R, and set T := φ−1 S (T ). Assume S ⊂ T . Then (T ′ )−1 (S −1 R) = T −1 R. Proof: Let’s check (T ′ )−1 (S −1 R) has the UMP characterizing T −1 R. Clearly ( )× φT ′ φS carries T into (T ′ )−1 (S −1 R) . Next, let ψ : R → R′ be a map carrying T into R′× . We must show ψ factors uniquely through (T ′ )−1 (S −1 R). First, ψ carries S into R′× since S ⊂ T . So ψ factors through a unique map ρ : S −1 R → R′ . Now, given r ∈ T ′ , write r = x/s. Then x/1 = s/1 · r ∈ T ′ since S ⊂ T . So x ∈ T . Hence ρ(r) = ψ(x) · ρ(1/s) ∈ (R′ )× . So ρ factors through a unique map ρ′ : (T ′ )−1 (S −1 R) → R′ . Hence ψ = ρ′ φT ′ φS , and ρ′ is clearly unique, as required. □ Corollary (11.27). — Let R be a ring, p ⊂ q prime ideals. Then Rp is the localization of Rq at the prime ideal pRq . ′ Proof: Set S := R − q and T ′ := Rq − pRq . Set T := φ−1 S (T ). Then T = R − p by (11.18)(2). So S ⊂ T , and (11.26) yields the assertion. □ Exercise (11.28). — Let R be a ring, S and T multiplicative subsets. (1) Set T ′ := φS (T ) and assume S ⊂ T . Prove T −1 R = T ′−1 (S −1 R) = T −1 (S −1 R). September 3, 2012 11Nts.tex 60 11. Localization of Rings (2) Set U := {st ∈ R | s ∈ S and t ∈ T }. Prove T −1 (S −1 R) = S −1 (T −1 R) = U −1 R. (3) Let S ′ := {t′ ∈ R | t′ t ∈ S for some t ∈ R}. Prove S ′−1 R = S −1 R. Proposition (11.29). (— Let) R be a ring, S a multiplicative subset, X a variable. Then (S −1 R)[X] = S −1 R[X] . Proof: Let’s check (S −1 R)[X] and S −1 (R[X]) have the same UMP: a ring map ψ : R → R′ factors uniquely through either one if ψ(S) ⊂ (R′ )× and if an x ∈ R′ is preassigned as the image of X. First, since ψ(S) ⊂ (R′ )× , there is a unique R-algebra map S −1 R → R′ , so a unique (S −1 R)-algebra map (S −1 R)[X] → R′ sending X to x. Second, there is a unique R-algebra map R[X] → R′ sending X to x, so a unique R[X]-algebra map R[X] → R′ sending X to x, and so a unique R[X]-algebra map S −1 (R[X]) → R′ since ψ(S) ⊂ (R′ )× , as required. □ Corollary (11.30). — Let R be a ring, S a multiplicative subset, X a variable, p an ideal of R[X]. Set R′ := S −1 R, and let φ : R[X] → R′ [X] be the canonical map. ( ) Then p is prime and p ∩ S = ∅ if and only if pR′ [X] is prime and p = φ−1 pR′ [X] . Proof: The assertion results directly from (11.30) and (11.18)(2). □ Exercise (11.31). — Let R be a domain, S a multiplicative subset with 0 ∈ / S. Assume R is normal. Show that S −1 R is normal. September 3, 2012 11Nts.tex 12. Localization of Modules Formally, we localize a module just as we do a ring. However, the result is a module over the localized ring, and comes equipped with a linear map from the original module; in fact, the result is the universal module with these two properties. Further, as a functor, localization is the left adjoint of restriction of scalars. Hence, localization preserves direct limits, or equivalently, direct sums and cokernels. Therefore, by Watts’ Theorem, localization is naturally isomorphic to tensor product with the localized ring. Moreover, localization is exact; so the localized ring is flat. We end by discussing various compatibilities and examples. Proposition (12.1). — Let R be a ring, S a multiplicative subset. Then a module M has a compatible S −1 R-module structure if and only if, for all s ∈ S, the multiplication map µs : M → M is bijective; if so, then the S −1 R-structure is unique. Proof: Assume M has a compatible S −1 R-structure, and take s ∈ S. Then µs = µs/1 . So µs · µ1/s = µ(s/1)(1/s) = 1. Similarly, µ1/s · µs = 1. So µs is bijective. Conversely, assume µs is bijective for all s ∈ S. Then µR : R → EndZ (M ) sends S into the units of EndZ (M ). Hence µR factors through a unique ring map µS −1 R : S −1 R → EndZ (M ) by (11.6). Thus M has a unique compatible S −1 Rstructure by (4.5). □ (12.2) (Localization of modules). — Let R be a ring, S a multiplicative subset, M a module. Define a relation on M × S by (m, s) ∼ (n, t) if there is u ∈ S such that utm = usn. As in (11.1), this relation is an equivalence relation. Denote by S −1 M the set of equivalence classes, and by m/s the class of (m, s). Then S −1 M is an S −1 R-module with addition given by m/s + n/t := (tm + sn)/st and scalar multiplication by a/s · m/t := am/st similar to (11.1). We call S −1 M the localization of M at S. For example, let a be an ideal. Then S −1 a = aS −1 R by (11.12)(1). Similarly, −1 S (aM ) = S −1 aS −1 M = aS −1 M . Further, given an R-algebra R′ , the S −1 Rmodule S −1 R′ constructed here underlies the S −1 R-algebra S −1 R′ of (11.22). Define φS : M → S −1 M by φS (m) := m/1. Clearly, φS is R-linear. Note that µs : S −1 M → S −1 M is bijective for all s ∈ S by (12.1). If S = {f n | n ≥ 0} for some f ∈ R, then we call S −1 M the localization of M at f , and set Mf := S −1 M and φf := φS . Similarly, if S = R−p for some prime ideal p, then we call S −1 M the localization of M at p, and set Mp := S −1 M and φp := φS . Theorem (12.3) (UMP). — Let R be a ring, S a multiplicative subset, and M a module. Then S −1 M is the universal example of an S −1 R-module equipped with an R-linear map from M . Proof: The proof is like that of (11.6): given an R-linear map ψ : M → N where N is an S −1 R-module, it is easy to prove that ψ factors uniquely via the S −1 R-linear map ρ : S −1 M → N well defined by ρ(m/s) := 1/s · ψ(m). □ Exercise (12.4). — Let R be a ring, S a multiplicative subset, and M a module. Show that M = S −1 M if and only if M is an S −1 R-module. September 3, 2012 11Nts.tex 62 12. Localization of Modules Exercise (12.5). — Let R be a ring, S ⊂ T multiplicative subsets, M a module. Set T1 := φS (T ) ⊂ S −1 R. Show T −1 M = T −1 (S −1 M ) = T1−1 (S −1 M ). Exercise (12.6). — Let R be a ring, S a multiplicative subset. Show that S becomes a filtered category when equipped as follows: given s, t ∈ S, set Hom(s, t) := {x ∈ R | xs = t}. Given a module M , define a functor S → ((R-mod)) as follows: for s ∈ S, set Ms := M ; to each x ∈ Hom(s, t), associate µx : Ms → Mt . Define βs : Ms → S −1 M ∼ S −1 M . by βs (m) := m/s. Show the βs induce an isomorphism lim Ms −→ −→ Exercise (12.7). — Let R be a ring, S a multiplicative subset, M a module. Prove S −1 M = 0 if Ann(M ) ∩ S ̸= ∅. Prove the converse if M is finitely generated. (12.8) (Functoriality). — Let R be a ring, S a multiplicative subset, α : M → N an R-linear map. Then φS α carries M to the S −1 R-module S −1 N . So (12.3) yields a unique S −1 R-linear map S −1 α making the following diagram commutative: φS −1 M  −−→ S M α  −1 y yS α φS N −−→ S −1 N The construction in the proof of (12.3) yields (S −1 α)(m/s) = α(m)/s. (12.8.1) Thus, canonically, we obtain the following map, and clearly, it is R-linear: HomR (M, N ) → HomS −1 R (S −1 M, S −1 N ). −1 −1 (12.8.2) −1 Any R-linear map β : N → P yields S (βα) = (S β)(S α) owing to uniqueness or to (12.8.1). Thus S −1 (•) is a linear functor from ((R-mod)) to ((S −1 R-mod)). Theorem (12.9). — Let R be a ring, S a multiplicative subset. Then the functor S −1 (•) is the left adjoint of the functor of restriction of scalars. Proof: Let N be an S −1 R-module. Then N = S −1 N by (12.4), and the map (12.8.2) is bijective with inverse taking β : S −1 M → N to βφS : M → N . And (12.8.2) is natural in M and N by (6.3). Thus the assertion holds. □ Corollary (12.10). — Let R be a ring, S a multiplicative subset. Then the functor S −1 (•) preserves direct limits, or equivalently, direct sums and cokernels. Proof: By (12.9), the functor is a left adjoint. Hence it preserves direct limits by (6.12); equivalently, it preserves direct sums and cokernels by (6.10). □ Exercise (12.11). — Let R be a ring, S a multiplicative subset, P a projective module. Then S −1 P is a projective S −1 R-module. Corollary (12.12). — Let R be a ring, S a multiplicative subset. Then the functors S −1 (•) and S −1 R ⊗R • are canonically isomorphic. Proof: As S −1 (•) preserves direct sums and cokernels by (12.10), the assertion is an immediate consequence of Watts Theorem (8.15). Alternatively, both functors are left adjoints of the same functor by (12.9) and by (8.10). So they are canonically isomorphic by (6.4). □ September 3, 2012 11Nts.tex 12. Localization of Modules 63 Exercise (12.13). — Let R be a ring, S a multiplicative subset, M and N modules. Show S −1 (M ⊗R N ) = S −1 M ⊗R N = S −1 M ⊗S −1 R S −1 N = S −1 M ⊗R S −1 N. Definition (12.14). — Let R be a ring, S a multiplicative subset, M a module. Given a submodule N , its saturation N S is defined by N S := {m ∈ M | there is s ∈ S with sm ∈ N }. If N = N S , then we say N is saturated. Proposition (12.15). — Let R be a ring, M a module, N and P submodules. Let S be a multiplicative subset, and K an S −1 R-submodule of S −1 M . (1) Then (a) N S is a submodule of M , and (b) S −1 N is a submodule of S −1 M . −1 S −1 (2) Then (a) φ−1 (φ−1 S K = (φS K) and (b) K = S S K). −1 −1 S (3) Then φS (S N ) = N ; in particular, Ker(φS ) = 0S . (4) Then (a) (N ∩ P )S = N S ∩ P S and (b) S −1 (N ∩ P ) = S −1 N ∩ S −1 P . (5) Then (a) (N + P )S ⊃ N S + P S and (b) S −1 (N + P ) = S −1 N + S −1 P . Proof: Assertion (1)(b) holds because N ×S is a subset of M ×S and is equipped with the induced equivalence relation. Assertions (1)(a), (2) and (3) can be proved as in (11.14)(3) and (11.17)(1), (2). As to (4)(a), clearly (N ∩ P )S ⊂ N S ∩ P S . Conversely, given m ∈ N S ∩ P S , there are s, t ∈ S with sm ∈ N and tm ∈ P . Then stm ∈ N ∩ P and st ∈ S. So m ∈ (N ∩ P )S . Thus (a) holds. Alternatively, (4)(b) and (3) yield (4)(a). As to (4)(b), since N ∩ P ⊂ N, P , using (1) yields S −1 (N ∩ P ) ⊂ S −1 N ∩ S −1 P . But, given m/s = n/t ∈ S −1 N ∩ S −1 P , there is a u ∈ S with utm = usn ∈ N ∩ P . Hence utm/uts = usn/uts ∈ S −1 (N ∩ P ). Thus (b) holds. As to (5)(a), given n ∈ N S and p ∈ P S , there are s, t ∈ S with sn ∈ N and tp ∈ P . Then st ∈ S and st(n + p) ∈ N + P . Thus (5)(a) holds. As to (5)(b), note N, P ⊂ N + P . So (1)(b) yields S −1 (N + P ) ⊃ S −1 N + S −1 P . But the opposite inclusion holds as (n + p)/s = n/s + p/s. Thus (5)(b) holds. □ Theorem (12.16) (Exactness of Localization). — Let R be a ring, and S a multiplicative subset. Then the functor S −1 (•) is exact. Proof: As S −1 (•) preserves injections by (12.15)(1) and cokernels by (12.10), it is exact by (9.3). β α →M − → M ′′ , for each s ∈ S, take a Alternatively, given an exact sequence M ′ − ′ ′′ copy Ms → Ms → Ms . Using (12.6), make S into a filtered category, and make these copies into a functor from S to the category of 3-term exact sequences; its limit is the following sequence, which is exact by (7.10), as desired: S −1 α S −1 β S −1 M ′ −−−→ S −1 M −−−→ S −1 M ′′ . The latter argument can be made more direct as follows. Since βα = 0, we have (S −1 β)(S −1 α) = S −1 (βα) = 0. Hence Ker(S −1 β) ⊃ Im(S −1 α). Conversely, given m/s ∈ Ker(S −1 β), there is t ∈ S with tβ(m) = 0. So β(tm) = 0. So exactness yields m′ ∈ M ′ with α(m′ ) = tm. So (S −1 α)(m′ /ts) = m/s. Hence Ker(S −1 β) ⊂ Im(S −1 α). Thus Ker(S −1 β) = Im(S −1 α), as desired. □ Corollary (12.17). — Let R be a ring, S a multiplicative subset. Then S −1 R is flat over R. September 3, 2012 11Nts.tex 64 12. Localization of Modules Proof: The functor S −1 (•) is exact by (12.16), and is isomorphic to S −1 R⊗R • by (12.12). Thus S −1 R is flat. Alternatively, using (12.6), write S −1 R as a filtered direct limit of copies of R. But R is flat by (9.6). Thus S −1 R is flat by (9.13). □ Corollary (12.18). — Let R be a ring, S a an ideal, and / a multiplicative subset, / M a module. Then S −1 (M/aM ) = S −1 M S −1 (aM ) = S −1 M aS −1 M . Proof: The assertion results from (12.16) and (12.2). □ Corollary (12.19). — Let R be a ring, p a prime. Then Frac(R/p) = Rp /pRp . Proof: We have Frac(R/p) = (R/p)p = Rp /pRp by (11.22) and (12.18). □ Proposition (12.20). — Let R be a ring, M a module, S a multiplicative subset. (1) Let m1 , . . . , mn ∈ M . If M is finitely generated and if the mi /1 ∈ S −1 M generate over S −1 R, then there’s f ∈ S so that the mi /1 ∈ Mf generate over Rf . (2) Assume M is finitely presented and S −1 M is a free S −1 R-module of rank n. Then there is h ∈ S such that Mh is a free Rh -module of rank n. Proof: To prove (1), define α : Rn → M by α(ei ) := mi with ei the ith standard basis vector. Set C := Coker(α). Then S −1 C = Coker(S −1 α) by (12.10). Assume the mi /1 ∈ S −1 M generate over S −1 R. Then S −1 α is surjective by (4.10)(1) as S −1 (Rn ) = (S −1 R)n by (12.10). Hence S −1 C = 0. In addition, assume M is finitely generated. Then so is C. Hence, (12.7) yields f ∈ S such that Cf = 0. Hence αf is surjective. So the mi /1 generate Mf over Rf again by (4.10)(1). Thus (1) holds. For (2), let m1 /s1 , . . . , mn /sn be a free basis of S −1 M over S −1 R. Then so is m1 /1, . . . , mn /1 as the 1/si are units. Form α and C as above, and set K := Ker(α). Then (12.16) yields S −1 K = Ker(S −1 α) and S −1 C = Coker(S −1 α). But S −1 α is bijective. Hence S −1 K = 0 and S −1 C = 0. Since M is finitely generated, C is too. Hence, as above, there is f ∈ S such αf that Cf = 0. Then 0 → Kf → Rfn −−→ Mf → 0 is exact by (12.16). Take a finite presentation Rp → Rq → M → 0. By (12.16), it yields a finite presentation Rfp → Rfq → Mf → 0. Hence Kf is a finitely generated Rf -module by (5.24). Let S1 ⊂ Rf be the image of S. Then (12.5) yields S1−1 (Kf ) = S −1 K. But −1 S K = 0. Hence there is g/1 ∈ S1 such that (Kf )g/1 = 0. Set h := f g. Let’s show Kh = 0. Let x ∈ K. Then there is a such that (g a x)/1 = 0 in Kf . Hence there is b such that f b g a x = 0 in K. Take c ≥ a, b. Then hc x = 0. Thus Kh = 0. But Cf = 0 implies Ch = 0. Hence αh : Rhn → Mh is an isomorphism, as desired. □ Proposition (12.21). — Let R be a ring, S a multiplicative subset, and M and N modules. Then there is a canonical homomorphism σ : S −1 HomR (M, N ) → HomS −1 R (S −1 M, S −1 N ). Further, σ is injective if M is finitely generated; σ is an isomorphism if M is finitely presented. Proof: The assertions result from (9.15) with R′ := S −1 R, since S −1 R is flat by (12.17) and since S −1 R ⊗ P = S −1 P for every R-module P by (12.12). □ September 3, 2012 11Nts.tex 12. Localization of Modules 65 Example (12.22). — Set R := Z and S := Z − ⟨0⟩ and M := Q/Z. Then M is faithful since z ∈ S implies z · (1/2z) = 1/2 ̸= 0; thus, µR : R → HomR (M, M ) is injective. But S −1 R = Q. So (12.16) yields S −1 HomR (M, M ) ̸= 0. On the other hand, S −1 M = 0 as s · r/s = 0 for any r/s ∈ M . So the map σ(M, M ) of (12.21) is not injective. Thus (12.21)(2) can fail if M is not finitely generated. Example (12.23). — Take R := Z and S := Z − 0 and Mn := Z/⟨n⟩ for n ≥ 2. Then S −1 Mn = 0 for all n as(∏ nm ≡) 0 (mod n) for all m. On the other hand, m · (1, 1, . . . ) is (1, 1, . . . )/1 is nonzero in S −1 Mn as the kth component ) of ∏ (∏ ∏ −1 −1 nonzero in Mn for k > m if m is nonzero. (∏Thus) S ∏ Mn ̸= (S Mn ). −1 Also S (Z = Q.)So (12.12) yields Q ⊗ Mn ̸= (Q ⊗ Mn ), whereas (8.11) ⊕ ⊕ yields Q ⊗ Mn = (Q ⊗ Mn ). ⊕ Exercise (12.24). — Set R := Z and S = Z − ⟨0⟩. Set M := n≥2 Z/⟨n⟩ and N := M . Show that the map σ of (12.21) is not injective. September 3, 2012 11Nts.tex 66 13. Support 13. Support The spectrum of a ring is this topological space: its points are all the prime ideals; each closed set consists of those primes containing a given ideal. The support of a module is this subset: its points are the primes at which the localized module is nonzero. We relate the support to the closed set of the annihilator. We prove that a sequence is exact if and only if it is exact after localizing at each maximal ideal. Lastly, we prove that a module is finitely generated and projective if and only if it is locally free of finite rank. (13.1) (Spectrum of a ring). — Let R be a ring. Its set of prime ideals is denoted Spec(R), and is called the (prime) spectrum of R. Let a be an ideal. Let V(a) denote the subset of Spec(R) of those primes that contain a. We call V(a) the variety of a. Let b be another ideal. Then by the Scheinnullstellensatz, V(a) = V(b) if and √ √ only if a = b. Further, (2.2) yields V(a) ∪ V(b) = V(a ∩ b) = V(ab). A prime ideal p contains the ideals aλ in an arbitrary collection if and only if p ∑ contains their sum aλ ; hence, (∑ ) ∩ V(aλ ) = V aλ . Finally, V(R) = ∅, and V(⟨0⟩) = Spec(R). Thus the subsets V(a) of Spec(R) are the closed sets of a topology; it is called the Zariski topology. Given an element f ∈ R, we call the open set D(f ) := Spec(R) − V(⟨f ⟩) a principal open set. These sets form a basis for the topology of Spec(R); indeed, given any prime p ̸⊃ a, there is an f ∈ a − p, and so p ∈ D(f ) ⊂ Spec(R) − V(a). Further, f, g ∈ / p if and only if f g ∈ / p, for any f, g ∈ R and prime p; in other words, D(f ) ∩ D(g) = D(f g). A ring map φ : R → R′ induces a set map Spec(φ) : Spec(R′ ) → Spec(R) by (13.1.1) Spec(φ)(p′ ) := φ−1 (p′ ). Clearly, Spec(φ)−1 V(a) = V(aR′ ); hence, Spec(φ) is continuous. Thus Spec(•) is a contravariant functor from ((Rings)) to ((Top spaces)). For example, the quotient map R → R/a induces a closed embedding Spec(R/a) ֒→ Spec(R), whose image is V(a), owing to (1.8) and (2.8). Furthermore, the localization map R → Rf induces an open embedding Spec(Rf ) ֒→ Spec(R), whose image is D(f ), owing to (11.18). Exercise (13.2). — Let R be a ring, p ∈ Spec(R). Show that p is a closed point — that is, {p} is a closed set — if and only if p is a maximal ideal. September 3, 2012 11Nts.tex 13. Support 67 Exercise (13.3). — Let R be a ring, R′ a flat algebra with structure map φ. Show that R′ is faithfully flat if and only if Spec(φ) is surjective. Proposition (13.4). Then X is quasi∪ — Let R be a ring, X := Spec(R). ∪n compact: if X = λ∈Λ Uλ with Uλ open, then X = i=1 Uλi for some λi ∈ Λ. (∑ ) ∩ ∑ aλ lies Proof: Say Uλ = X − V(aλ ). Then V aλ = V(aλ ) = ∅. So in no prime ideal. Hence there are λ , . . . , λ ∈ Λ and f ∈ a for all i with 1 n λ λ i i ) ∩ (∑ ∪ ∑ ∑ aλi = V(aλi ) = ∅; so X = Uλi . □ fλi = 1. Hence aλi = R; so V Exercise (13.5). — Let R be a ring, X := Spec(R), and U an open subset. Show U is quasi-compact if and only if X − U = V (a) where a is finitely generated. Exercise (13.6). — Let B be a Boolean ring, and set X := Spec(B). Show X is a compact Hausdorff space. (Following Bourbaki, “quasi-compact” is shortened to “compact” when the space is Hausdorff.) Further, show a subset U ⊂ X is both open and closed if and only if U = D(f ) for some f ∈ B. Exercise (13.7) (Stone’s Theorem). — Show every Boolean ring B is isomorphic to the ring of continuous functions from a compact Hausdorff space X to F2 with the discrete topology. Equivalently, show B is isomorphic to the ring R of open and ∼ R is given by f 7→ D(f ). closed subsets of X; in fact, X := Spec(B), and B −→ Definition (13.8). — Let R be a ring, M a module. Its support is the set Supp(M ) := { p ∈ Spec(R) | Mp ̸= 0 }. Proposition (13.9). — Let R be a ring, M a module. (1) Let 0 → L → M → N → 0 be ∑exact. Then Supp(L) ∪ ∪ Supp(N ) = Supp(M ). (2) Let Mλ be submodules with Mλ = M . Then Supp(Mλ ) = Supp(M ). (3) Then Supp(M ) ⊂ V(Ann(M )), with equality if M is finitely generated. Proof: Consider (1). For every prime p, the sequence 0 → Lp → Mp → Np → 0 is exact by (12.16). Hence Mp ̸= 0 if and only if Lp ̸= 0 or Np ̸= 0. Thus (1) holds. ∪ In (2), Mλ ⊂ M . So ∪ (1) yields Supp(Mλ ) ⊂ Supp(M ). To prove the opposite inclusion, take p ⊕ ∈ / Supp(Mλ ). Then (Mλ )p ⊕ = 0 for all λ. By hypothesis, the natural map Mλ → M is surjective. So (Mλ )p → Mp is surjective by (12.10). Hence M = 0. Alternatively, given m/s ∈ Mp , express m as a finite sum p ∑ m= m with m ∈ M . For each such λ, there is tλ ∈ R − p with tλ mλ = 0. λ λ λ ∏ Set t := tλ . Then tm = 0 and t ∈ / p. So m/s = 0 in Mp . Hence again, Mp = 0. Thus p ∈ / Supp(M ), and so (2) holds. Consider (3). Let p be a prime. By (12.7), Mp = 0 if Ann(M ) ∩ (R − p) ̸= ∅, and the converse holds if M is finitely generated. But Ann(M ) ∩ (R − p) ̸= ∅ if and only if Ann(M ) ̸⊂ p. The assertion follows directly. □ Definition (13.10). — Let R be a ring, x ∈ R. We say x is nilpotent on a √ module M if there is n ≥ 1 with xn m = 0 for all m ∈ M , that is, if x √ ∈ Ann(M ). We denote the set of nilpotents on M by nil(M ); that is, nil(M ) := Ann(M ). Proposition (13.11). — Let R be a ring, M a finitely generated module. Then ∩ nil(M ) = p∈Supp(M ) p. September 3, 2012 11Nts.tex 68 13. Support ∩ Proof: First, nil(M ) = p⊃Ann(M ) p by the Scheinnullstellensatz (3.22). But p ⊃ Ann(M ) if and only if p ∈ Supp(M ) by (13.9)(3). □ Proposition (13.12). — Let R be a ring, M and N modules. Then Supp(M ⊗R N ) ⊂ Supp(M ) ∩ Supp(N ), (13.12.1) with equality if M and N are finitely generated. Proof: First, (M ⊗R N )p = Mp ⊗Rp Np by (12.13); whence, (13.12.1) holds. The opposite inclusion follows from (10.14) if M and N are finitely generated. □ Corollary (13.13). — Let R be a ring, a an ideal, M a module. Then Supp(M/aM ) ⊂ Supp(M ) ∩ V(a). with equality if M is finitely generated. Proof: First, (8.13)(1) yields M/aM = M ⊗ R/a. But Ann(R/a) = a; hence (13.9)(3) yields Supp(R/a) = V(a). Thus (13.12) yields the assertion. □ Exercise (13.14). — Let R be a ring, M a module, p ∈ Supp(M ). Prove V(p) ⊂ Supp(M ). Exercise (13.15). — Let Z be the integers, Q the rational numbers, and set M := Q/Z. Find Supp(M ), and show that it is not Zariski closed. Proposition (13.16). — Let R be a ring, M a module. These conditions are equivalent: (1) M = 0; (2) Supp(M ) = ∅; (3) Mm = 0 for every maximal ideal m. Proof: Trivially, if (1) holds, then S −1 M = 0 for any multiplicative subset S. In particular, (2) holds. Trivially, (2) implies (3). Finally, assume M ̸= 0, and take a nonzero m ∈ M , and set a := Ann(m). Then 1∈ / a, so a lies in some maximal ideal m. Then, for all f ∈ R − m, we have f m ̸= 0. Hence m/1 ̸= 0 in Mm . Thus (3) implies (1). □ Exercise (13.17). — Let R be a ring, P a module, and M, N submodules. Show M = N if Mm = Nm for every maximal ideal m. First assume M ⊂ N . Exercise (13.18). — Prove these three conditions on a ring R are equivalent: (1) R is reduced. (2) S −1 R is reduced for all multiplicatively closed sets S. (3) Rm is reduced for all maximal ideals m. Exercise (13.19). — Let R be a ring, Σ the set of minimal primes. Prove this: (1) If Rp ∏ is a domain for any prime p, then the p ∈ Σ are pairwise comaximal. n (2) R = i=1 Ri where Ri is a domain if and only if Rp is a domain for any prime p and Σ is finite. If so, then Ri = R/pi with {p1 , . . . , pn } = Σ. α β Proposition (13.20). — A sequence of modules L − →M − → N is exact if and βm αm Mm −−→ Nm is exact at each maximal ideal m. only if its localization Lm −−→ Proof: If the sequence is exact, then so is its localization by (12.16). ( ) Consider the converse. First Im(βm αm ) = 0. But Im(βm αm ) = Im(βα) m by (12.16) and (9.3). Hence Im(βα) = 0 by (13.16). So βα = 0. Thus Im(α) ⊂ Ker(β). / / Set H := Ker(β) Im(α). Then Hm = Ker(βm ) Im(αm ) by (12.16) and (9.3). So Hm = 0 owing to the hypothesis. Hence H = 0 by (13.16), as required. □ September 3, 2012 11Nts.tex 13. Support 69 Exercise (13.21). — Let R be a ring, M a module. Prove elements mλ ∈ M generate M if and only if, at every maximal ideal m, their images mλ generate Mm . Proposition (13.22). — Let A be a semilocal ring, m1 , . . . , mn its maximal ideals, M, N finitely presented modules. Assume Mmi ∼ = Nmi for each i. Then M ∼ = N. ∼ N . Then (12.21) Proof: For each i, take an isomorphism ψi : Mmi −→ mi yields s ∈ A − m and φ : M → N with (φ ) = s ψ . However, (2.2) implies i i i i m i i i ∩ ∩ ∑ m ⊂ ̸ m ; so there’s x ∈ m with x ∈ / m . Set γ := x φ . i i i i j̸=i j j̸=i j i i i∑ ∼ For each i, set αi := xi si ψi . Then αi : Mmi −→ Nmi . Set βi := j̸=i xj sj ψj . Then βi (Mmi ) ⊂ mi Nmi as xj ∈ mi for j ̸= i. Further, γmi = αi + βi . So γmi is an ∼ N. isomorphism by (10.12). Hence (13.20) implies γ : M −→ □ Proposition (13.23). — Let R be a ring, S a multiplicative subset, M a module. Then the following conditions are equivalent: (1) M is flat over R. (2) S −1 M is flat over S −1 R and over R. (3) At every maximal ideal m, the localization Mm is flat over Rm . Proof: Assume (1) holds. Let α : N ′ → N be an injection of S −1 R-modules. Then M ⊗R α : M ⊗R N ′ → M ⊗R N is injective. Now, (12.4) yields α = S −1 α. So (12.13) yields M ⊗R α = S −1 M ⊗S −1 R α. Hence S −1 M ⊗S −1 R α is injective; that is, S −1 M is flat over S −1 R. Therefore, S −1 M is flat over R by (12.17) and (9.7). Thus (2) holds. Trivially, (2) implies (3). Assume (3) holds. Let α : N ′ → N be an injection of R-modules. Then αm is injective by (13.20). So Mm ⊗Rm αm is injective. Now, that map is equal to (M ⊗ α)m by (12.13), hence is injective. Therefore, M ⊗ α is injective by (13.20); that is, (1) holds. □ Exercise (13.24). — Let R be a ring, R′ a flat algebra, p′ a prine in R′ , and p its contraction in R. Prove that Rp′ ′ is a faithfully flat Rp -algebra. Definition (13.25). — Let R be a ring, M a module. We say M is locally finitely generated if each p ∈ Spec(R) has a neighborhood on which M becomes finitely generated; more precisely, there exists f ∈ R − p such that Mf is finitely generated over Rf . Similarly, we define the properties locally finitely presented, locally free of finite rank, and locally free of rank n. Proposition (13.26). — Let R be a ring, M a module. (1) If M is locally finitely generated, then it is finitely generated. (2) If M is locally finitely presented, then it is finitely presented. ∪ Proof: By (13.4), there are f1 , . . . , fn ∈ R with D(fi ) = Spec(R) and finitely n many mi,j ∈ M such that, for some ni,j ≥ 0, the mi,j /fi i,j generate Mfi . Clearly, for each i, the mi,j /1 also generate Mfi . Given any maximal ideal m, there is i such that fi ∈ / m. Let S1 be the image of R − m in Rfi . Then (12.5) yields Mm = S1−1 (Mfi ). Hence the mi,j /1 generate Mm . Thus (13.21) yields (1). Assume M is locally finitely presented. Then M is finitely generated by (1). So there is a surjection Rk → → M . Let K be its kernel. Then K is locally finitely generated owing to (5.24). Hence K too is finitely generated by (1). So there is a surjection Rℓ → → K. It yields the desired finite presentation Rℓ → Rk → M → 0. Thus (2) holds. □ September 3, 2012 11Nts.tex 70 13. Support Theorem (13.27). — These conditions on an R-module P are equivalent: (1) P is finitely generated and projective. (2) P is finitely presented and flat. (3) P is finitely presented, and Pm is free over Rm at each maximal ideal m. (4) P is locally free of finite rank. (5) P is finitely generated, and for each p ∈ Spec(R), there are f and n such that p ∈ D(f ) and Pq is free of rank n over Rq at each q ∈ D(f ). Proof: Condition (1) implies (2) by (10.15). Let m be a maximal ideal. Then Rm is local by (11.20). If P is finitely presented, then Pm is finitely presented, because localization preserves direct sums and cokernels by (12.10). Assume (2). Then Pm is flat by (13.23), so free by (10.15). Thus (3) holds. Assume (3). Fix a surjective map α : M → N . Then αm : Mm → Nm is surjective. So Hom(Pm , αm ) : Hom(Pm , Mm ) → Hom(Pm , Nm ) is surjective by (5.22) and (5.21). But Hom(Pm , αm ) = Hom(P, α)m by (12.21) as P is finitely presented. Further, m is arbitrary. Hence Hom(P, α) is surjective by (13.20). Therefore, P is projective by (5.22). Thus (1) holds. Again assume (3). Given any prime p, take a maximal ideal m containing it. By hypothesis, Pm is free; its rank is finite as Pm is finitely generated. By (12.20)(2), there is f ∈ R − m such that Pf is free of finite rank over Rf . Thus (4) holds. Assume (4). Then P is locally finitely presented. So P is finitely presented by (13.26)(2). Further, given p ∈ Spec(R), there are f ∈ R − p and n such that Mf is locally free of rank n over Rf . Given q ∈ D(f ), let S1 be the image of R − q in Rf . Then (12.5) yields Mq = S1−1 (Mf ). Hence Mq is locally free of rank n over Rq . Thus (5) holds. Further, (3) results from taking p := m and q := m. Finally, assume (5), and let’s prove (4). Given p ∈ Spec(R), let f and n be provided by (5). Take a free basis p1 /f k1 , . . . , pn /f kn of Pp over Rp . The pi define a map α : Rn → P , and αp : Rpn → Pp is bijective, in particular, surjective. As P is finitely generated, (12.20)(1) provides g ∈ R − p such that αg : Rgn → Pg is surjective. It follows that αq : Rqn → Pq is surjective for every q ∈ D(g). If also q ∈ D(f ), then by hypothesis Pq ≃ Rqn . So αq is bijective by (10.4). Set h := f g. Clearly, D(f ) ∩ D(g) = D(h). By (13.1), D(h) = Spec(Rh ). Clearly, αq = (αh )(qRh ) for all q ∈ D(h). Hence αh : Rhn → Ph is bijective owing to (13.20) with Rh for R. Thus (4) holds. □ Exercise (13.28). — Given n, prove an R-module P is locally free of rank n if n and only if P is finitely generated and Pm ≃ Rm holds at each maximal ideal m. Exercise (13.29). — Let A be a semilocal ring, P a locally free module of rank n. Show that P is free of rank n. September 3, 2012 11Nts.tex 14. Krull–Cohen–Seidenberg Theory Krull–Cohen–Seidenberg Theory relates the prime ideals in a ring to those in an integral extension. We prove each prime has at least one prime lying over it — that is, contracting to it. The overprime can be taken to contain any ideal that contracts to an ideal contained in the given prime; this stronger statement is known as the Going-up Theorem. Further, one prime is maximal if and only if the other is, and two overprimes cannot be nested. On the other hand, the Going-down Theorem asserts that, given nested primes in the subring and a prime lying over the larger, there is a subprime lying over the smaller, either if the subring is normal and the overring is a domain or if the extension is flat even if it’s not integral. Lemma (14.1). — Let R ⊂ R′ be an integral extension of domains. Then R′ is a field if and only if R is. Proof: First, suppose R′ is a field. Let x ∈ R be nonzero. Then 1/x ∈ R′ , so satisfies an equation of integral dependence: (1/x)n + a1 (1/x)n−1 + · · · + an = 0 with n ≥ 1 and ai ∈ R. Multiplying the equation by xn−1 , we obtain 1/x = −(a1 + an−2 x + · · · + an xn−1 ) ∈ R. Conversely, suppose R is a field. Let y ∈ R′ be nonzero. Then y satisfies an equation of integral dependence y n + a1 y n−1 + · · · + an−1 y + an = 0 with n ≥ 1 and ai ∈ R. Rewriting the equation, we obtain y(y n−1 + · · · + an−1 ) = −an . Take n minimal. Then an ̸= 0 as R′ is a domain. So dividing by −an y, we obtain 1/y = (−1/an )(y n−1 + · · · + an−1 ) ∈ R′ . □ Definition (14.2). — Let R be a ring, R′ an R-algebra, p a prime of R, and p′ a prime of R′ . We say p′ lies over p if p′ contracts to p. Theorem (14.3). — Let R ⊂ R′ be an integral extension of rings, and p a prime of R. Let p′ ⊂ q′ be nested primes of R′ , and a′ an arbitrary ideal of R′ . (1) (Maximality) Suppose p′ lies over p. Then p′ is maximal if and only if p is. (2) (Incomparability) Suppose both p′ and q′ lie over p. Then p′ = q′ . (3) (Lying over) Then there is a prime r′ of R′ lying over p. (4) (Going up) Suppose a′ ∩ R ⊂ p. Then in (3) we can take r′ to contain a′ . Proof: Assertion (1) follows from (14.1) applied to the extension R/p ⊂ R′ /p′ , which is integral as R ⊂ R′ is, since, if y ∈ R′ satisfies y n + a1 y n−1 + · · · + an = 0, then reduction modulo p′ yields an equation of integral dependence over R/p. To prove (2), localize at R − p, and form this commutative diagram: ′ R → Rxp′ x −     R −→ Rp September 3, 2012 11Nts.tex 72 14. Krull–Cohen–Seidenberg Theory Here Rp → Rp′ is injective by (12.15)(1), and the extension is integral by (11.23). Here p′ Rp′ and q′ Rp′ are nested primes of Rp′ by (11.18)(2). By the same token, both lie over pRp , because both their contractions in Rp contract to p in R. Thus we may replace R by Rp and R′ by Rp′ , and so assume R is local with p as maximal ideal by (11.20). Then p′ is maximal by (1); whence, p′ = q′ . To prove (3), again we may replace R by Rp and R′ by Rp′ : if r′′ is a prime ideal of Rp′ lying over pRp , then the contraction r′ of r′′ in R′ lies over p. So we may assume R is local with p as unique maximal ideal. Now, R′ has a maximal ideal r′ by 2.28; further, r′ contracts to a maximal ideal r of R by (1). Thus r = p. Finally, (4) follows from (3) applied to the extension R/(a′ ∩ R) ⊂ R′ /a′ . □ Exercise (14.4). — Let R ⊂ R′ be an integral extension of rings, and p a prime of R. Suppose R′ has just one prime p′ over p. Show (a) that p′ Rp′ is the only maximal ideal of Rp′ , (b) that Rp′ ′ = Rp′ , and (c) that Rp′ ′ is integral over Rp . Exercise (14.5). — Let R ⊂ R′ be an integral extension of domains, and p a prime of R. Suppose R′ has at least two distinct primes p′ and q′ lying over p. Show that Rp′ ′ is not integral over Rp . Show that, in fact, if y lies in q′ , but not in p′ , then 1/y ∈ Rp′ ′ is not integral over Rp . Exercise (14.6). — Let k be a field, and X an indeterminate. Set R′ := k[X], and Y := X 2 , and R := k[Y ]. Set p := (Y − 1)R and p′ := (X − 1)R′ . Is Rp′ ′ integral over Rp ? Explain. Lemma (14.7). — Let R ⊂ R′ be a ring extension, X a variable, f ∈ R[X] a monic polynomial. Suppose f = gh with g, h ∈ R′ [X] monic. Then the coefficients of g and h are integral over R. Proof: Set R1 := R′ [X]/⟨g⟩. Let x1 be the residue of X. Then 1, x1 , x21 , . . . form a free basis of R1 over R′ by (10.20) as g is monic; hence, R′ ⊂ R1 . Now, g(x1 ) = 0; so g factors as (X − x1 )g1 with g1 ∈ R1 [X] monic of degree ∏ 1 less than g. Repeat this process, extending R . Continuing, obtain g(X) = (X − xi ) 1 ∏ and h(X) = (X − yj ) with all xi and yj in an extension of R′ . The xi and yj are integral over R as they are roots of f . But the coefficients of g and h are polynomials in the xi and yj ; so they too are integral over R. □ Proposition (14.8). — Let R be a normal domain, K := Frac(R), and L/K a field extension. Let y ∈ L be integral over R, and p ∈ K[X] its monic minimal polynomial. Then p ∈ R[X], and so p(y) = 0 is an equation of integral dependence. Proof: Since y is integral, there is a monic polynomial f ∈ R[X] with f (y) = 0. Write f = pq with q ∈ K[X]. Then by (14.7) the coefficients of p are integral over R, so in R since R is normal. □ Theorem (14.9) (Going down for integral extensions). — Let R ⊂ R′ be an integral extension of domains, p ⫋ q nested primes of R, and q′ a prime of R′ lying over q. If R is normal, then there is a prime p′ lying over p and contained in q′ . Proof: First, let us show pRq′ ′ ∩ R = p. Take y ∈ pRq′ ′ ∩ R. Say y = x/s with ∑m x ∈ pR′ and s ∈ R′ − q′ . Say x = i=1 yi xi with yi ∈ p and xi ∈ R′ , and set R′′ := R[x1 , . . . , xm ]. Then R′′ is a finite R-module by (10.23) and xR′′ ⊂ pR′′ . Let f (X) = X n + a1 X n−1 + · · · + an be the characteristic polynomial of µx : R′′ → September 3, 2012 11Nts.tex 14. Krull–Cohen–Seidenberg Theory 73 R′′ . Then ai ∈ pi ⊂ p by (10.1), and f (x) = 0 by the Determinant Trick (10.2). Set K := Frac(R). Suppose f = gh with g, h ∈ K[X] monic. By (14.7) the coefficients of g, h lie in R as R is normal. Further, f ≡ X n (mod p). So g ≡ X r (mod p) and h ≡ X n−r (mod p) for some r by unique factorization in Frac(R/p)[X]. Hence g and h have all nonleading coefficients in p. Replace f by a monic factor of minimal degree. Then f is the minimal polynomial of x over K. Recall s = x/y. So s satisfies the equation sn + b1 sn−1 + · · · + bn = 0 with bi := ai /y i ∈ K. This equation is of minimal degree since y ∈ R ⊂ K and deg(f ) is minimal for x. But s is integral over R. So all bi are in R by (14.8). Assume y ∈ / p. Then bi ∈ p since ai = bi y i ∈ p. So sn ∈ pR′ ⊂ qR′ ⊂ q′ . So s ∈ q′ , a contradiction. Hence y ∈ p. Thus pRq′ ′ ∩R ⊂ p. But the opposite inclusion holds trivially. Thus pRq′ ′ ∩ R = p. Hence, there is a prime p′′ of Rq′ ′ with p′′ ∩ R = p by (3.10). Set p′ := p′′ ∩ R′ . Then p′ ∩ R = p, and p′ ⊂ q′ by (11.18)(2), as desired. □ Lemma (14.10). — Always, a minimal prime consists entirely of zerodivisors. Proof: Let R be the ring, p the minimal prime. Then Rp has only one prime pRp by (11.18)(2). So by the Scheinnullstellensatz, pRp consists entirely of nilpotents. Hence, given x ∈ p, there is s ∈ R − p with sxn = 0 for some n ≥ 1. Take n minimal. Then sxn−1 ̸= 0, but (sxn−1 )x = 0. Thus x is a zerodivisor. □ Theorem (14.11) (Going down for Flat Algebras). — Let R be a ring, R′ a flat algebra. Let p ⫋ q be nested primes of R, and q′ a prime of R′ lying over q. Then there is a prime p′ of R′ that lies over p and is contained in q′ . Proof: The canonical map Rq → Rq′ ′ is faithfully flat by (13.24). Therefore, Spec(Rq′ ) → Spec(Rq ) is surjective by (13.3). Thus (11.18) yields the desired p′ . Alternatively, R′ ⊗R (R/p) is flat over R/p by (9.8). Also, R′ /pR′ = R′ ⊗R R/p by (8.13)(1). Hence, owing to (1.8), we may replace R by R/p and R′ by R′ /pR′ , and thus assume R is a domain and p = 0. By (3.11), q′ contains a minimal prime p′ of R′ . Let’s show that p′ lies over ⟨0⟩. Let x ∈ R be nonzero. Then the multiplication map µx : R → R is injective. Since R′ is flat, µx : R′ → R′ is also injective. Hence, (14.10) implies that x does not belong to the contraction of p′ , as desired. □ Exercise (14.12). ∪ — Let R be a reduced ring, Σ the set of minimal primes. Prove that z.div(R) = p∈Σ p and that Rp = Frac(R/p) for any p ∈ Σ. Exercise (14.13). — Let R be a ring, Σ the set of minimal primes, and K the total quotient ring. Assume Σ is finite. Prove these three conditions are equivalent: (1) R is reduced. ∪ (2) z.div(R) = p∈Σ p, and Rp = Frac(R/p) for each p ∈ Σ. ∏ (3) K/pK = Frac(R/p) for each p ∈ Σ, and K = p∈Σ K/pK. Exercise (14.14). — Let A be a reduced local ring with residue field k and finite set Σ of minimal primes. For each p ∈ Σ, set K(p) := Frac(A/p). Let P be a finitely generated module. Show that P is free of rank r if and only if dimk (P ⊗A k) = r and dimK(p) (P ⊗A K(p)) = r for each p ∈ Σ. September 3, 2012 11Nts.tex 74 14. Krull–Cohen–Seidenberg Theory Exercise (14.15). — Let A be a reduced local ring with residue field k and a finite set of minimal primes. Let P be a finitely generated module, B an A-algebra with Spec(B) → Spec(A) surjective. Show that P is a free A-module of rank r if and only if P ⊗ B is a free B-module of rank r. (14.16) (Arbitrary normal rings). — An arbitrary ring R is said to be normal if Rp is a normal domain for every prime p. If R is a domain, then this definition recovers that in (10.24), owing to (11.31). Exercise (14.17). — Let R be a ring, p1 . . . , pr all its minimal primes, and K the total quotient ring. Prove that these three conditions are equivalent: (1) R is normal. (2) R is reduced and integrally closed in K. (3) R is a finite product of normal domains Ri . Assume the conditions hold. Prove the Ri are equal to the R/pj in some order. September 3, 2012 11Nts.tex 15. Noether Normalization The Noether Normalization Lemma describes the basic structure of a finitely generated algebra over a field; namely, given a chain of ideals, there is a polynomial subring over which the algebra is module finite, and the ideals contract to ideals generated by initial segments of variables. After proving this lemma, we derive several versions of the Nullstellensatz. The most famous is Hilbert’s; namely, the radical of any ideal is the intersection of all the maximal ideals containing it. Then we study the (Krull) dimension: the maximal length of any chain of primes. We prove our algebra is catenary; that is, if two chains have the same ends and maximal lengths, then the lengths are the same. Further, if the algebra is a domain, then its dimension is equal to the transcendence degree of its fraction field. In an appendix, we give a simple direct proof of the Hilbert Nullstellensatz. At the same time, we prove it in significantly greater generality: for Jacobson rings. Lemma (15.1) (Noether Normalization). — Let k be a field, R := k[x1 , . . . , xn ] a finitely generated k-algebra, and a1 ⊂ · · · ⊂ ar a chain of proper ideals of R. Then there are algebraically independent elements t1 , . . . , tν ∈ R such that (1) R is module finite over P := k[t1 , . . . tν ] and (2) for i = 1, · · · , r, there is an hi such that ai ∩ P = ⟨t1 , . . . , thi ⟩. If k is infinite, then we may choose the ti to be k-linear combinations of the xi . Proof: Let R′ := k[X1 , . . . , Xn ] be the polynomial ring, and φ : R′ → R the k-algebra map with φXi := xi . Set a′0 := Ker φ and a′i := φ−1 ai for i = 1, · · · , r. It suffices to prove the lemma for R′ and a′0 ⊂ · · · ⊂ a′r : if t′i ∈ R′ and h′i work here, then ti := φt′i+h′ and hi := h′i − h′0 work for R and the ai , because the ti 0 are algebraically independent by (1.9), and clearly (1) and (2) hold. Thus we may assume the xi are algebraically independent. The proof proceeds by induction on r (and shows ν := n works now). First, assume r = 1 and a1 = t1 R for some nonzero t1 . Then t1 ∈ / k because a1 is proper. Suppose we have found t2 , . . . , tn ∈ R so that x1 is integral over P := k[t1 , t2 , . . . , tn ] and so that P [x1 ] = R. Then (10.23) yields (1). Further, by the theory of transcendence bases [1, (8.3), p. 526], [6, Thm. 1.1, p. 356], the elements t1 , . . . , tn are algebraically independent. Now, take x ∈ a1 ∩ P . Then x = t1 x′ where x′ ∈ R ∩ Frac(P ). Further, R ∩ Frac(P ) = P because P is normal by (10.29) as P is a polynomial algebra. Hence a1 ∩ P = t1 P . Thus (2) holds too. To find t2 , . . . , tn , we are going to choose ℓi and set ti := xi − xℓ1i . Then clearly ∑ P [x1 ] = R. Now, say t1 = a(j) xj11 · · · xjnn with (j) := (j1 , . . . , jn ) and a(j) ∈ k. Recall t1 ∈ / k, and note that x1 satisfies this equation: ∑ a(j) xj11 (t2 + xℓ12 )j2 · · · (tn + xℓ1n )jn = t1 . Set e(j) := j1 + ℓ2 j2 + · · · + ℓn jn . Take ℓ > max{ji } and ℓi := ℓi . Then the e(j) are distinct. Let e(j ′ ) be largest among the e(j) with a(j) ̸= 0. Then e(j ′ ) > 0, and the above equation may be rewritten as follows: ∑ e(j ′ ) + e<e(j ′ ) pe xe1 = 0 a(j ′ ) x1 September 3, 2012 11Nts.tex 76 15. Noether Normalization where pe ∈ P . Thus x1 is integral over P , as desired. Suppose k is infinite. We are going to reorder the xi , choose ai ∈ k, and set ti := xi − ai x1 . Then clearly P [x1 ] = R. Now, say t1 = Hd + · · · + H0 where Hi is homogeneous of degree i in x1 , . . . , xn and where Hd ̸= 0. Then d > 0 as t1 ∈ / k. Since k is infinite, we may reorder the xi and take ai ∈ k with Hd (1, a2 , . . . , an ) ̸= 0. Then Hd (1, a2 , . . . , an ) is the coefficient of xd1 in Hd (x1 , t2 + a2 x1 , . . . , tn + an x1 ). So after we collect like powers of x1 , the equation Hd (x1 , t2 + a2 x1 , . . . , tn + an x1 ) + · · · + H0 (x1 , t2 + a2 x1 , . . . , tn + an x1 ) + t1 = 0 becomes an equation of integral dependence for x1 over P , as desired. Second, assume r = 1 and a1 is arbitrary. We may assume a1 ̸= 0. The proof proceeds by induction on n. The case n = 1 follows from the first case (but is simpler) because k[x1 ] is a PID. Let t1 ∈ a1 be nonzero. By the first case, there exist elements u2 , . . . , un such that t1 , u2 , . . . un are algebraically independent and satisfy (1) and (2) with respect to R and t1 R. By induction, there are t2 , . . . , tn satisfying (1) and (2) with respect to k[u2 , . . . , un ] and a1 ∩ k[u2 , . . . , un ]. Set P := k[t1 , . . . , tn ]. Since R is module finite over k[t1 , u2 , . . . , un ] and the latter is module finite over P , the former is module finite over P by (10.22). Thus (1) holds, and so t1 , . . . , tn are algebraically independent. Further, by assumption, a1 ∩ k[t2 , . . . , tn ] = ⟨t2 , . . . , th ⟩ for some h. But t1 ∈ a1 . So a1 ∩ P ⊃ ⟨t1 , . . . , th ⟩. ∑d Conversely, given x ∈ a1 ∩ P , say x = i=0 fi ti1 with fi ∈ k[t2 , . . . , tn ]. Since t1 ∈ a1 , we have f0 ∈ a1 ∩ k[t2 , . . . , tn ]; so f0 ∈ ⟨t2 , . . . , th ⟩. Hence x ∈ ⟨t1 , . . . , th ⟩. Thus a1 ∩ P = ⟨t1 , . . . , th ⟩. Thus (2) holds for r = 1. Finally, assume the lemma holds for r − 1. Let u1 , . . . , un ∈ R be algebraically independent elements satisfying (1) and (2) for the sequence a1 ⊂ · · · ⊂ ar−1 , and set h := hr−1 . By the second case, there exist elements th+1 , . . . , tn satisfying (1) and (2) for k[uh+1 , . . . , un ] and ar ∩ k[uh+1 , . . . , un ]. Then, for some hr , ar ∩ k[th+1 , . . . , tn ] = ⟨th+1 , . . . , thr ⟩. Set ti := ui for 1 ≤ i ≤ h. Set P := k[t1 , . . . , tn ]. Then, by assumption, R is module finite over k[u1 , . . . , un ], and k[u1 , . . . , un ] is module finite over P ; hence, R is module finite over P by (10.22). Thus (1) holds, and t1 , . . . , tn are algebraically independent over k. Fix∑i with 1 ≤ i ≤ r. Set m := hi . Then t1 , . . . , tm ∈ ai . Given x ∈ ai ∩ P , say x= f(v) tv11 · · · tvmm with (v) = (v1 , . . . , vm ) and f(v) ∈ k[tm+1 , . . . , tn ]. Then f(0) lies in ai ∩ k[tm+1 , . . . , tn ]. We are going to see the latter intersection is equal to ⟨0⟩. It is so if i ≤ r − 1 because it lies in ai ∩ k[um+1 , . . . , un ], which is equal to ⟨0⟩. Further, if i = r, then, by assumption, ai ∩ k[tm+1 , . . . , tn ] = ⟨tm+1 , . . . , tm ⟩ = 0. Thus f(0) = 0. Hence x ∈ ⟨t1 , . . . , thi ⟩. Thus ai ∩ P ⊂ ⟨t1 , . . . , thi ⟩. So the two are equal. Thus (2) holds, and the proof is complete. □ Exercise (15.2). — Let k := Fq be the finite field with q elements, and k[X, Y ] / the polynomial ring. Set f := X q Y − XY q and R := k[X, Y ] ⟨f ⟩. Let x, y ∈ R be the residues of X, Y . For every a ∈ k, show that R is not module finite over P := k[y −ax]. (Thus, in (15.1), no k-linear combination works.) First, take a = 0. September 3, 2012 11Nts.tex 15. Noether Normalization 77 Exercise (15.3). — Let k be a field, and X, Y, Z variables. Set / R := k[X, Y, Z] ⟨X 2 − Y 3 − 1, XZ − 1⟩, and let x, y, z ∈ R be the residues of X, Y, Z. Fix a, b ∈ k, and set t := x + ay + bz and P := k[t]. Show that x and y are integral over P for any a, b and that z is integral over P if and only if b ̸= 0. Theorem (15.4) (Weak Nullstellensatz). — Let k be a field, and R a finitely generated k-algebra. Suppose R is a field. Then R is a finite extension field of k. Proof: By the Noether Normalization Lemma (15.1), R is module finite over a polynomial subring P := k[t1 , . . . , tν ]. Then P ⊂ R is an integral extension by (10.18). Since R is a field, so is P by (14.1). Hence ν = 0. So P = k. Thus R is module finite over k, as asserted. □ Corollary (15.5). — Let k be a field, R := k[x1 , . . . , xn ] a finitely generated k-algebra, and m a maximal ideal of R. Assume k is algebraically closed. Then there are a1 , . . . , an ∈ k such that m = ⟨x1 − a1 , . . . , xn − an ⟩. Proof: Set K := R/m. Then K is a finite extension field of k by the Weak Nullstellensatz (15.4). But k is algebraically closed. Hence k = K. Let ai ∈ k be the residue of xi , and set n := ⟨x1 − a1 , . . . , xn − an ⟩. Then n ⊂ m. Let R′ := k[X1 , . . . , Xn ] be the polynomial ring, and φ : R′ → R the k-algebra map with φXi := xi . Set n′ := ⟨X1 − a1 , . . . , Xn − an ⟩. Then φ(n′ ) = n. But n′ is maximal by (2.19). So n is maximal. Hence n = m, as desired. □ Theorem (15.6) (Hilbert Nullstellensatz). — Let k be a field, and R a finitely generated k-algebra. Let a be a proper ideal of R. Then ∩ √ a = m⊃a m where m runs through all maximal ideals containing a. √ ∩ Proof: We may assume a = 0 by replacing R by R/a. Clearly 0 ⊂ m. √ Conversely, take f ∈ / 0. Then Rf ̸= 0 by (11.2). So Rf has a maximal ideal n by (2.28). Let m be its contraction in R. Now, R is a finitely generated k-algebra by hypothesis; hence, Rf is one too owing to (11.11). Therefore, by the weak Nullstellensatz, Rf /n is a finite extension field of k. Set K := R/m. By construction, K is a k-subalgebra of Rf /n. Therefore, K is a finite-dimensional k-vector space. So k ⊂ K is an integral extension by (10.18). Since k is a field, so is K by (14.1). Thus m √ is maximal. But f /1 is a unit in Rf ; ∩ ∩ □ so f /1 ∈ / n. Hence f ∈ / m. So f ∈ / m. Thus 0 = m. Exercise (15.7). — Let k be a field, K an algebraically closed extension field. (So K contains a copy of every finite extension field.) Let P := k[X1 , . . . , Xn ] be the polynomial ring, and f, f1 , . . . , fr ∈ P . Assume f vanishes at every zero in K n of f1 , . . . , fr ; in other words, if (a) := (a1 , . . . , an ) ∈ K n and f1 (a) = 0, . . . , fr (a) = 0, then f (a) = 0 too. Prove that there are polynomials g1 , . . . , gr ∈ P and an integer N such that f N = g1 f1 + · · · + gr fr . Lemma (15.8). — Let k be a field, R a finitely generated k-algebra. Assume R is a domain. Let p0 ⫋ · · · ⫋ pr be a chain of primes. Set K := Frac(R) and d := tr. degk K. Then r ≤ d, with equality if and only if the chain cannot be lengthened. September 3, 2012 11Nts.tex 78 15. Noether Normalization Proof: By the Noether Normalization Lemma (15.1), R is module finite over a polynomial subring P := k[t1 , . . . , tm ] such that pi ∩P = ⟨t1 , . . . , thi ⟩ for suitable hi . Set M := Frac(P ). Then m = tr. degk M . But P ⊂ R is an integral extension by (10.18). So M ⊂ K is algebraic. Hence m = d. Now, Incomparability (14.3)(2) yields hi < hi+1 for all i. Hence r ≤ hr . But hr ≤ m and m = d. Thus r ≤ d. If r = d, then r is maximal, as it was just proved that no chain can be longer. Conversely, assume r is maximal. Then p0 = 0 since R is a domain. So h0 = 0. Further, pr is maximal since pr is contained in some maximal ideal and it is prime. So pr ∩ P is maximal by Maximality (14.3)(1). Hence hr = m. Suppose there is an i such that hi + 1 < hi+1 . Then (pi ∩ P ) ⫋ ⟨t1 , . . . , thi +1 ⟩ ⫋ (pi+1 ∩ P ). Now, P/(pi ∩ P ) is, by (1.9), equal to k[thi +1 , . . . , tm ]; the latter is a polynomial ring, so normal by (10.29)(1). Also, the extension P/(pi ∩ P ) ⊂ R/pi is integral as P ⊂ R is. Hence, the Going-down Theorem (14.9) yields a prime p with pi ⊂ p ⊂ pi+1 and p ∩ P = ⟨t1 , . . . , thi +1 ⟩. Then pi ⫋ p ⫋ pi+1 , contradicting the maximality of r. Thus hi + 1 = hi+1 for all i. But h0 = 0. Hence r = hr . But hr = m and m = d. Thus r = d, as desired. □ Definition (15.9). — Given a ring R, its (Krull) dimension dim(R) is defined to be the supremum of the lengths r of all strictly ascending chains of primes: dim(R) := sup{ r | there’s a chain of primes p0 ⫋ · · · ⫋ pr in R }. Exercise (15.10). — Let R be a domain of (finite) dimension r, and p a nonzero prime. Prove that dim(R/p) < r. Exercise (15.11). — Let R′ /R be an integral extension of rings. Prove that dim(R) = dim(R′ ). Theorem (15.12). — Let k be a field, R a finitely generated k-algebra. If R is a domain, then dim(R) = tr. degk (Frac(R)). Proof: The assertion is an immediate consequence of (15.8). □ Corollary (15.13). — Let k be a field, R a finitely generated k-algebra, and p a prime of R. Suppose R is a domain. Then dim(Rp ) + dim(R/p) = dim(R). If also p is maximal, then dim(Rp ) = dim(R). Proof: A chain of primes p0 ⫋ · · · ⫋ p ⫋ · · · ⫋ pr in R gives rise to a pair of chains of primes, one in Rp and one in R/p, p0 Rp ⫋ · · · ⫋ pRp and 0 = p/p ⫋ · · · ⫋ pr /p, owing to (11.18) and to (1.8) and (2.7); conversely, every such pair of chains arises from a unique chain in R through p. But by (15.8), every strictly ascending chain through p of maximal length is of length dim(R). The asserted equation follows. If also p is maximal, then clearly dim(R/p) = 0, and so dim(Rp ) = dim(R). □ Definition (15.14). — We call a ring catenary if, given any two nested primes q ⊂ p, there exists a chain of primes p0 ⫋ · · · ⫋ pr of maximal length r with p0 = q and pr = p, and any two such chains have the same length r. September 3, 2012 11Nts.tex 15. Noether Normalization 79 Theorem (15.15). — Over a field, a finitely generated algebra is catenary. Proof: Let R be the algebra, and q ⊂ p two nested primes. Replacing R by R/q, we may assume R is a domain. Then the proof of (15.13) shows that any chain of primes 0 ⫋ · · · ⫋ p of maximal length is of length dim(R) − dim(R/p). □ Exercise (15.16). — Let k be a field, R a finitely generated k-algebra, f ∈ R nonzero. Assume R is a domain. Prove that dim(R) = dim(Rf ). Exercise (15.17). — Let k be a field, P := k[f ] the polynomial ring in one variable f . Set p := ⟨f ⟩ and R := Pp . Find dim(R) and dim(Rf ). Exercise (15.18). — Let R be a ring, R[X] the polynomial ring. Prove 1 + dim(R) ≤ dim(R[X]) ≤ 1 + 2 dim(R). September 3, 2012 11Nts.tex 15. Appendix: Jacobson Rings (15.19) (Jacobson Rings). — We call a ring R Jacobson if, given any ideal a, its radical is equal to the intersection of all maximal ideals containing it; that is, ∩ √ (15.19.1) a = m⊃a m. Plainly, the nilradical of a Jacobson ring is equal to its Jacobson radical. Also, any quotient ring of a Jacobson ring is Jacobson too. In fact, a ring is Jacobson if and only if the the nilradical of every quotient ring is equal to its Jacobson radical. In general, the right-hand side √ of (15.19.1) contains the left. So (15.19.1) holds if and only if every f outside a lies outside some√maximal ∩ ideal m containing a. Recall the Scheinnullstellensatz, (3.22): it says a = p⊃a p with p prime. Thus R is Jacobson if and only if (15.19.1) holds whenever a is prime. For example, a field k is Jacobson. More generally, a local ring A is Jacobson if and only if its maximal ideal is its only prime. Further, a Boolean ring B is Jacobson, as every prime is maximal by (2.16), and so trivially (15.19.1) holds whenever a is prime. Moreover, the polynomial ring k[X1 , . . . , Xn ] is Jacobson by (2.21). Finally, owing to the next lemma, both Z and k[X1 ] are Jacobson. Lemma (15.20). — Let R be a 1-dimensional domain. Assume every nonzero element lies in only finitely many maximal ideals. Then R is Jacobson if and only if the set {mλ }λ∈Λ of maximal ideals is infinite. ∏ Proof: If {mλ } is finite, take√ a nonzero xλ ∈ mλ for each λ. Set x := x . √ ∩ λ ∩ Then x ̸= 0 and x ∈ mλ . But ⟨0⟩ = ⟨0⟩ as R is a domain. So ⟨0⟩ ̸= mλ . Thus R is not Jacobson. ∩ If {mλ } is infinite, then mλ = ⟨0⟩ by hypothesis. But every nonzero prime is maximal as R is 1-dimensional. Thus (15.19.1) holds whenever a is prime. □ Proposition (15.21). — A ring R is Jacobson if and only if, for any nonmaximal prime p and any f ∈ / p, the extension pRf is not maximal. Proof: Assume R is Jacobson. Take a nonmaximal prime p and an f ∈ / p. Then f∈ / m for some maximal ideal m containing p. So pR is not maximal by (11.18). f √ Conversely, let a be an ideal, f ∈ / a. Then (R/a)f ̸= 0. So there is a maximal ideal n in (R/a)f . Let m be its contraction / m. Further, / in R. Then m ⊃/a and f ∈ (4.8) and (12.18) yield Rf /mRf = (R/a m/a)f = (R/a)f n. Since n is maximal, Rf /mRf is a field. So m is maximal by hypothesis. Thus R is Jacobson. □ Exercise (15.22). — Let X be a topological space. We say a subset Y is locally closed if Y is the intersection of an open set and a closed set; equivalently, Y is open in its closure Y ; equivalently, Y is closed in an open set containing it. We say a subset X0 of X is very dense if X0 meets every nonempty locally closed subset Y . We say X is Jacobson if its set of closed points is very dense. Show that the following conditions on a subset X0 of X are equivalent: (1) X0 is very dense. (2) Every closed set F of X satisfies F ∩ X0 = F . (3) The map U 7→ U ∩ X0 from the open sets of X to those of X0 is bijective. 80 15. Appendix: Jacobson Rings 81 Exercise (15.23). — Let R be a ring, X := Spec(R), and X0 the set of closed points of X. Show that the following conditions are equivalent: (1) R is a Jacobson ring. (2) X is a Jacobson space. (3) If y ∈ X is a point such that {y} is locally closed, then y ∈ X0 . Theorem (15.24) (Generalized Hilbert Nullstellensatz). — Let R be a Jacobson ring, R′ a finitely generated algebra, m′ a maximal ideal of R′ , and m its contraction. Then (1) m is maximal, and R′ /m′ is algebraic over R/m, and (2) R′ is Jacobson. Proof: To prove (1), replace R by R/m and R′ by R′ /m′ . Then R is Jacobson, R is a field as well as a finitely generated algebra, and R ⊂ R′ . We must show R is a field and R′ /R is a finite field extension. Write R′ = R[x1 , . . . , xn ] with xi ̸= 0. Then R′ = R[x1 , . . . , xn−1 ][xn ]. So the tower property for finite extensions (10.22) implies it suffices to prove (1) for n = 1. Set x := x1 and Q := Frac(R). Then Q[x] = R′ as R ⊂ Q. But R′ is a field, and x ̸= 0; so 1/x ∈ Q[x]. Say 1/x = q0 xm + · · · + qm with qi ∈ Q and q0 ̸= 0. Then ′ axm+1 + a1 xm + · · · + am x + am+1 = 0 ′ with ′ a, ai ∈ R and a ̸= 0. So x is integral over Ra . Further Ra [x] = R . Also R is a field. Hence Ra is a field by (14.1); so ⟨0⟩ ⊂ Ra is maximal. But R is a Jacobson domain. Hence, ⟨0⟩ ⊂ R is maximal by (15.21). So R is a field. So R = Ra . So R[x] = R′ . Thus (1) holds. To prove (2), let p′ ⊂ R′ be prime, and p its contraction. Given a′ ∈ R′ − p′ with ′ ′ p Ra′ maximal, apply (1) to Ra′ ′ /R; thus p is maximal and Ra′ ′ /p′ Ra′ ′ is integral (algebraic) over R/p. But Ra′ ′ /p′ Ra′ ′ ⊃ R′ /p′ . Hence R′ /p′ is integral over R/p. So R′ /p′ is a field by (14.1). So p′ is maximal. Thus (15.21) yields (2). □ Exercise (15.25). — Let P := Z[X1 , . . . , Xn ] be the polynomial ring. Assume f ∈ P vanishes at every zero in K n of f1 , . . . , fr ∈ P for every finite field K; that is, if (a) := (a1 , . . . , an ) ∈ K n and f1 (a) = 0, . . . , fr (a) = 0 in K, then f (a) = 0 too. Prove there are g1 , . . . , gr ∈ P and N ≥ 1 such that f N = g1 f1 + · · · + gr fr . Exercise (15.26). — Let R be a ring, R′ an algebra. Prove that if R′ is integral over R and R is Jacobson, then R′ is Jacobson. Exercise (15.27). — Let R be a Jacobson ring, S a multiplicative subset, f ∈ R. True or false: prove or give a counterexample to each of the following statements: (1) The localized ring Rf is Jacobson. (2) The localized ring S −1 R is Jacobson. (3) The filtered direct limit lim Rλ of Jacobson rings Rλ is Jacobson. −→ Exercise (15.28). — Let R be a reduced Jacobson ring with a finite set Σ of minimal primes, and P a finitely generated module. Show that P is locally free of rank r if and only if dimR/m (P/mP ) = r for any maximal ideal m. 82 16. Chain Conditions 16. Chain Conditions In a ring, often every ideal is finitely generated; if so, we call the ring Noetherian. Examples include the ring of integers and any field. We characterize Noetherian rings as those in which every ascending chain of ideals stabilizes, or equivalently, in which every set of ideals has one member maximal under inclusion. We prove the Hilbert Basis Theorem: if a ring is Noetherian, then so is any finitely generated algebra over it. We define and characterize Noetherian modules similarly, and we prove that, over a Noetherian ring, a module is Noetherian if and only if it is finitely generated. Lastly, we study Artinian rings and modules; in them, by definition, every descending chain of ideals, respectively of submodules, stabilizes. (16.1) (Noetherian rings). — We call a ring Noetherian if every ideal is finitely generated. A PID is, trivially, Noetherian. Examples include a field k, the polynomial ring k[X] in one variable, and the ring of integers Z. Here are two standard examples of non-Noetherian rings. A third is given below in (16.6), and a fourth later in (18.26). First, form the polynomial ring k[X1 , X2 , . . . ] in infinitely many variables. It is non-Noetherian as ⟨X1 , X2 , . . . ⟩ is not finitely generated (but the ring is a UFD). Second, in the polynomial ring k[X, Y ], form this subring R and its ideal a: { } R := f := a + Xg | a ∈ k and g ∈ k[X, Y ] and a := ⟨X, XY, XY 2 , . . . ⟩. Then a is not generated by any f1 , . . . , fm ∈ a. Indeed, let n be the highest power of Y occurring in any fi . Then XY n+1 ∈ / ⟨f1 , . . . , fm ⟩. Thus R is non-Noetherian. Exercise (16.2). — Let a be a finitely generated ideal in an arbitrary ring. Show every set that generates a contains a finite subset that generates a. Definition (16.3). — We say the ascending chain condition (acc) is satisfied if every ascending chain of ideals a0 ⊂ a1 ⊂ · · · stabilizes; that is, there is a j ≥ 0 such that aj = aj+1 = · · · . We say the maximal condition (maxc) is satisfied if every nonempty set of ideals S contains ones maximal for inclusion, that is, properly contained in no other in S. Lemma (16.4). — Acc is satisfied if and only if maxc is. Proof: Let a0 ⊂ a1 ⊂ · · · be a chain of ideals. If aj is maximal, then trivially aj = aj+1 = · · · . Thus maxc implies acc. Conversely, given a nonempty set of ideals S with no maximal member, there’s a0 ∈ S; for each j ≥ 0, there’s aj+1 ∈ S with aj ⫋ aj+1 . So the Axiom of Countable Choice provides an infinite chain a0 ⫋ a1 ⫋ · · · . Thus acc implies maxc. □ Proposition (16.5). — Given a ring R, the following conditions are equivalent: (1) R is Noetherian; (2) acc is satisfied; (3) maxc is satisfied. September 3, 2012 11Nts.tex 16. Chain Conditions 83 ∪ Proof: Assume (1) holds. Let a0 ⊂ a1 ⊂ · · · be a chain of ideals. Set a := an . Clearly, a is an ideal. So by hypothesis, a is finitely generated, say by x1 , . . . , xr . For each i, there is an ji such that xi ∈ aji . Set j := max{ji }. Then xi ∈ aj for all i. So a ⊂ aj ⊂ aj+1 ⊂ · · · ⊂ a. So aj = aj+1 = · · · . Thus (2) holds. Assume (2) holds. Then (3) holds by (16.4). Assume (3) holds. Let a be an ideal, aλ for λ ∈ Λ generators, S the set of ideals generated by finitely many aλ . Let b be a maximal element of S; say b is generated by aλ1 , . . . , aλm . Then b ⊂ b + ⟨aλ ⟩ for any λ. So by maximality, b = b + ⟨aλ ⟩. Hence aλ ∈ b. So b = a; whence, a is finitely generated. Thus (1) holds. □ Example (16.6). — In the field of rational functions k(X, Y ), form this ring: R := k[X, Y, X/Y, X/Y 2 , X/Y 3 , . . . ]. Then R is non-Noetherian by (16.5). Indeed, X does not factor into irreducibles: X = (X/Y ) · Y and X/Y = (X/Y 2 ) · Y and so on. Correspondingly, there is an ascending chain of ideals that does not stabilize: ⟨X⟩ ⫋ ⟨X/Y ⟩ ⫋ ⟨X/Y 2 ⟩ ⫋ · · · . Proposition (16.7). — Let R be a Noetherian ring, S a multiplicative subset, a an ideal. Then R/a and S −1 R are Noetherian. Proof: If R satisfies the acc, so do R/a and S −1 R by (1.8) and by (11.18)(1). Alternatively, any ideal b/a of R/a is, clearly, generated by the images of generators of b. Similarly, any ideal b of S −1 R is generated by the images of generators □ of φ−1 S b by (11.17)(1)(b). Exercise (16.8). — Let R be a ring, X a variable, R[X] the polynomial ring. Prove this statement or find a counterexample: if R[X] is Noetherian, then so is R. Theorem (16.9) (Cohen). — A ring is Noetherian if every prime ideal is finitely generated. Proof: Let R be a ring. Suppose there are non-finitely-generated ideals. Given ∪ a nonempty set of them {aλ } that is linearly ordered by inclusion, set a := aλ . If a is finitely generated, then all the generators lie in some aλ , so generate aλ , a contradiction. Thus a is non-finitely-generated. Hence, by Zorn’s Lemma, there is a maximal non-finitely-generated ideal p. In particular, p ̸= R. Assume every prime is finitely generated. Then there are a, b ∈ R−p with ab ∈ p. So p + ⟨a⟩ is finitely generated, say by x1 + w1 a, . . . , xn + wn a with xi ∈ p. Then {x1 , . . . , xn , a} (generate p + ) ⟨a⟩. Set b = Ann (p + ⟨a⟩)/p . Then b ⊃ ⟨b⟩ + p and b ∈ / p. So b is finitely generated, say by y1 , . . . , ym . Take z ∈ p. Then z ∈ p + ⟨a⟩, so write z = a1 x1 + · · · + an xn + ya with ai , y ∈ R. Then ya ∈ p. So y ∈ b. Hence y = b1 y1 + · · · + bm ym with bj ∈ R. Thus p is generated by {x1 , . . . , xn , ay1 , . . . , aym }, a contradiction. Thus there are no non-finitely-generated ideals; in other words, R is Noetherian. □ Lemma (16.10). — If a ring R is Noetherian, then so is the polynomial ring R[X]. September 3, 2012 11Nts.tex 84 16. Chain Conditions Proof: By way of contradiction, assume there is an ideal a of R[X] that is not finitely generated. Set a0 := ⟨0⟩. For each i ≥ 1, choose inductively fi ∈ a − ai−1 of least degree di , and set ai := ⟨f1 , . . . , fi ⟩. Let ai be the leading coefficient of fi , and b the ideal generated by all the ai . Since R is Noetherian, b = ⟨a1 , . . . , an ⟩ for some n by (16.2). Then an+1 = r1 a1 + · · · + rn an with ri ∈ R. By construction, di ≤ di+1 for all i. Set f := fn+1 − (r1 f1 X dn+1 −d1 + · · · + rn fn X dn+1 −dn ). Then deg(f ) < dn+1 , so f ∈ an . Therefore, fn+1 ∈ an , a contradiction. □ Theorem (16.11) (Hilbert Basis). — Let R be a Noetherian ring, R′ a finitely generated algebra. Then R′ is Noetherian. Proof: Say x1 , . . . , xr generate R′ over R, and let P := R[X1 , . . . , Xr ] be the polynomial ring in r variables. Then P is Noetherian by (16.10) and induction on r. Assigning xi to Xi defines an R-algebra map P → R′ , and obviously, it is surjective. Hence R′ is Noetherian by (16.7). □ (16.12) (Noetherian modules). — We call a module M Noetherian if every submodule is finitely generated. In particular, a ring is Noetherian as a ring if and only if it is Noetherian as a module, because its submodules are just the ideals. We say the ascending chain condition (acc) is satisfied in M if every ascending chain of submodules M0 ⊂ M1 ⊂ · · · stabilizes. We say the maximal condition (maxc) is satisfied in M if every nonempty set of submodules contains ones maximal under inclusion. It is simple to generalize (16.5): These conditions are equivalent: (1) M is Noetherian; (2) acc is satisfied in M ; (3) maxc is satisfied in M . Lemma (16.13). — Let R be a ring, M a module. Nested submodules M1 ⊂ M2 of M are equal if both these equations hold: M1 ∩ N = M2 ∩ N and (M1 + N )/N = (M2 + N )/N. Proof: Given m2 ∈ M2 , there is m1 ∈ M1 with n := m2 − m1 ∈ N . Then n ∈ M2 ∩ N = M1 ∩ N . Hence m2 ∈ M1 . Thus M1 = M2 . □ α β Exercise (16.14). — Let 0 → L − →M − → N → 0 be a short exact sequence of R-modules, and M1 , M2 two submodules of M . Prove or give a counterexample to this statement: if β(M1 ) = β(M2 ) and α−1 (M1 ) = α−1 (M2 ), then M1 = M2 . Proposition (16.15). — Let R be a ring, M a module, N a submodule. (1) Then M is finitely generated if N and M/N are finitely generated. (2) Then M is Noetherian if and only if N and M/N are Noetherian. Proof: Assertion (1) is equivalent to (5.6) owing to (5.2). To prove (2), first assume M is Noetherian. A submodule N ′ of N is also a submodule of M , so N ′ is finitely generated; thus N is Noetherian. A submodule of M/N is finitely generated as its inverse image in M is so; thus M/N is Noetherian. Conversely, assume N /and M/N are Noetherian. Let P be a submodule of M . ∼ (P + N )/N Then P ∩ N and (P + N ) N are finitely generated. But P/(P ∩ N ) −→ by (4.8.2). So (1) implies P is finitely generated. Thus M is Noetherian. Here is a second proof of (2). First assume M is Noetherian. Then any ascending chain in N is also a chain in M , so it stabilizes. And any chain in M/N is the image of a chain in M , so it too stabilizes. Thus N and M/N are Noetherian. September 3, 2012 11Nts.tex 16. Chain Conditions 85 Conversely, assume N and M/N are Noetherian. Given M1 ⊂ M2 ⊂ · · · ⊂ M , both (M1 ∩ N ) ⊂ (M2 ∩ N ) ⊂ · · · and (M1 + N )/N ⊂ (M2 + N )/N ⊂ · · · stabilize, say Mj ∩ N = Mj+1 ∩ N = · · · and (Mj + N )/N = (Mj+1 + N )/N = · · · . Then Mj = Mj+1 = · · · by (16.13). Thus M is Noetherian. □ Corollary (16.16). — Modules M1 , . . . , Mr are Noetherian if and only if their direct sum M1 ⊕ · · · ⊕ Mr is Noetherian. Proof: The sequence 0 → M1 → M1 ⊕ (M2 ⊕ · · · ⊕ Mr ) → M2 ⊕ · · · ⊕ Mr → 0 is exact. So the assertion results from (16.15)(2) by induction on r. □ Exercise (16.17). — Let R be⊕ a ring, a1 , . . . , ar ideals such that each R/ai is a Noetherian ring. Prove (1) that R/ai is a Noetherian R-module, and (2) that, ∩ if ai = 0, then R too is a Noetherian ring. Theorem (16.18). — Let R be a Noetherian ring, and M a module. Then the following conditions on M are equivalent: (1) M is Noetherian; (2) M is finitely generated; (3) M is finitely presented. Proof: Assume (2). Then there is an exact sequence 0 → K → Rn → M → 0. Now, Rn is Noetherian by (16.16) and by (16.12). Hence K is finitely generated, so (3) holds; further, (1) holds by (16.15)(2). Trivially, (1) or (3) implies (2). □ Theorem (16.19) (E. Artin–Tate). — Let R ⊂ R′ ⊂ R′′ be rings. Assume R is Noetherian. Assume R′′ is module finite over R′ , and R′′ is algebra finite over R. Then R′ is algebra finite over R. Proof: Say x1 , . . . , xm generate R′′ as an R-algebra, and y1 , . . . , yn generate R′′ as an R′ -module. Then there exist zij ∈ R′ and zijk ∈ R′ with ∑ ∑ xi = zij yj and yi yj = zijk yk . (16.19.1) Let R0′ be the R-algebra generated by the zij and the zijk . Since R is Noetherian, so is R0′ by the Hilbert Basis Theorem, (16.11). Any x ∈ R′′ is a polynomial in the xi with coefficients in R. Therefore, (16.19.1) implies that x is a linear combination of the yj with coefficients in R0′ . But R0′ is a Noetherian ring, and R′ is an R0′ -submodule of R′′ . Hence R′ is module finite over R0′ by (16.15). Since R0′ is algebra finite over R, it follows that R′ is too. □ Exercise (16.20). — Let G be a finite group acting on a domain R, and R′ the subring of invariants. Let k ⊂ R′ be a field. Using (10.17), prove this celebrated theorem of E. Noether (1926): if R is algebra finite over k, then so is R′ . √ Example (16.21). — Set δ := −5, set R := Z[δ], and set p := (2, 1 + δ). Let’s prove that p is finitely presented and that pRq is free of rank 1 over Rq for every maximal ideal q of R, but that p is not free. Thus the equivalent conditions of (13.27) do not imply that P is free. Since Z is Noetherian and since R is generated over Z, the Hilbert Basis Theorem (16.11) yields that R is Noetherian. So since p is generated by two elements, (16.18) yields that p is finitely presented. Recall from [1, pp. 417, 421, 425] that p is maximal in R, but not principal. Now, 3∈ / p; otherwise, 1 ∈ p as 2 ∈ p, but p ̸= R. So (1 − δ)/3 ∈ Rp . Hence (1 + δ)Rp contains (1 + δ)(1 − δ)/3, or 2. So (1 + δ)Rp = pRp . Since Rp is a domain, the map µ1+δ : Rp → pRp is injective, so bijective. Thus pRp is free of rank 1. September 3, 2012 11Nts.tex 86 16. Chain Conditions Let q be a maximal ideal distinct from p. Then p ∩ (R − q) ̸= ∅; so, pRq = Rq by (11.12)(2). Thus pRq is free of rank 1. Finally, suppose p ≃ Rn . Set S := R − 0. Then S −1 R is the fraction field, K say, of R. So S −1 p ≃ K n . But the inclusion p ֒→ R yields an injection S −1 p ֒→ K. ∼ K, since S −1 p is a nonzero K-vector space. Therefore, n = 1. So Hence S −1 p −→ p ≃ R. Hence p is generated by one element. But p is not principal. So there is a contradiction. Thus p is not free. Definition (16.22). — We say a module M is Artinian or the descending chain condition (dcc) is satisfied in M if every descending chain of submodules stabilizes. We say the ring itself is Artinian if it is an Artinian module. We say the minimal condition (minc) is satisfied in M if every nonempty set of submodules has a minimal member. Proposition (16.23). — Let M1 , . . . , Mr , M be modules, N a submodule of M . (1) Then M is Artinian if and only if minc is satisfied in M . (2) Then M is Artinian if and only if N and M/N are Artinian. (3) Then M1 , . . . , Mr are Artinian if and only if M1 ⊕ · · · ⊕ Mr is Artinian. Proof: It is easy to adapt the proof of (16.4), the second proof of (16.15)(2), and the proof of (16.16). □ Exercise (16.24). — Let k be a field, R an algebra. Assume that R is finite dimensional as a k-vector space. Prove that R is Noetherian and Artinian. / Exercise (16.25). — Let p be a prime number, and set M := Z[1/p] Z ⊂ Q/Z. Prove that any Z-submodule N ⊂ M is either finite or all of M . Deduce that M is an Artinian Z-module, and that it is not Noetherian. Exercise (16.26). — Let R be an Artinian ring. Prove that R is a field if it is a domain. Deduce that, in general, every prime ideal p of R is maximal. September 3, 2012 11Nts.tex 17. Associated Primes Given a module, a prime is associated to it if the prime is equal to the annihilator of an element. Given a subset of the set of all associated primes, we prove there is a submodule whose own associated primes constitute that subset. If the ring is Noetherian, then the set of annihilators of elements has maximal members; we prove the latter are prime, so associated. Then the union of all the associated primes is the set of zerodivisors on the module. If also the module is finitely generated, then the intersection is the set of nilpotents. Lastly, we prove there is then a finite chain of submodules whose successive quotients are cyclic with prime annihilators; these primes include all associated primes, which are, therefore, finite in number. Definition (17.1). — Let R be a ring, M a module. A prime ideal p is said to be associated to M if there is a (nonzero) m ∈ M with p = Ann(m). The set of associated primes is denoted by Ass(M ) or AssR (M ). The primes that are minimal in Ass(M ) are called the minimal primes of M ; the others, the embedded primes. Warning: following a old custom, we mean by the associated primes of an ideal a not those of a viewed as an abstract module, but rather those of R/a. Lemma (17.2). — Let R be a ring, M a module, and p a prime ideal. Then p ∈ Ass(M ) if and only if there is an R-injection R/p ֒→ M . Proof: Assume p = Ann(m) with m ∈ M . Define a map R → M by x 7→ xm. This map induces an R-injection R/p ֒→ M . Conversely, suppose there is an R-injection R/p ֒→ M , and let m ∈ M be the image of 1. Then p = Ann(m), so p ∈ Ass(M ). □ Proposition (17.3). — Let M be a module. Then Ass(M ) ⊂ Supp(M ). Proof: Let p ∈ AssR (M ). Say p = Ann(m). Then m/1 ∈ Mp is nonzero as no x ∈ (R−p) satisfies xm = 0. Alternatively, (17.2) yields an R-injection R/p ֒→ M . It induces an injection (R/p)p ֒→ Mp by (12.16). But (R/p)p = Frac(R/p) by (12.19). Thus Mp ̸= 0 and so p ∈ Supp(M ). □ Lemma (17.4). — Let R be a ring, p a prime ideal, m ∈ R/p a nonzero element. Then (1) Ann(m) = p and (2) Ass(R/p) = {p}. Proof: To prove (1), say m is the residue of y ∈ R. Let x ∈ R. Then xm = 0 if and only if xy ∈ p, so if and only if x ∈ p, as p is prime and m ̸= 0. Thus (1) holds. Trivially, (1) implies (2). □ Proposition (17.5). — Let M be a module, N a submodule. Then Ass(N ) ⊂ Ass(M ) ⊂ Ass(N ) ∪ Ass(M/N ). Proof: Take m ∈ N . Then the annihilator of m is the same whether m is regarded as an element of N or of M . So Ass(N ) ⊂ Ass(M ). Let p ∈ Ass(M ). Then (17.2) yields an R-injection R/p ֒→ M . Denote its image by E. If E ∩ N = 0, then the composition R/p → M → M/N is injective; hence, p ∈ Ass(M/N ) by (17.2). Else, take a nonzero m ∈ E ∩ N . Then Ann(m) = p by (17.4)(1). Thus p ∈ Ass(N ). □ September 3, 2012 11Nts.tex 88 17. Associated Primes Exercise (17.6). — Given modules M1 , . . . , Mr , set M := M1 ⊕ · · · ⊕ Mr . Prove Ass(M ) = Ass(M1 ) ∪ · · · ∪ Ass(Mr ). Exercise (17.7). — Take R := Z and M := Z/⟨2⟩ ⊕ Z. Find Ass(M ) and find two submodules L, N ⊂ M with L + N = M but Ass(L) ∪ Ass(N ) ⫋ Ass(M ). Proposition (17.8). — Let M be a module, and Ψ a subset of Ass(M ). Then there is a submodule N of M with Ass(M/N ) = Ψ and Ass(N ) = Ass(M ) − Ψ. ∪ Proof: Given submodules Nλ of M totally ordered by inclusion, set N := Nλ . Given p ∈ Ass(N ), say p = Ann(m). Then m ∈ Nλ for some ∪ λ; so p ∈ Ass(Nλ ). Conversely, Ass(Nλ ) ⊂ Ass(N ) by (17.5). Thus Ass(N ) = Ass(Nλ ). So we may apply Zorn’s Lemma to obtain a submodule N of M that is maximal with Ass(N ) ⊂ Ass(M ) − Ψ. By (17.5), it suffices to show that Ass(M/N ) ⊂ Ψ. Take p ∈ Ass(M/N ). Then M/N has a submodule N ′ /N isomorphic to R/p by (17.2). So Ass(N ′ ) ⊂ Ass(N ) ∪ {p} by (17.5) and (17.4)(2). Now, N ′ ⫌ N and N is maximal with Ass(N ) ⊂ Ass(M ) − Ψ. Hence p ∈ Ass(N ′ ) ⊂ Ass(M ), but p∈ / Ass(M ) − Ψ. Thus p ∈ Ψ. □ Proposition (17.9). — Let R be a ring, S a multiplicative subset, M a module, and p a prime ideal. If p ∩ S = ∅ and p ∈ Ass(M ), then S −1 p ∈ Ass(S −1 M ); the converse holds if p is finitely generated. Proof: Assume p ∈ Ass(M ). Then (17.2) yields an injection R/p ֒→ M . It induces an injection S −1 (R/p) ֒→ S −1 M by (12.16). But S −1 (R/p) = S −1 R/S −1 p by (12.18). Assume p ∩ S = ∅ also. Then pS −1 R is prime by (11.17)(3)(b). But pS −1 R = S −1 p by (12.2). Thus S −1 p ∈ Ass(S −1 M ). Conversely, assume S −1 p ∈ Ass(S −1 M ). Then there are m ∈ M and t ∈ S with −1 S p = Ann(m/t). Say p = ⟨x∏ 1 , . . . , xn ⟩. Fix i. Then xi m/t = 0. So there is si ∈ S with si xi m = 0. Set s := si . Then xi ∈ Ann(sm). Thus p ⊂ Ann(sm). Take b ∈ Ann(sm). Then bsm/st = 0. So b/1 ∈ S −1 p. So b ∈ p by (11.17)(1)(a) and (11.17)(3)(a). Thus p ⊃ Ann(sm). So p = Ann(sm). Thus p ∈ Ass(M ). Finally, p ∩ S = ∅ by (11.18)(2), as S −1 p is prime. □ Exercise (17.10). — Let R be a ring, and suppose Rp is a domain for every prime p. Prove every associated prime of R is minimal. Lemma (17.11). — Let R be a ring, M a module, and a an ideal. Suppose a is maximal in the set of annihilators of nonzero elements m of M . Then a ∈ Ass(M ). Proof: Say a := Ann(m) with m ̸= 0. Then 1 ∈ / a as m ̸= 0. Now, take b, c ∈ R with bc ∈ a, but c ∈ / a. Then bcm = 0, but cm ̸= 0. Plainly, a ⊂ Ann(cm). So a = Ann(cm) by maximality. But b ∈ Ann(cm), so b ∈ a. Thus a is prime. □ Proposition (17.12). — Let R be a Noetherian ring, M a module. Then M = 0 if and only if Ass(M ) = ∅. Proof: Obviously, if M = 0, then Ass(M ) = ∅. Conversely, suppose M ̸= 0. Let S be the set of annihilators of nonzero elements of M . Then S has a maximal element a by (16.5). By (17.11), a ∈ Ass(M ). Thus Ass(M ) ̸= ∅. □ Definition (17.13). — Let R be a ring, M a module, x ∈ R. We say x is a zerodivisor on M if there is a nonzero m ∈ M with xm = 0; otherwise, we say x is a nonzerodivisor. We denote the set of zerodivisors by z.div(M ). September 3, 2012 11Nts.tex 17. Associated Primes 89 Proposition (17.14). — Let R be a Noetherian ring, M a module. Then ∪ z.div(M ) = p∈Ass(M ) p. Proof: Given x ∈ z.div(M ), say xm = 0 where m ∈ M and m ̸= 0. Then x ∈ Ann(m). But Ann(m) is contained in an ideal p that is maximal among annihilators of nonzero elements because of (16.5); hence, p ∈ Ass(M ) by (17.11). ∪ Thus z.div(M ) ⊂ p. The opposite inclusion results from the definitions. □ Exercise (17.15). — Let R be a Noetherian ring, M a module, N a submodule, x ∈ R. Show that, if x ∈ / p for any p ∈ Ass(M/N ), then xM ∩ N = xN . Lemma (17.16). — Let R be a Noetherian ring, M a module. Then ∪ Supp(M ) = q∈Ass(M ) V(q) ⊃ Ass(M ). Proof: Let p be a prime. Then Rp is Noetherian by (16.7) as R is. So Mp ̸= 0 if and only if AssRp (Mp ) ̸= ∅ by (17.12). But R is Noetherian; so AssRp (Mp ) ̸= ∅ if and only if there is q ∈ Ass(M ) with q∩(R−p) = ∅, or q ⊂ p, owing to (11.18)(2) and (17.9). Thus p ∈ Supp(M ) if and only if p ∈ V(q) for some q ∈ Ass(M ). □ Theorem (17.17). — Let R be a Noetherian ring, M a module, p ∈ Supp(M ). Then p contains some q ∈ Ass(M ); if p is minimal in Supp(M ), then p ∈ Ass(M ). Proof: By (17.16), q exists. Also, q ∈ Supp(M ); so q = p if p is minimal. □ Theorem (17.18). — Let R be a Noetherian ring, and M a finitely generated module. Then ∩ nil(M ) = p∈Ass(M ) p. ∩ Proof: Since M is finitely generated, nil(M ) = p∈Supp(M ) p by (13.11). Since R is Noetherian, given p ∈ Supp(M ), there is q ∈ Ass(M ) with q ⊂ p by (17.16). The assertion follows. □ Lemma (17.19). — Let R be a Noetherian ring, M a finitely generated module. Then there exists a chain of submodules 0 = M0 ⊂ M1 ⊂ · · · Mn−1 ⊂ Mn = M with Mi /Mi−1 ≃ R/pi for some prime pi for i = 1, . . . , n. For any such chain, Ass(M ) ⊂ {p1 , . . . , pn } ⊂ Supp(M ). (17.19.1) Proof: Among all submodules of M having such a chain, there is a maximal submodule N by (16.18) and (16.12). Suppose M/N ̸= 0. Then by (17.12), the quotient M/N contains a submodule N ′ /N isomorphic to R/p for some prime p. Then N ⫋ N ′ , contradicting maximality. Hence N = M . Thus a chain exists. The first inclusion of (17.19.1) follows by induction from (17.5) and (17.4)(2). Now, pi ∈ Supp(R/pi ) owing to (12.19). Thus (17.19.1) follows from (13.9)(1). □ Theorem (17.20). — Let R be a Noetherian ring, and M a finitely generated module. Then the set Ass(M ) is finite. Proof: The assertion follows directly from (17.19). □ Exercise (17.21). — Let R be a Noetherian ring, a an ideal. Prove the primes minimal containing a are associated to a. Prove such primes are finite in number. September 3, 2012 11Nts.tex 90 17. Associated Primes Exercise (17.22). — Take R := Z and M := Z in (17.19). Determine when a chain 0 ⊂ M1 ⫋ M is acceptable, and show that then p2 ∈ / Ass(M ). Exercise (17.23). — Take R := Z and M := Z/⟨12⟩ in (17.19). Find all three acceptable chains, and show that, in each case, {pi } = Ass(M ). Proposition (17.24). — Let R be a Noetherian ring, and M and N finitely generated modules. Then ∩ Ass(Hom(M, N )) = Supp(M ) Ass(N ). ( ) Proof: Take p ∈ Ass Hom(M, N ) . Then (17.2) yields an injective R-map R/p ֒→ Hom(M, N ). Set k(p) := Frac(R/p). Then k(p) = Rp /pRp by (12.19). Now, M is finitely presented by (16.18) as R is Noetherian; hence, Hom(M, N )p = HomRp (Mp , Np ) (17.24.1) by (12.21)(2). Therefore, by exactness, localizing yields an injection φ : k(p) ֒→ HomRp (Mp , Np ). Hence Mp ̸= 0; so p ∈ Supp(M ). For any m ∈ Mp with φ(1)(m) ̸= 0, the map k(p) → Np given by x 7→ φ(x)(m) is nonzero, so an injection. Hence by (17.2), we have pRp ∈ Ass(Np ). Therefore, also p ∈ Ass(N ) by (17.9). Conversely, take p ∈ Supp(M ) ∩ Ass(N ). Then Mp ̸= 0. So by Nakayama’s Lemma, Mp /pMp is a nonzero vector space over k(p). Take any nonzero R-map Mp /pMp → k(p), precede it by the canonical map Mp → Mp /pMp , and follow it by an R-injection k(p) ֒→ Np , which exists by (17.2) and (17.9). We obtain a nonzero element of HomRp (Mp , Np ), annihilated by pRp . )But pRp is maximal, so is the) ( ( entire annihilator. So pRp ∈ Ass HomRp (Mp , Np ) . Hence p ∈ Ass Hom(M, N ) by (17.24.1) and (17.9). □ Proposition (17.25). — Let R be a Noetherian ring, p a prime, M a finitely generated module, and x, y ∈ p nonzerodivisors on M . Then p ∈ Ass(M/xM ) if and only if p ∈ Ass(M/yM ). µy Proof: Form the sequence 0 → K → M/xM −−→ M/xM with K := Ker(µy ). Apply the functor Hom(R/p, •) to that sequence, and get this one: µy 0 → Hom(R/p, K) → Hom(R/p, M/xM ) −−→ Hom(R/p, M/xM ). It is exact by (5.17). But y ∈ p; so the right-hand map vanishes. Thus ∼ Hom(R/p, M/xM ). Hom(R/p, K) −→ Form the following commutative diagram with exact rows: µx → M/xM − →0 0− → M −−→ M −       µy y µy y µy y µx 0− → M −−→ M − → M/xM − →0 µx The Snake Lemma (5.12) yields an exact sequence 0 → K → M/yM −−→ M/yM . ∼ Hom(R/p, M/yM ). Therefore, Hence, similarly, Hom(R/p, K) −→ Hom(R/p, M/yM ) = Hom(R/p, M/xM ). (17.25.1) Finally, p ∈ Supp(R/p) by (13.9)(3). Hence (17.24) yields the assertion. September 3, 2012 11Nts.tex □ 18. Primary Decomposition Primary decomposition of a submodule generalizes factorization of an integer into powers of primes. A submodule is called primary if the quotient module has only one associated prime. We characterize these submodules in various ways over a Noetherian ring, emphasizing the case of ideals. A primary decomposition is a representation of a submodule as a finite intersection of primary submodules. The decomposition is called irredundant, or minimal, if cannot be reduced. We consider several illustrative examples in a polynomial ring. Then we prove existence and uniqueness theorems for a proper submodule of a finitely generated module over a Noetherian ring. The celebrated Lasker–Noether Theorem asserts the existence of an irredundant primary decomposition. The First Uniqueness Theorem asserts the uniqueness of the primes that arise; they are just the associated primes of the quotient. The Second Uniqueness Theorem asserts the uniqueness of the primary components whose primes are minimal among these associated primes; the other primary components may vary. Definition (18.1). — Let R be a ring, M a module, Q a submodule. If Ass(M/Q) consists of a single prime p, we say Q is primary or p-primary in M . Example (18.2). — A prime p is p-primary, as Ass(R/p) = {p} by (17.4)(2). Proposition (18.3). — Let R be a Noetherian ring, M a finitely generated module, Q a submodule. If Q is p-primary, then p = nil(M/Q). ∩ Proof: The assertion holds as nil(M/Q) = q∈Ass(M/Q) q by (17.18). □ Theorem (18.4). — Let R be a Noetherian ring, M a nonzero finitely generated module, Q a submodule. Set p := nil(M/Q). Then these conditions are equivalent: (1) p is prime and Q is p-primary. (2) p = z.div(M/Q). (3) Given x ∈ R and m ∈ M with xm ∈ Q but m ∈ / Q, necessarily x ∈ p. ∩ ∪ Proof: Recall p = q∈Ass(M/Q) q by (17.18), and z.div(M/Q) = q∈Ass(M/Q) q by (17.14). Thus p ⊂ z.div(M/Q). Further, (2) holds if Ass(M/Q) = {p}, that is, if (1) holds. Conversely, if x ∈ q ∈ Ass(M/Q), but x ∈ / q′ ∈ Ass(M/Q), then x ∈ / p, but x ∈ z.div(M/Q); hence, (2) implies (1). Thus (1) and (2) are equivalent. Clearly, (3) means every zerodivisor on M/Q is nilpotent, or p ⊃ z.div(M/Q). But the opposite inclusion always holds. Thus (2) and (3) are equivalent. □ Corollary (18.5). — Let R be a Noetherian ring, and q a proper ideal. Set √ p := q. Then q is primary (in R) if and only if, given x, y ∈ R with xy ∈ q but x∈ / q, necessarily y ∈ p; if so, then p is prime and q is p-primary. Proof: Clearly q = Ann(R/q). So p = nil(R/q). So the assertions result directly from (18.4) and (18.3). □ Exercise (18.6). — Let R be a ring, and p = ⟨p⟩ a principal prime generated by a nonzerodivisor p. Show every positive power pn is p-primary. Show conversely, if R is Noetherian, then every p-primary ideal q is equal to some power pn . September 3, 2012 11Nts.tex 92 18. Primary Decomposition Exercise (18.7). — Let k be a field, and k[X, √ Y ] the polynomial ring. Let a be the ideal ⟨X 2 , XY ⟩. Show a is not primary, but a is prime. Show a satisfies this condition: ab ∈ a implies a2 ∈ a or b2 ∈ a. Exercise (18.8). — Let φ : R → R′ be a homomorphism of Noetherian rings, and q ⊂ R′ a p-primary ideal. Show that φ−1 q ⊂ R is φ−1 p-primary. Show that the converse holds if φ is surjective. Proposition (18.9). — Let R be a Noetherian ring, M a finitely generated module, Q a submodule. Set p := nil(M/Q). If p is maximal, then Q is p-primary. ∩ Proof: Since p = q∈Ass(M/Q) q by (17.18), if p is maximal, then p = q for any q ∈ Ass(M/Q), or {p} = Ass(M/Q), as desired. □ √ Corollary (18.10). — Let R be a Noetherian ring, q an ideal. Set p := q. If p is maximal, then q is p-primary. Proof: Since p = nil(R/q), the assertion is a special case of (18.9). □ Corollary (18.11). — Let R be a Noetherian ring, m a maximal ideal. An ideal q is m-primary if and only if there exists n ≥ 1 such that mn ⊂ q ⊂ m. √ Proof: The condition mn ⊂ q ⊂ m just means that m := q by (3.26). So the assertion results from (18.5) and (18.10). □ Lemma (18.12). — Let R be a Noetherian ring, p a prime ideal, M a module. Let Q1 and Q2 be p-primary submodules; set Q := Q1 ∩ Q2 . Then Q is p-primary. Proof: Form the canonical map M → M/Q1 ⊕ M/Q2 . Its kernel is Q, so it induces an injection M/Q ֒→ M/Q1 ⊕ M/Q2 . Hence (17.12) and (17.5) yield ∅= ̸ Ass(M/Q) ⊂ Ass(M/Q1 ) ∪ Ass(M/Q2 ). Since the latter two sets are each equal to {p}, so is Ass(M/Q), as desired. □ (18.13) (Primary decomposition). — Let R be a ring, M a module, and N a submodule. A primary decomposition of N is a decomposition N = Q1 ∩ · · · ∩ Qr with the Qi primary. We call the decomposition irredundant or minimal if these conditions are satisfied: ∩ ∩ (1) N ̸= j̸=i Qj , or equivalently, j̸=i Qj ̸⊂ Qi for i = 1, . . . , r. (2) Say Qi is pi -primary for i = 1, . . . , r. Then p1 , . . . , pr are distinct. If R is Noetherian, then owing to (18.12), any primary decomposition can be made irredundant by intersecting all the primary submodules with the same prime and then discarding those of them that are not needed. Example (18.14). — Let k be a field, R := k[X, Y ] the polynomial ring, and a := ⟨X 2 , XY ⟩. Below, it is proved that, for any n ≥ 1, 2 n a = ⟨X⟩ ∩ ⟨X 2 , XY, Y n ⟩ = ⟨X⟩ ∩ ⟨X 2 , Y ⟩. 2 n (18.14.1) Here ⟨X , XY, Y ⟩ and ⟨X , Y ⟩ contain ⟨X, Y ⟩ , so are ⟨X, Y ⟩-primary by (18.11). Thus (18.14.1) shows infinitely many distinct primary decompositions of a. They are clearly irredundant. Note: the ⟨X, Y ⟩-primary component is not unique! Plainly, a ⊂ ⟨X⟩ and a ⊂⟨X 2 , XY, Y n ⟩ ⊂ ⟨X 2 , Y ⟩. To see a ⊃ ⟨X⟩ ∩ ⟨X 2 , Y ⟩, September 3, 2012 11Nts.tex 18. Primary Decomposition 93 take F ∈ ⟨X⟩ ∩ ⟨X 2 , Y ⟩. Then F = GX = AX 2 + BY where A, B, G ∈ R. Then X(G − AX) = BY . So X | B. Say B = B ′ X. Then F = AX 2 + B ′ XY ∈ a. Example (18.15). — Let k be a field, P := k[X, Y, Z] the polynomial ring. Set R := P/⟨XZ − Y 2 ⟩. Let x, y, z be the residues of X, Y, Z in R. Set p := ⟨x, y⟩. Clearly p2 = ⟨x2 , xy, y 2 ⟩ = x⟨x, y, z⟩. Let’s show that p2 = ⟨x⟩ ∩ ⟨x2 , y, z⟩ is an irredundant primary decomposition. First note the inclusions x⟨x, y, z⟩ ⊂ ⟨x⟩ ∩ ⟨x, y, z⟩2 ⊂ ⟨x⟩ ∩ ⟨x2 , y, z⟩. Conversely, given f ∈ ⟨x⟩ ∩ ⟨x2 , y, z⟩, represent f by GX with G ∈ P . Then GX = AX 2 + BY + CZ + D(XZ − Y 2 ) with A, B, C, D ∈ P. So (G − AX)X = B ′ Y + C ′ Z with B ′ , C ′ ∈ P . Say G − AX = A′′ + B ′′ Y + C ′′ Z with A′′ ∈ k[X] and B ′′ , C ′′ ∈ P . Then A′′ X = −B ′′ XY − C ′′ XZ + B ′ Y + C ′ Z = (B ′ − B ′′ X)Y + (C ′ − C ′′ X)Z; whence, A′′ = 0. Therefore, GX ∈ X⟨X, Y, Z⟩. Thus p2 =⟨x⟩ ∩ ⟨x2 , y, z⟩. The ideal ⟨x⟩ is ⟨x, y⟩-primary in R by (18.8). Indeed, the preimage in P of ⟨x⟩ is ⟨X, Y 2 ⟩ and of ⟨x, y⟩ is ⟨X, Y ⟩. Further, ⟨X, Y 2 ⟩ is ⟨X, Y ⟩-primary, as under the map φ : P → k[Y, Z] with φ(X) = 0, clearly ⟨X, Y 2 ⟩ = φ−1 ⟨Y 2 ⟩ and ⟨X, Y ⟩ = φ−1 ⟨Y ⟩; moreover, ⟨Y 2 ⟩ is ⟨Y ⟩-primary by (18.5), or by (18.6). Finally ⟨x, y, z⟩2 ⊂ ⟨x2 , y, z⟩ ⊂ ⟨x, y, z⟩ and ⟨x, y, z⟩ is maximal. So ⟨x2 , y, z⟩ is ⟨x, y, z⟩-primary by (18.11). Thus p2 = ⟨x⟩ ∩ ⟨x2 , y, z⟩ is a primary decomposition. It is clearly irredundant. Exercise (18.16). — Let k be a field, R := k[X, Y, Z] be the polynomial ring. Set a := ⟨XY, X − Y Z⟩, set q1 := ⟨X, Z⟩ and set q2 := ⟨Y 2 , X − Y Z⟩. Show that a = q1 ∩ q2 and that this expression is an irredundant primary decomposition. Exercise (18.17). — Let R := R′ × R′′ be a product of two domains. Find an irredundant primary decomposition of ⟨0⟩. Lemma (18.18). — Let R be a ring, M a module, N = Q1 ∩ · · · ∩ Qr a primary decomposition in M . Say Qi is pi -primary for i = 1, . . . , r. Then Ass(M/N ) ⊆ {p1 , . . . , pr }. (18.18.1) If equality holds and if p1 , . . . , pr are distinct, then the decomposition is irredundant; the converse holds if R is Noetherian. ∩ ⊕ Proof: Since N = Qi , the canonical M/Qi . So ∪ map is injective: M/N ֒→ (17.5) and (17.6) yield Ass(M/N ) ⊆ Ass(M/Qi ). Thus (18.18.1) holds. If N = Q2 ∩ · · · ∩ Qr , then Ass(M/N ) ⊆ {p2 , . . . , pr } too. Thus if equality holds in (18.18.1) and if p1 , . . . , pr are distinct, then N = Q1 ∩ · · · ∩ Qr is irredundant. ∩ Conversely, assume N = Q1 ∩ · · · ∩ Qr is irredundant. Given i, set Pi := j̸=i Qj . Then Pi ∩ Qi = N and Pi /N ̸= 0. Consider these two canonical injections: Pi /N ֒→ M/Qi and Pi /N ֒→ M/N. Assume R is Noetherian. Then Ass(Pi /N ) ̸= ∅ by (17.12). So the first injection yields Ass(Pi /N ) = {pi } by (17.5); then the second yields pi ∈ Ass(M/N ). Thus Ass(M/N ) ⊇ {p1 , . . . , pr }, and (18.18.1) yields equality, as desired. □ September 3, 2012 11Nts.tex 94 18. Primary Decomposition Theorem (18.19) (First Uniqueness). — Let R be a Noetherian ring, and M a module. Let N = Q1 ∩ · · · ∩ Qr be an irredundant primary decomposition in M ; say Qi is pi -primary for i = 1, . . . , r. Then p1 , . . . , pr are uniquely determined; in fact, they are just the distinct associated primes of M/N . Proof: The assertion is just part of (18.18). □ Theorem (18.20) (Lasker–Noether). — Over a Noetherian ring, each proper submodule of a finitely generated module has an irredundant primary decomposition. Proof: Let M be the module, N the submodule. By (17.20), M/N has finitely many distinct associated primes, say p1 , . . . , pr . Owing to (17.8), for each i, there is a p∩ i -primary submodule Qi of M with Ass(Qi /N ) = Ass(M/N ) − {pi }. Set P := Qi . Fix i. Then P/N ⊂ Qi /N . So Ass(P/N ) ⊂ Ass(Qi /N ) by (17.5). But i is arbitrary. Hence∩Ass(P/N ) = ∅. Therefore, P/N = 0 by (17.12). Finally, the decomposition N = Qi is irredundant by (18.18). □ Exercise (18.21). — Let R be a Noetherian ring, a an ideal, and M a finitely generated module. Consider the following submodule of M : ∪ Γa (M ) := n≥1 {m ∈ M | an m = 0 for some n ≥ 1}. ∩ ∩ (1) For any decomposition 0 = Qi with Qi pi -primary, show Γa (M ) = a̸⊂pi Qi . (2) Show Γa (M ) is the set of all m ∈ M such that m/1 ∈ Mp vanishes for every prime p with a ̸⊂ p. (Thus Γa (M ) is the set of all m whose support lies in V(a).) Lemma (18.22). — Let R be a Noetherian ring, S a multiplicative subset, p a prime ideal, M a module, and Q a p-primary submodule. If S ∩ p ̸= ∅, then S −1 Q = S −1 M and QS = M . If S ∩ p = ∅, then S −1 Q is S −1 p-primary and −1 Q) = Q. QS = φ−1 S (S Proof: Every prime of S −1 R is of the form S −1 q where q is a prime of R with S ∩ q = ∅ by (11.18)(2) and (12.2). And S −1 q ∈ Ass(S −1 (M/Q)) if and only if q ∈ Ass(M/Q), that is, q = p, by (17.9). However, S −1 (M/Q) = S −1 M/S −1 Q by (12.16). Therefore, if S ∩ p ̸= ∅, then Ass(S −1 M/S −1 Q) = ∅; whence, (17.12) yields S −1 M/S −1 Q = 0. Otherwise, if S ∩ p = ∅, then Ass(S −1 M/S −1 Q) = {S −1 p}; whence, S −1 Q is S −1 p-primary. −1 Q) by (12.15)(3). So if S −1 Q = S −1 M , then QS = M . Finally, QS = φ−1 S (S Now, suppose S ∩ p = ∅. Given m ∈ QS , there is s ∈ S with sm ∈ Q. But s ∈ / p. Further, p = z.div(M/Q) owing to (17.14). Therefore, m ∈ Q. Thus QS ⊂ Q. But QS ⊃ Q as 1 ∈ S. Thus QS = Q. □ Proposition (18.23). — Let R be a Noetherian ring, S a multiplicative subset, M a finitely generated module. Let N = Q1 ∩ · · · ∩ Qr ⊂ M be an irredundant primary decomposition. Say Qi is pi -primary for all i, and S ∩ pi = ∅ just for i ≤ h. Then S −1 N = S −1 Q1 ∩ · · · ∩ S −1 Qh ⊂ S −1 M are irredundant primary decompositions. and N S = Q1 ∩ · · · ∩ Qh ⊂ M Proof: By (12.15)(4)(b), S −1 N = S −1 Q1 ∩ · · · ∩ S −1 Qr . Further, by (18.22), S Qi is S −1 pi -primary for i ≤ h, and S −1 Qi = S −1 M for i > h. Therefore, S −1 N = S −1 Q1 ∩ · · · ∩ S −1 Qh is a primary decomposition. It is irredundant by (18.18). Indeed, Ass(S −1 M/S −1 N ) = {S −1 p1 , . . . , S −1 ph } −1 September 3, 2012 11Nts.tex 18. Primary Decomposition 95 by an argument like that in the first part of (18.22). Further, S −1 p1 , . . . , S −1 ph are distinct by (11.18)(2) as the pi are distinct. −1 N = S −1 Q1 ∩ · · · ∩ S −1 Qh . Owing to (12.15)(3), we get Apply φ−1 S to S S S S N = Q1 ∩ · · · ∩ Qh . But QSi = Qi by (18.22). So N S = Q1 ∩ · · · ∩ Qh is a primary decomposition. It is irredundant as, clearly, (18.13)(1) and (2) hold for it, since they hold for N = Q1 ∩ · · · ∩ Qr . □ Theorem (18.24) (Second Uniqueness). — Let R be a ring, M a module, N a submodule. Assume R is Noetherian and M is finitely generated. Let p be a minimal prime of M/N . Then, in any irredundant primary decomposition of N in M , the p-primary component Q is uniquely determined; in fact, Q = N S where S := R − p. Proof: In (18.23), take S := R − p. Then h = 1 as p is minimal. □ Exercise (18.25). — Let R∩be a Noetherian ring, M a finitely generated module, N a submodule. Prove N = p∈Ass(M/N ) φ−1 p (Np ). Exercise (18.26). — Let R be a Noetherian ring, p a prime. Its nth symbolic power p(n) is defined as the saturation (pn )S where S := R − p. (1) Show p(n) is the p-primary component of pn . (2) Show p(m+n) is the p-primary component of p(n) p(m) . (3) Show p(n) = pn if and only if pn is p-primary. (4) Given a p-primary ideal q, show q ⊃ p(n) for all large n. Exercise (18.27). — Let R be a Noetherian ring, ⟨0⟩ = q1 ∩· · ·∩qn an irredundant √ primary decomposition. Set pi := qi for i = 1, . . . , n. (r) (1) Suppose pi is minimal for some i. Show qi = pi for all large r. (r) (2) Suppose pi is not minimal for some i. Show that replacing qi by pi for large r gives infinitely many distinct irredundant primary decompositions of ⟨0⟩. Theorem (18.28) (Krull Intersection). — Let∩R be a Noetherian ring, a an ideal, and M a finitely generated module. Set N := n≥0 an M . Then there exists x ∈ a such that (1 + x)N = 0. Proof: By (16.18), N is finitely generated. So the desired x ∈ a exists by (10.3) provided N = aN . Clearly ∩ N ⊃ aN . To prove N ⊂ aN , use (18.20): take a primary decomposition aN = Qi with Qi pi -primary. Fix i. If there’s a ∈ a − pi , then aN ⊂ Qi , and so (18.4) yields N ⊂ Qi . If a ⊂ pi , then there’s ∩ ni with ani M ⊂ Qi by (18.3) and (3.25), and so again N ⊂ Qi . Thus N ⊂ Qi = aN , as desired. □ Exercise (18.29). — Let R be a Noetherian ring, m ⊂ rad(R) an ideal, M a finitely generated module, and M ′ a submodule. Considering M/M ′ , show that ∩ M ′ = n≥0 (mn M + M ′ ). Example (18.30) (Another non-Noetherian ring). — Let R denote the ring of C ∞ functions on the real line, m the ideal of all f ∈ R that vanish at the origin. ∼ R. Note that m is maximal, as f 7→ f (0) defines an isomorphism R/m −→ Let f ∈ R and n ≥ 1. Then, Taylor’s Theorem yields f (x) = f (0) + f ′ (0)x + · · · + where f (n−1) (0) n−1 (n−1)! x fn (x) := September 3, 2012 ∫1 0 + xn fn (x) (1−t)n−1 (n) (xt) dt. (n−1)! f 11Nts.tex 96 18. Primary Decomposition Here fn is C ∞ too, since we can differentiate under the integral sign by [5, (7.1), p. 276]. So, if f ∈ m, then f (x) = xf1 (x). Thus m ⊂ ⟨x⟩. But, obviously, m ⊃ ⟨x⟩. Hence m = ⟨x⟩. Therefore, mn = ⟨xn ⟩. If the first n − 1 derivatives of f vanish at 0, then Taylor’s Theorem yields f ∈ ⟨xn ⟩. Conversely, assume f (x) = xn g(x) for some g ∈ R. By Leibniz’s Rule, ∑k ( ) n! f (k) (x) = j=0 kj (n−j+1)! xn−j+1 g (k−j) (x). Hence f (k) vanishes at 0 if n > k. Thus ⟨xn ⟩ consists of the f ∈ R whose first n − 1 ∩ derivatives vanish at 0. But ⟨xn ⟩ = mn . Thus n≥0 mn consists of those f ∈ R all of whose derivatives vanish at 0. There is a well-known nonzero C ∞ -function all of whose derivatives vanish at 0: { 2 e−1/x if x ̸= 0, h(x) := 0 if x = 0; ∩ n see [5, Ex. 7, p. 82]. Thus n≥0 m ̸= 0. Given g ∈ m, let’s show (1 + g)h ̸= 0. Since g(0) = 0 and g is continuous, there is that |g(x)| < 1/2 if |x| < δ. Hence 1 + g(x) ≥ ∩ 1/2 if |x| < δ. Hence ( δ > 0 such ) 1 + g(x) h(x) > (1/2)h(x) > 0 if 0 < |x| < δ. Thus (1 + g)( mn ) ̸= 0. Thus the Krull Intersection Theorem (18.28) fails for R, and so R is non-Noetherian. September 3, 2012 11Nts.tex 19. Length The length of a module is a generalization of the dimension of a vector space. The length is the number of links in a composition series, which is a finite chain of submodules whose successive quotients are simple — that is, their only proper submodules are zero. Our main result is the Jordan–Hölder Theorem: any two composition series do have the same length and even the same successive quotients; further, their annihilators are just the primes in the support of the module, and the module is equal to the product of its localizations at these primes. Consequently, the length is finite if and only if the module is both Artinian and Noetherian. We also prove Akizuki’s Theorem: a ring is Artinian if and only if it is Noetherian and every prime is maximal. Consequently, a ring is Artinian if and only if its length is finite; if so, then it is the product of Artinian local rings. (19.1) (Length). — Let R be a ring, and M a module. We call M simple if it is nonzero and its only proper submodule is 0. We call a chain of submodules, M = M0 ⊃ M1 ⊃ · · · ⊃ Mm = 0 (19.1.1) a composition series of length m if each successive quotient Mi−1 /Mi is simple. Finally, we define the length ℓ(M ) to be the infimum of all those lengths: ℓ(M ) := inf{ m | M has a composition series of length m }. (19.1.2) By convention, if M has no composition series, then ℓ(M ) := ∞; further, ℓ(0) := 0. For example, if R is a field, then M is a vector space and ℓ(M ) = dimR (M ). Further, the chains in (17.23) are composition series, but those in (17.22) are not. Exercise (19.2). — Let R be a ring, M a module. Prove these statements: (1) If M is simple, then any nonzero element m ∈ M generates M . (2) M is simple if and only if M ≃ R/m for some maximal ideal m, and if so, then m = Ann(M ). (3) If M has finite length, then M is finitely generated. Theorem (19.3) (Jordan–Hölder). — Let R be a ring, and M a module with a composition series (19.1.1). Then any chain of submodules can be refined to a composition series, and every composition series is of the same length ℓ(M ). Also, Supp(M ) = { m ∈ Spec(R) | m = Ann(Mi−1 /Mi ) for some i }; the m ∈ Supp(M ) are maximal; there is a canonical isomorphism ∏ ∼ M −→ m∈Supp(M ) Mm ; and ℓ(Mm ) is equal to the number of i with m = Ann(Mi−1 /Mi ). Proof: First, let M ′ be a proper submodule of M . Let’s show that ℓ(M ′ ) < ℓ(M ). ′ To do so, set Mi′ := Mi ∩ M ′ . Then Mi−1 ∩ Mi = Mi′ . So ′ ′ + Mi )/Mi ⊂ Mi−1 /Mi . /Mi′ = (Mi−1 Mi−1 September 3, 2012 11Nts.tex (19.3.1) 98 19. Length ′ ′ Since Mi−1 /Mi is simple, either Mi−1 /Mi′ = 0, or Mi−1 /Mi′ = Mi−1 /Mi and so ′ Mi−1 + Mi = Mi−1 . ′ (19.3.2) ′ If (19.3.2) holds and if Mi ⊂ M , then Mi−1 ⊂ M . Hence, if (19.3.2) holds for ′ all i, then M ⊂ M ′ , a contradiction. Therefore, there is an i with Mi−1 /Mi′ = 0. ′ ′ ′ ′ ′ ′ Now, M = M0 ⊃ · · · ⊃ Mm = 0. Omit Mi if Mi−1 /Mi = 0. Thus M ′ has a composition series of length strictly less than m. Therefore, ℓ(M ′ ) < m for any choice of (19.1.1). Thus (19.3.1) holds. Next, given a chain N0 ⫌ · · · ⫌ Nn = 0, let’s prove n ≤ ℓ(M ) by induction on ℓ(M ). If ℓ(M ) = 0, then M = 0; so also n = 0. Assume ℓ(M ) ≥ 1. If n = 0, then we’re done. If n ≥ 1, then ℓ(N1 ) < ℓ(M ) by (19.3.1); so n − 1 ≤ ℓ(N1 ) by induction. Thus n ≤ ℓ(M ). If Ni−1 /Ni is not simple, then there is N ′ with Ni−1 ⫌ N ′ ⫌ Ni . The new chain can have length at most ℓ(M ) by the previous paragraph. Repeating, we can refine the given chain into a composition series in at most ℓ(M ) − n steps. Suppose the given chain is a composition series. Then ℓ(M ) ≤ n by (19.1.2). But we proved n ≤ ℓ(M ) above. Thus n = ℓ(M ), and the first assertion is proved. To proceed, fix a prime p. Exactness of Localization, (12.16), yields this chain: Mp = (M0 )p ⊃ (M1 )p ⊃ · · · ⊃ (Mm )p = 0. (19.3.3) Now, consider a maximal ideal m. If p = m, then (R/m)p ≃ R/m by (12.4). If p ̸= m, then there is s ∈ m − p; so (R/m)p = 0. Set mi := Ann(Mi−1 /Mi ). So Mi−1 /Mi ≃ R/mi and mi is maximal by (19.2)(2). Then Exactness of Localization yields (Mi−1 /Mi )p = (Mi−1 )p /(Mi )p . Hence { 0, if p ̸= mi ; (Mi−1 )p /(Mi )p = Mi−1 /Mi ≃ R/mi , if p = mi . Thus Supp(M ) = {m1 , . . . , mm }. If we omit the duplicates from the chain (19.3.3), then we get a composition series from the (Mi )p with Mi−1 /Mi ≃ R/p. Thus ∏ the number of such i is ℓ(Mp ). Finally, consider the canonical map φ : M → m∈Supp(M ) Mm . To prove φ is an isomorphism, it suffices, by (13.20), to prove φp is for each maximal ideal p. Now, localization commutes with finite product by (12.10). Therefore, (∏ ) ∏ φp : Mp −→ m Mm p = m (Mm )p = Mp as (Mm )p = 0 if m ̸= p and (Mm )p = Mp if m = p by the above. Thus φp = 1. □ Exercise (19.4). — Let R be a Noetherian ring, M a finitely generated module. Prove the equivalence of the following three conditions: (1) that M has finite length; (2) that Supp(M ) consists entirely of maximal ideals; (3) that Ass(M ) consists entirely of maximal ideals. Prove that, if the conditions hold, then Ass(M ) and Supp(M ) are equal and finite. Exercise (19.5). — Let R be a Noetherian ring, q a p-primary ideal. Consider chains of primary ideals from q to p. Show (1) all such chains have length at most ℓ(A) where A := (R/q)p and (2) all maximal chains have length exactly ℓ(A). Corollary (19.6). — A module M is both Artinian and Noetherian if and only if M is of finite length. September 3, 2012 11Nts.tex 19. Length 99 Proof: Any chain M ⊃ N0 ⫌ · · · ⫌ Nn = 0 has n < ℓ(M ) by the Jordan–Hölder Theorem, (19.3). So if ℓ(M ) < ∞, then M satisfies both the dcc and the acc. Conversely, assume M is both Artinian and Noetherian. Form a chain as follows. Set M0 := M . For i ≥ 1, if Mi−1 ̸= 0, take a maximal Mi ⫋ Mi−1 by the maxc. By the dcc, this recursion terminates. Then the chain is a composition series. □ Example (19.7). — Any simple Z-module is finite owing to (19.2)(2). Hence, a Z-module is of finite length if and only if it is finite. In particular, ℓ(Z) = ∞. Of course, Z is Noetherian, but not Artinian. / Let p ∈ Z be a prime, and set M := Z[1/p] Z. Then M is an Artinian Z-module, but not Noetherian by (16.25). Since M is infinite, ℓ(M ) = ∞. Exercise (19.8). — Let k be a field, and R a finitely generated k-algebra. Prove that R is Artinian if and only if R is a finite-dimensional k-vector space. Theorem (19.9) (Additivity of Length). — Let M be a module, and M ′ a submodule. Then ℓ(M ) = ℓ(M ′ ) + ℓ(M/M ′ ). Proof: If M has a composition series, then the Jordan–Hölder Theorem yields another one of the form M = M0 ⊃ · · · ⊃ M ′ ⊃ · · · ⊃ Mm = 0. The latter yields a pair of composition series: M/M ′ = M0 /M ′ ⊃ · · · ⊃ M ′ /M ′ = 0 and M ′ ⊃ · · · ⊃ Mm = 0. Conversely, every such pair arises from a unique composition series in M through M ′ . Therefore, ℓ(M ) < ∞ if and only if ℓ(M/M ′ ) < ∞ and ℓ(M ′ ) < ∞; furthermore, if so, then ℓ(M ) = ℓ(M ′ ) + ℓ(M/M ′ ), as desired. □ Exercise (19.10). — Let k be a field, A a local k-algebra. Assume the map from k to the residue field is bijective. Given an A-module M , prove ℓ(M ) = dimk (M ). Theorem (19.11) (Akizuki). — A ring R is Artinian if and only if R is Noetherian and dim(R) = 0. If so, then R has only finitely many primes. Proof: If dim(R) = 0, then every prime is maximal. If also R is Noetherian, then R has finite length by (19.4). Thus R is Artinian by (19.6). Conversely, suppose R is Artinian. Let m be a minimal product of maximal ideals of R. Then m2 = m. Let S be the set of ideals a contained in m such that am ̸= 0. If S ̸= ∅, take a ∈ S minimal. Then am2 = am ̸= 0; hence, am = a by minimality of a. For any x ∈ a, if xm ̸= 0, then a = ⟨x⟩ by minimality of a. Let n be any maximal ideal. Then nm = m by minimality of m. But nm ⊂ n. Thus m ⊂ rad(R). But a = ⟨x⟩. So Nakayama’s Lemma yields a = 0, a contradiction. So xm = 0 for any x ∈ a. Thus am = 0, a contradiction. Hence S = ∅. Therefore, m2 = 0. But m2 = m. Thus m = 0. Say m = m1 · · · mr with mi maximal. Set ai := m1 · · · mi for 1 ≤ i ≤ r. Consider the chain R =: a0 ⊃ a1 ⊃ · · · ⊃ ar = 0. Fix i. Set Vi := ai−1 /ai . Then Vi is a vector space over R/mi . Given linearly independent elements x1 , x2 , . . . ∈ Vi , let Wj ⊂ Vi be the subspace spanned by xj , xj+1 , . . . . The Wj form a descending chain. It must stabilize as R is Artinian. Thus dim(Vi ) < ∞. Hence ℓ(R) < ∞ by (19.9). So R is Noetherian by (19.6). So, by (19.4), every prime is maximal, and there are only finitely many primes. □ Exercise (19.12). — Prove these conditions on a Noetherian ring R equivalent: (1) that R is Artinian; September 3, 2012 11Nts.tex 100 19. Length (2) that Spec(R) is discrete and finite; (3) that Spec(R) is discrete. Exercise (19.13). — Let R be an Artinian ring. Show that rad(R) is nilpotent. Corollary (19.14). — Let R be an Artinian ring, and M a finitely generated module. Then M has finite length, and Ass(M ) and Supp(M ) are equal and finite. Proof: By (19.11) every prime is maximal, so Supp(M ) consists of maximal ideals. Also R is Noetherian by (19.11). Hence (19.4) yields the assertions. □ Corollary (19.15). — A ring R is Artinian if and only if ℓ(R) < ∞. Proof: Simply take M := R in (19.14) and (19.6). □ Exercise (19.16). — Let R be a ring, p a prime ideal, and R′ a module-finite R-algebra. Show that R′ has only finitely many primes p′ over p, as follows: reduce to the case that R is a field by localizing at p and passing to the residue rings. Corollary (19.17). — A ring R is Artinian if and only if R is a finite product ∏ of Artinian local rings; if so, then R = m∈Spec(R) Rm . Proof: A finite product of rings is Artinian if and only if each factor is Artinian ∏ by (16.23)(3). If R is Artinian, then ℓ(R) < ∞ by (19.15); whence, R = Rm by the Jordan–Hölder Theorem. Thus the assertion holds. □ Exercise (19.18). — Let R be a Noetherian ring, and M a finitely generated module. Prove the following four conditions are equivalent: (1) that M has finite length; ∏ (2) that M is annihilated by some finite product of maximal ideals mi ; (3) that every prime p containing Ann(M ) is maximal; (4) that R/Ann(M ) is Artinian. September 3, 2012 11Nts.tex 20. Hilbert Functions The Hilbert Function of a graded module lists the lengths of its components. The corresponding generating function is called the Hilbert Series. This series is, under suitable hypotheses, a rational function, according to the Hilbert–Serre Theorem, which we prove. Passing to an arbitrary module, we study its Hilbert– Samuel Series, namely, the generating function of the colengths of the submodules in a filtration. We prove Samuel’s Theorem: if the ring is Noetherian, if the module is finitely generated, and if the filtration is stable, then the Hilbert–Samuel Series is a rational function with poles just at 0 and 1. In the same setup, we prove the Artin–Rees Lemma: given any submodule, its induced filtration is stable. In a brief appendix, we study further one notion that arose: homogeneity. (20.1) (Graded rings and modules). — We ⊕ call a ring R graded if there are additive subgroups Rn for n ≥ 0 with R = Rn and Rm Rn ⊂ Rm+n for all m, n. For example, a polynomial ring R with coefficient ring R0 is graded if Rn is the R0 -submodule generated by the monomials of (total) degree n. In general, R0 is a subring. Obviously, R0 is closed ∑under addition and under multiplication, but we must check 1 ∈ R . So say 1 = xm with xm ∑ ∈ Rm . Given 0 ∑ z ∈ R, say z = z∑ xm zn with n with zn ∈ Rn . Fix n. Then zn = 1 · zn = xm zn ∈ Rm+n . So m>0 xm zn = zn − x0 zn ∈ Rn . Hence xm zn = 0 for m > 0. But n is arbitrary. So xm z = 0 for m > 0. But z is arbitrary. Taking z := 1 yields xm = xm · 1 = 0 for m > 0. Thus 1 = x0 ∈ R0 . We call an R-module ⊕ M (compatibly) graded if there are additive subgroups Mn for n ∈ Z with M = Mn and Rm Mn ⊂ Mm+n for all m, n. We call Mn the nth homogeneous component; we say its elements are homogeneous. Obviously, Mn is an R0 -module. ⊕ Given m ∈ Z, set M (m) := Mm+n . Then M (m) is another graded module; its nth graded component M (m)n is Mm+n . Thus M (m) is obtained from M by shifting m places to the left. ⊕ ⊕ Lemma (20.2). — Let R = Rn be a graded ring, and M = Mn a graded R-module. If R is a finitely generated R0 -algebra and if M is a finitely generated R-module, then each Mn is a finitely generated R0 -module. ∑ Proof: Say R = R0 [x1 , . . . , xr ]. If xi = j xij with xij ∈ Rj , then replace the xi by the nonzero xij . Similarly, say M is generated over R by m1 , . . . , ms∑ with ∑ mi ∈ Mli . Then any m ∈ Mn is a sum m = fi mi where fi ∈ R. Say fi = fij with fij ∈ Rj , and replace fi by fik with k := n − li or by 0 if n < li . Then fi is an R0 -linear combination of monomials xi11 · · · xirr ∈ Rk ; hence, m is one of the □ products xi11 · · · xirr mi ∈ Mn . Thus Mn is a finitely generated R0 -module. ⊕ ⊕ (20.3) (Hilbert functions). — Let R = Rn be a graded ring, and M = Mn a graded R-module. Assume R0 is Artinian, R is a finitely generated R0 -algebra, and M is a finitely generated R-module. Then each Mn is a finitely generated R0 -module by (20.2), so is of finite length ℓ(Mn ) by (19.14). We call n 7→ ℓ(Mn ) the Hilbert Function of M and its generating function ∑ H(M, t) := n∈Z ℓ(Mn )tn September 3, 2012 11Nts.tex 102 20. Hilbert Functions the Hilbert Series of M . This series is a rational function by (20.7) below. If R = R0 [x1 , . . . , xr ] with xi ∈ R1 , then by (20.8) below, the Hilbert Function is, for n ≫ 0, a polynomial h(M, n), which we call the Hilbert Polynomial of M. Example (20.4). — Let R := R0 [X1 , . . . , Xr ] be the polynomial ring, graded by ( ) degree. Then Rn is free over R0 on the monomials of degree n, so of rank r−1+n . r−1 (r−1+n) Assume R0 is Artinian. Then ℓ(Rn ) = ℓ(R0 ) r−1 by Additivity of Length, (19.9). Thus the Hilbert Function is,) for n ≥ 0, ( a )polynomial of degree r − 1. ( n −r = (−1) Formal manipulation yields r−1+n n . Therefore, Newton’s binomial r−1 theorem for negative exponents yields this computation for the Hilbert Series: )n ∑ / ( (−r) ∑ n r H(R, t) = n≥0 ℓ(R0 ) r−1+n n≥0 ℓ(R0 ) n (−t) = ℓ(R0 ) (1 − t) . r−1 t = Exercise (20.5). — Let k be a field, k[X, Y ] the polynomial ring. Show ⟨X, Y 2 ⟩ and ⟨X 2 , Y 2 ⟩ have different Hilbert Series, but the same Hilbert Polynomial. ⊕ ⊕ Exercise (20.6).⊕ — Let R = Rn be a graded ring, M = Mn a graded Rmodule. Let N = Nn be a homogeneous submodule; that is, Nn = N ∩ Mn . Assume R0 is Artinian, R is a finitely generated R0 -algebra, and M is a finitely generated R-module. Set N ′ := { m ∈ M | there is k0 such that Rk m ∈ N for all k ≥ k0 }. (1) Prove that N ′ is a homogeneous submodule of M with the same Hilbert ′ Polynomial as N ∩, and that N is the largest such submodule. ⊕ (2) Let N = ∩ Qi be a decomposition with Qi pi -primary. Set R+ := n>0 Rn . Prove that N ′ = pi ̸⊃R+ Qi . ⊕ Theorem Rn be a graded ring, and let ⊕ (20.7) (Hilbert–Serre). — Let R = M= Mn be a graded R-module. Assume R0 is Artinian, R is a finitely generated R0 -algebra, and M is a finitely generated R-module. Then / H(M, t) = e(t) tl (1 − tk1 ) · · · (1 − tkr ) with e(t) ∈ Z[t], with l ≥ 0, and with k1 , . . . , kr ≥ 1. Proof: Say R = R0 [x1 , . . . , xr ] with xi ∈ Rki . First, assume r = 0. Say M is generated over R by m1 , . . . , ms with mi ∈ Mli . Then R = R0 . So Mn = 0 for n < l0 := min{li } and for n > max{li }. Hence t−l0 H(M, t) is a polynomial. Next, assume r ≥ 1 and form the exact sequence µx 1 0 → K → M (−k1 ) −−→ M →L→0 where µx1 is the map of multiplication by x1 . Since x1 ∈ Rk1 , the grading on M induces a grading on K and on L. Further, µx1 acts as 0 on both K and L. Since R0 is Artinian, R0 is Noetherian by Akizuki’s Theorem, (19.11). So, since R is a finitely generated R0 -algebra, R is Noetherian by (16.11). Since M is a finitely generated R-module, obviously so is M (−k1 ). Hence, so are both K and L by (16.15)(2). Set R′ := R0 [x2 , . . . , xr ]. Since x1 acts as 0 on K and L, they are finitely generated R′ -modules. Therefore, H(K, t) and H(L, t) are defined, and they may be written in the desired form by induction on r. By definition, M (−k1 )n := Mn−k1 ; hence, H(M (−k1 ), t) = tk1 H(M, t). Therefore, Additivity of Length, (19.9), and the previous paragraph yield / (1 − tk1 )H(M, t) = H(L, t) − H(K, t) = e(t) tl (1 − tk2 ) · · · (1 − tkr ). September 3, 2012 11Nts.tex 20. Hilbert Functions 103 Thus the assertion holds. □ Corollary (20.8). — Under the conditions of (20.7), assume that M ̸= 0 and R = R0 [x/1 , . . . , xr ] with xi ∈ R1 . Then H(M, t) can be written uniquely in the form e(t) tl (1 − t)d where e(t) ∈ Z[t] with e(0) ̸= 0 and e(1) ̸= 0 and where l ∈ Z and r ≥ d ≥ 0; further, there is a polynomial h(M, n) ∈ Q[n] with degree d − 1 and leading coefficient e(1)/(d − 1) ! such that ℓ(Mn ) = h(M, n) for n ≥ deg(e(t)) − l. Proof: We may / take ki = 1 for all i in the proof of (20.7). Hence H(M, t) is of the form e(t)(1 − t)s tl (1 − t)r with e(0) ̸= 0 and e(1) ̸= 0 and l ∈ Z. Set d := r − s. Then d ≥ 0 since H(M, 1) > 0 as M ̸= 0. Thus H(M, t) is of the asserted form. This form is unique owing to the uniqueness of factorization of polynomials. ∑ (d−1+n) n ∑N ∑ (−d) n Say e(t) = i=0 ei ti . Now, (1 − t)−d = d−1 t . Hence n (−t) = (d−1+n−i) (d−1+n+l−i) ∑N d−1 = n /(d − 1) ! + · · · . for n + l ≥ N . But ℓ(Mn ) = i=0 ei d−1 d−1 d−1 Therefore, ℓ(Mn ) = e(1)n /(d − 1) ! + · · · , as asserted. □ Exercise (20.9). — Let k be a field, P := k[X, Y, Z] the polynomial ring in three variables, f ∈ P a homogeneous polynomial of degree d ≥ 1. Set R := P/⟨f ⟩. Find the coefficients of the Hilbert Polynomial h(R, n) explicitly in terms of d. Exercise (20.10). — Under the conditions of (20.8),( assume) there is ( a homo) geneous nonzerodivisor f ∈ R with Mf = 0. Prove deg h(R, n) > deg h(M, n) ; start with the case M := R/⟨f k ⟩. (20.11) (Filtrations). — Let R be an arbitrary ring, q an ideal, and M a module. A filtration of M is an infinite descending chain of submodules: M = M0 ⊃ M1 ⊃ M2 ⊃ · · · . (20.11.1) We call it a q-filtration if qMn ⊂ Mn+1 for all n, and a stable q-filtration if also qMn = Mn+1 for n ≫ 0, or equivalently, if also there is an m with qn Mm = Mn+m for n ≥ 0. For example, setting Mn := qn M , we get a stable q-filtration; we call it the q-adic filtration. The q-adic filtration of R yields a graded ring Gq (R) or G(R), defined by ⊕ G(R) := n≥0 G(R)n where G(R)n := qn /qn+1 . We obtain the product of an element in qi /qi+1 and one in qj /qj+1 by choosing a representative of each, forming their product, and taking its residue in qi+j /qi+j+1 . We call G(R) the associated graded ring. Similarly, if (20.11.1) is a q-filtration, then we obtain a graded G(R)-module ⊕ Gq (M ) := G(M ) := n≥0 G(M )n where G(M )n := Mn /Mn+1 . If all the quotients M/Mn of the filtration (20.11.1) are of finite length, then we call n 7→ ℓ(M/Mn ) the Hilbert–Samuel Function, and the generating function ∑ P (M• , t) := n≥0 ℓ(M/Mn )tn the Hilbert–Samuel Series. If the function n 7→ ℓ(M/Mn ) is, for n ≫ 0, a polynomial p(M• , n), then we call it the Hilbert–Samuel Polynomial. If the filtration is the q-adic filtration, we also denote P (M• , t) and p(M• , n) by Pq (M, t) and pq (M, n). September 3, 2012 11Nts.tex 104 20. Hilbert Functions Lemma (20.12). — Let R be a Noetherian ring, q an ideal, M a finitely generated module with a stable q-filtration. Then G(R) is generated as an R/q-algebra by finitely many elements of q/q2 , and G(M ) is a finitely generated G(R)-module. Proof: Since R is Noetherian, q is a finitely generated ideal, say by x1 , . . . , xr . Then, clearly, the residues of the xi in q/q2 generate G(R) as a R/q-algebra. Say the filtration is M = M0 ⊃ M1 ⊃ · · · . Since it is stable, there is an m with qn Mm = Mn+m for n ≥ 0. Hence G(M ) is generated by M0 /M1 , . . . , Mm /Mm+1 over G(R). But R is Noetherian and M is finitely generated over R; hence, every Mi is finitely generated over R. Therefore, every Mn /Mn+1 is finitely generated over R/q. Thus G(M ) is a finitely generated G(R)-module. □ Theorem (20.13) (Samuel). — Let R be a Noetherian ring, q an ideal, M a finitely generated module with a stable q-filtration M = M0 ⊃ M1 ⊃ · · · . Assume ℓ(M/qM ) < ∞. Then ℓ(Mn /Mn+1 ) < ∞ and ℓ(M/Mn ) < ∞ for every n; further, P (M• , t) = H(G(M ), t) t/(1 − t). Proof: Set a := Ann(M ). Set R′ := R/a and q′ := (a + q)/a. Then R′ /q′ is Noetherian as R is. Further, M can be viewed as a finitely generated R′ -module, and M = M0 ⊃ M1 ⊃ · · · as a stable q′ -filtration. So G(R′ ) is generated as a R′ /q′ algebra by finitely many elements of degree 1, and G(M ) is a finitely generated G(R′ )-module by (20.12). Therefore, each Mn /Mn+1 is a finitely generated R′ /q′ module by (20.2) or by the proof of (20.12). On the other hand, (13.1) and (13.9)(3) and (13.13) yield, respectively, ∩ ∩ V(a + q) = V(a) V(q) = Supp(M ) V(q) = Supp(M/qM ). Hence V(a + q) consists entirely of maximal ideals, because Supp(M/qM ) does by (19.4) as ℓ(M/qM ) < ∞. Thus dim(R′ /q′ ) = 0. But R′ /q′ is Noetherian. Therefore, R′ /q′ is Artinian by Akizuki’s Theorem, (19.11). Therefore, ℓ(Mn /Mn+1 ) < ∞ for every n by (19.14). Form the exact sequence 0 → Mn /Mn+1 → M/Mn+1 → M/Mn → 0. Then Additivity of Length, (19.9), yields ℓ(Mn /Mn+1 ) = ℓ(M/Mn+1 ) − ℓ(M/Mn ). So induction on n yields ℓ(M/Mn+1 ) < ∞ for every n. Further, multiplying that equation by tn and summing over n yields the desired expression in another form: H(G(M ), t) = (t−1 − 1)P (M• , t) = P (M• , t) (1 − t)/t. □ Corollary (20.14). — Under the conditions of (20.13), assume q is generated by r/ elements and M ̸= 0. Then P (M• , t) can be written uniquely in the form e(t) tl−1 (1 − t)d+1 where e(t) ∈ Z[t] with e(0) ̸= 0 and e(1) ̸= 0 and where l ∈ Z and r ≥ d ≥ 0; further, there is a polynomial p(M• , n) ∈ Q[n] with degree d and leading coefficient e(1)/d ! such that ℓ(M/Mn ) = p(M• , n) for n ≥ deg(e(t)) − l. Finally, pq (M, n)−p(M• , n) is a polynomial with degree at most d−1 and nonnegative leading coefficient; further, d and e(1) are the same for every stable q-filtration. Proof: The proof of (20.13) shows that G(R′ ) and G(M ) satisfy the hypotheses of (20.8). So (20.8) yields a certain form for H(G(M ), t). Then (20.13) yields the asserted form for P (M• , t). In turn, that form yields the asserted polynomial p(M• , n) by the argument in the second paragraph of the proof of (20.8). September 3, 2012 11Nts.tex 20. Hilbert Functions 105 Finally, as M = M0 ⊃ M1 ⊃ · · · is a stable q-filtration, there’s an m such that Mn ⊃ qn M ⊃ qn Mm = Mn+m for all n ≥ 0. Dividing into M and extracting lengths yields ℓ(M/Mn ) ≤ ℓ(M/qn M ) ≤ ℓ(M/Mn+m ). Therefore, for large n, we get p(M• , n) ≤ pq (M, n) ≤ p(M• , n + m). The two extremes are polynomials in n with the same degree and leading coefficient, say d and c. Dividing by nd and letting n → ∞, we conclude that the polynomial pq (M, n) also has degree d and leading coefficient c. Thus the degree and leading coefficient are the same for every stable q-filtration. Further pq (M, n)−p(M• , n) has degree at most d−1 and positive leading coefficient, owing to cancellation of the two leading terms and to the first inequality. □ Exercise (20.15). — Let R be a Noetherian ring, q an ideal, and M a finitely √ generated module. Assume ℓ(M/qM ) < ∞. Set m := q. Show deg pm (M, n) = deg pq (M, n). (20.16) (Rees Algebras). — Let R be an arbitrary ring, q an ideal. The sum ⊕ R(q) := n≥0 qn is, canonically, a graded ring, with R as zeroth graded component and q as first. We call R(q) the Rees Algebra of q. Let M be a module with a q-filtration M = M0 ⊃ M1 ⊃ · · · . Then the sum ⊕ R(M• ) := n≥0 Mn is canonically a module over the Rees Algebra R(q). Lemma (20.17). — Let R be a Noetherian ring, q an ideal, and M a finitely generated module with a q-filtration. Then R(q) is generated as an R-algebra by finitely many elements of q, and R(M• ) is a finitely generated R(q)-module if and only if the filtration is stable. Proof: Say the filtration is M =∑ M0 ⊃ M1 ⊃ · · · . Suppose R(M• ) is generated µ over R(q) by m1 , . . . , ms . Say mi = j=0 mij with mij ∈ Mj . Then for any n ≥ 0, ∑ any m ∈ Mn+µ is a sum m = fij mij where fij ∈ qn+µ−j . But qn+µ−j = qn qµ−j . n Thus Mn+µ = q Mµ ; that is, the filtration is stable. The rest of the proof is similar to that of (20.12), but simpler. □ Lemma (20.18) (Artin–Rees). — Let R be a Noetherian ring, M a finitely generated module, N a submodule, q an ideal, M = M0 ⊃ M1 ⊃ · · · a stable q-filtration. For n ≥ 0, set Nn := N ∩ Mn . Then N = N0 ⊃ N1 ⊃ · · · is a stable q-filtration. Proof: By (20.17), the Rees Algebra R(q) is finitely generated over R, so Noetherian by (16.11). By (20.17), the module R(M• ) is finitely generated over R(q), so Noetherian by (16.18). Clearly, N = N0 ⊃ N1 ⊃ · · · is a q-filtration; hence, R(N• ) is a submodule of R(M• ), so Noetherian by (16.15)(2), so finitely generated by (16.18). Hence, N = N0 ⊃ N1 ⊃ · · · is stable by (20.17), as desired. □ September 3, 2012 11Nts.tex 106 20. Hilbert Functions Exercise (20.19). — Derive the Krull Intersection Theorem, (18.28), from the Artin–Rees Lemma, (20.18). Proposition (20.20). — Let R be a Noetherian ring, q an ideal, and 0 → M ′ → M → M ′′ → 0 an exact sequence of finitely generated modules. Then M/qM has finite length if and only if M ′ /qM ′ and M ′′ /qM ′′ do. If so, then the polynomial pq (M ′ , n) − pq (M, n) + pq (M ′′ , n) ) has degree at most deg pq (M ′ , n) − 1 and has positive leading coefficient; also then ( deg pq (M, n) = max{ deg pq (M ′ , n), deg pq (M ′′ , n) }. Proof: First off, (13.13) and (13.9)(1) and (13.13) again yield ( )∩ ∩ ∪ Supp(M/qM ) = Supp(M ) V(q) = Supp(M ′ ) Supp(M ′′ ) V(q) ( ) ( ) ∩ ∪ ∩ = Supp(M ′ ) V(q) Supp(M ′′ ) V(q) ∪ ′ ′ = Supp(M /qM ) Supp(M ′′ /qM ′′ ). Hence M/qM has finite length∩if and only if M ′ /qM ′ and M ′′ /qM ′′ do by (19.4). For n ≥ 0, set Mn′ := M ′ qn M . Then M ′ = M0′ ⊃ M1′ ⊃ · · · is a stable q-filtration by the Artin–Rees Lemma. Form this canonical commutative diagram: 0− → Mn′ − → qn M − → qn M ′′ − →0       y y y 0− → M ′ −−→ M −−−→ M ′′ −−→ 0 Its rows are exact. So the Nine Lemma yields this exact sequence: 0 → M ′ /Mn′ → M/qn M → M ′′ /qn M ′′ → 0. Assume M/qM has finite length. Then Additivity of Length and (20.14) yield ′ p(M•′ , n) − pq (M, n) + pq (M ′′ , n) = 0. ′′ ′ (20.20.1) ′ + pq (M , n) is equal to pq (M , n) − p(M ( • , n).′ But ) by a polynomial with degree at most deg pq (M , n) − 1 Hence pq (M , n) − pq (M, n) (20.14) again, the latter is and positive leading coefficient. Finally, deg pq (M, n) = max{deg p(M•′ , n), deg pq (M ′′ , n)} owing to (20.20.1), as the leading coefficients of p(M•′ , n) and pq (M ′′ , n) are both positive, so cannot cancel. But deg p(M•′ , n) = deg pq (M ′ , n) by (20.14), completing the proof. □ September 3, 2012 11Nts.tex 20. Appendix: Homogeneity ⊕ (20.21) (Homogeneity). — Let R be a graded ring, and M = Mn a graded module. We call the Mn the homogeneous components of M . ∑ Given m ∈ M , write m = mn with mn ∈ Mn . Call the finitely many nonzero mn the homogeneous components of m. Say that a component mn is homogeneous of degree n. If n is lowest, call mn the initial component of m. Call a submodule ⊕N ⊂ M homogeneous if, whenever m ∈ N , also mn ∈ N , or equivalently, N = (Mn ∩ N ). Call a map α : M ′ → M of graded modules with components Mn′ and Mn homogeneous of degree r if α(Mn′ ) ⊂ Mn+r for all n. If so, then clearly Ker(α) is a homogeneous submodule of M . Further, Coker(α) is canonically graded, and the quotient map M → Coker(α) is homogeneous of degree 0. ⊕ ⊕ Exercise (20.22). — Let R = Rn be a graded ring, M = n≥n0 Mn a graded ⊕ module, a ⊂ n>0 Rn a homogeneous ideal. Assume M = aM . Show M = 0. ⊕ ⊕ Exercise (20.23). — Let R = Rn be ⊕ a Noetherian graded ring, M = Mn a finitely generated graded R-module, N = Nn a homogeneous submodule. Set N ′ := { m ∈ M | Rn m ∈ N for all n ≫ 0 }. Show that N ′ is the largest homogeneous submodule of M containing N and having, for all n ≫ 0, its degree-n homogeneous component Nn′ equal to Nn . Proposition (20.24). — Let R be a Noetherian graded ring, M a nonzero finitely generated graded module, Q a homogeneous submodule. Suppose Q possesses this property: given any homogeneous x ∈ R and homogeneous m ∈ M with xm ∈ Q but m ∈ / Q, necessarily x ∈ p := nil(M/Q). Then p is prime, and Q is p-primary. Proof: Given x∑∈ R and m ∈ M ∑ , decompose them into their homogeneous components: x = x and m = xm ∈ Q, but m ∈ / Q. i i≥r j≥s mj . Suppose ∑ Then mt ∈ / Q for some t; take t minimal. Set m′ := j<t mj . Then m′ ∈ Q. Set m′′ := m − m′ . Then xm′′ ∈ Q. Either xs mt vanishes or it’s the initial component of xm′′ . But Q is homogeneous. So xs mt ∈ Q. But mt ∈ / Q.∑Hence xs ∈ p by the hypothesis. Say xs , . . . , xu ∈ p u with u maximal. Set x′ := i=s xi . Then x′ ∈ p. So x′k ∈ Ann(M/Q) for some k ≥ 1. So x′k m′′ ∈ Q. Set x′′ := x − x′ . Since xm′′ ∈ Q, also x′′k m′′ ∈ Q. Suppose x ∈ / p. Then x′′ ̸= 0. And its initial component is xv with v > u. Either ′′ ′′ xv mt vanishes or it is the initial component of xm. But Q is homogeneous. So xv mt ∈ Q. But mt ∈ / Q. Hence xv ∈ p by the hypothesis, contradicting v > u. Thus x ∈ p. Thus Q is p-primary by (18.4). □ Exercise (20.25). — Let √ R be a graded ring, a a homogeneous ideal, and M a graded module. Prove that a and Ann(M ) and nil(M ) are homogeneous. Exercise (20.26). — Let R be a graded ring, M a graded module, and Q a primary submodule. Let Q∗ ⊂ Q be the submodule generated by the homogeneous elements of Q. Then Q∗ is primary. 107 108 20. Appendix: Homogeneity Theorem (20.27). — Let R be a Noetherian graded ring, M a finitely generated graded module, N a homogeneous submodule. Then all the associated primes of M/N are homogeneous, and N admits an irredundant primary decomposition in which all the primary submodules are homogeneous. ∩ Proof: Let N = Qj be any primary decomposition; one exists by (18.20). Let Q∗j ⊂∩Qj be the by the homogeneous elements of Qj . ∩ submodule∩generated ∗ ∗ ∗ Trivially, Qj ⊂ Qj = N ⊂ Qj . Further, ∩ ∗ each Qj is clearly homogeneous, and is primary by (20.26). Thus N = Qj is a primary decomposition into homogeneous primary submodules. And, owing to (18.18), it is irredundant if ∩ N = Qj is, as both decompositions have minimal length. Finally, M/Q∗j is graded by (20.21); so each associated prime is homogeneous by (18.19) and (20.25). □ ⊕ (20.28) (Graded Domains). — Let R = n≥0 Rn be a graded domain, and set K := Frac(R). We call z ∈ K homogeneous of degree n ∈ Z if z = x/y with x ∈ Rm and y ∈ Rm−n . Clearly, n is well defined. Let Kn be the Km Kn ⊂ Km+n . Clearly, the ⊕ set of all such z, plus 0. Then⊕ canonical map n∈Z Kn → K is injective. Thus n≥0 Kn is a graded subring of K. Further, K0 is a field. The n with Kn ̸= 0 form a subgroup of Z. So by renumbering, we may assume K1 ̸= 0. Fix any nonzero x ∈ K1 . Clearly, x is transcendental over K0 . If z ∈ Kn , then z/xn ∈⊕ K0 . Hence R ⊂ K0 [x]. So (2.3) yields K = K0 (x). Any w ∈ Kn can be written w = a/b with a, b ∈ ∏ R and b homogeneous: say ∑ ∑ w = (an /bn ) with an , bn ∈ R homogeneous; set b := bn and a := (an b/bn ). Theorem (20.29). — Let R be a Noetherian graded domain, K := Frac(R), and R the integral closure of R in K. Then R is a graded subring of K. Proof: Use the setup of (20.28). Since K0 [x] is a polynomial ring over a field, it is normal by (10.29). Hence R ⊂ K0 [x]. So every y ∈ R can be written as ∑r+n y = i=r yi , with yi homogeneous and nonzero. Let’s show yi ∈ R for all i. Since y is integral over R, the R-algebra R[y] is module finite by (10.18). So (20.28) yields a homogeneous b ∈ R with bR[y] ⊂ R. Hence by j ∈ R for all j ≥ 0. But R is graded. Hence byrj ∈ R. Set z := 1/b. Then yrj ∈ Rz. Since R is Noetherian, the R-algebra R[yr ] is module finite. Hence yr ∈ R. Then y − yr ∈ R. □ Thus yi ∈ R for all i by induction on n. Thus R is graded. Exercise (20.30). — Under the conditions of (20.8), assume that R is a domain and that its integral closure R in Frac(R) is a finitely generated R-module. (1) Prove that there is a homogeneous f ∈ R with Rf = Rf . (2) Prove that the Hilbert Polynomials of R and R have the same degree and same leading coefficient. 21. Dimension The dimension of a module is defined as the sup of the lengths of the chains of primes in its support. The Dimension Theorem, which we prove, characterizes the dimension of a nonzero finitely generated semilocal module over a Noetherian ring in two ways. First, the dimension is the degree of the Hilbert–Samuel Polynomial formed with the radical of the ring. Second, the dimension is the smallest number of elements in the radical that span a submodule of finite colength. Next, in an arbitrary Noetherian ring, we study the height of a prime: the length of the longest chain of subprimes. We bound the height by the minimal number of generators of an ideal over which the prime is minimal. In particular, when this number is 1, we obtain Krull’s Principal Ideal Theorem. Finally, we study regular local rings: Noetherian local rings whose maximal ideal has the minimum number of generators, namely, the dimension. (21.1) (Dimension of a module). — Let R be a ring, and M a nonzero module. The dimension of M , denoted dim(M ), is defined by this formula: dim(M ) := sup{ r | there’s a chain of primes p0 ⫋ · · · ⫋ pr in Supp(M ) }. Assume R is Noetherian, and M is finitely generated. Then M has finitely many minimal (associated) primes by (17.19). They are also the minimal primes p0 ∈ Supp(M ) by (17.16). Thus (1.8) yields dim(M ) = max{ dim(R/p0 ) | p0 ∈ Supp(M ) is minimal }. (21.1.1) (21.2) (Parameters). — Let R be a ring, M a nonzero module. Denote the intersection of the maximal ideals in Supp(M ) by rad(M ), and call it the radical of M . If there are only finitely many such maximal ideals, call M semilocal.Call an ideal q a parameter ideal of M if q ⊂ rad(M ) and M/qM is Artinian. Assume M is finitely generated. Then Supp(M ) = V(Ann(M )) by (13.9)(3). Hence M is semilocal if and only if R/ Ann(M ) is a semilocal ring. Assume, in addition, R is Noetherian; so M is Noetherian by (16.18). Fix an ideal q. Then (19.6) yields that M/qM is Artinian if and only if ℓ(M/qM ) < ∞. However, ℓ(M/qM ) < ∞ if and only if Supp(M/qM ) consists of finitely many maximal ideals by (19.4) and (17.20). Further, (13.13), (13.9)(3), and (13.1) yield ∩ ∩ Supp(M/qM ) = Supp(M ) V(q) = V(Ann(M )) V(q) = V(Ann(M ) + q). Set q′ := Ann(M ) + q. Thus M/qM is Artinian if and only if V(q′ ) consists of finitely many maximal ideals; so by (19.11), if and only if R/q′ is Artinian. But (19.18) implies that R/q′ is Artinian if and only if q′ contains a product of maximal ideals each of which contains q′ . Then each lies in Supp(M ), so contains rad(M ). Set m := rad(M ). Thus if R/q′ is Artinian, then q′ ⊃ mn for some n > 0. Assume, in addition, M is semilocal, so that Supp(M ) contains only finitely many maximal ideals. Then their product is contained in m. Thus, conversely, if q′ ⊃ mn for some n > 0, then R/q′ is Artinian. Thus q is a parameter ideal if and only if m ⊃ q′ ⊃ mn for some n, September 3, 2012 11Nts.tex (21.2.1) 110 21. Dimension √ or by (3.26) if and only if m = q′ , or by (13.1) if and only if V(m) = V(q′ ). In particular, mn is a parameter ideal for any n. Assume q is a parameter ideal. Then the Hilbert–Samuel polynomial pq (M, n) exists by (20.14). Similarly, pm (M, √ n) exists, and the two polynomials have the same degree by (20.15) since m = q′ and pq′ (M, n) = pq (M, n). Thus the degree is the same for every parameter ideal. Denote this common degree by d(M ). Alternatively, d(M ) can be viewed as the order of pole at 1 of the Hilbert series H(Gq (M ), t). Indeed, that order is 1 less than the order of pole at 1 of the Hilbert– Samuel series Pq (M, t) by (20.13). In turn, the latter order is d(M )+1 by (20.14). Denote by s(M ) the smallest s such that there are x1 , . . . , xs ∈ m with ℓ(M/⟨x1 , . . . , xs ⟩M ) < ∞. (21.2.2) By convention, if ℓ(M ) < ∞, then s(M ) = 0. We say that x1 , . . . , xs ∈ m form a system of parameters (sop) for M if s = s(M ) and (21.2.2) holds. Note that a sop generates a parameter ideal. Lemma module, (1) (2) (21.3). — Let R be a Noetherian ring, M a nonzero Noetherian semilocal µx q a parameter ideal of M , and x ∈ rad(M ). Set K := Ker(M −−→ M ). Then s(M ) ≤ s(M/xM ) + 1. Then dim(M/xM ) ≤ dim(M ) − 1 if x ∈ / p for any p ∈ Supp(M ) with dim(R/p)(= dim(M ). ) (3) Then deg pq (K, n) − pq (M/xM, n) ≤ d(M ) − 1. Proof: For (1), set s := s(M/xM ). There are x1 , . . . , xs ∈ rad(M/xM ) with ℓ(M/⟨x, x1 , . . . , xs ⟩M ) < ∞. Now, Supp(M/xM ) = Supp(M ) ∩ V(⟨x⟩) by (13.13). However, x ∈ rad(M ). Hence, Supp(M/xM ) and Supp(M ) have the same maximal ideals. Therefore, rad(M/xM ) = rad(M ). Hence s(M ) ≤ s + 1. Thus (1) holds. To prove (2), take a chain of primes p0 ⫋ · · · ⫋ pr in Supp(M/xM ). Now, Supp(M/xM ) = Supp(M ) ∩ V(⟨x⟩) by (13.13). So x ∈ p0 ∈ Supp(M ). So, by hypothesis, dim(R/p0 ) < dim(M ). Hence r ≤ dim(M ) − 1. Thus (2) holds. To prove (3), set xM := Im(µx ), and form these two exact sequences: 0 → K → M → xM → 0, and 0 → xM → M → M/xM → 0. Then (20.20) yields d(K) ≤ d(M ) and d(xM ) ≤ d(M ). So by (20.20) again, both pq (K, n) + pq (xM, n) − pq (M, n) and pq (xM, n) + pq (M/xM, n) − pq (M, n) are of degree at most d(M ) − 1. So their difference is too. Thus (3) holds. □ Theorem (21.4) (Dimension). — Let R be a Noetherian ring, M a nonzero finitely generated semilocal module. Then dim(M ) = d(M ) = s(M ) < ∞. Proof: Let’s prove a cycle of inequalities. Set m := rad(M ). First, let’s prove dim(M ) ≤ d(M ). We proceed by induction on d(M ). Suppose d(M ) = 0. Then ℓ(M/mn M ) stabilizes. So mn M = mn+1 M for some n. Hence mn M = 0 by Nakayama’s Lemma applied over the semilocal ring R/ Ann(M ). Hence ℓ(M ) < ∞. So dim(M ) = 0 by (19.4). Suppose d(M ) ≥ 1. Take p0 ∈ Ass(M ) with dim(R/p0 ) = dim(M ). Then M has a submodule N isomorphic to R/p0 by (17.2). Further, d(N ) ≤ d(M ) by (20.20). Take a chain of primes p0 ⫋ · · · ⫋ pr in Supp(N ). If r = 0, then r ≤ d(M ). September 3, 2012 11Nts.tex 21. Dimension 111 Suppose r ≥ 1. Then there’s an x1 ∈ p1 − p0 . Further, since p0 is not maximal, for ∏ each maximal ideal n in Supp(M ), there is an xn ∈ n − p∩0 . Set x := x1 xn . Then x ∈ (p1 ∩ m) − p0 . Then p1 ⫋ · · · ⫋ pr lies in Supp(N ) V(⟨x⟩). But the latter is equal to Supp(N/xN ) by (13.13). So r − 1 ≤ dim(N/xN ). However, µx is injective on N as N ≃ R/p0 and x ∈ / p0 . So (21.3)(3) yields d(N/xN ) ≤ d(N ) − 1. But d(N ) ≤ d(M ). So dim(N/xN ) ≤ d(N/xN ) by the induction hypothesis. Therefore, r ≤ d(M ). Thus dim(M ) ≤ d(M ). Second, let’s prove d(M ) ≤ s(M ). Let q be a parameter ideal of M with s(M ) generators. Then d(M ) := deg pq (M, n). But deg pq (M, n) ≤ s(M ) owing to (20.14). Thus d(M ) ≤ s(M ). Finally, let’s prove s(M ) ≤ dim(M ). Set r := dim(M ), which is finite since r ≤ d(M ) by the first step. The proof proceeds by induction on r. If r = 0, then M has finite length by (19.4); so s(M ) = 0. Suppose r ≥ 1. Let p1 , . . . , pk be the primes of Supp(M ) with dim(R/pi ) = r. No pi is maximal as r ≥ 1. So m lies in no pi . Hence, by Prime Avoidance (3.15), there is an x ∈ m such that x ∈ / pi for all i. So (21.3)(1), (2) yield s(M ) ≤ s(M/xM ) + 1 and dim(M/xM ) + 1 ≤ r. By the induction hypothesis, s(M/xM ) ≤ dim(M/xM ). Hence s(M ) ≤ r, as desired. □ Corollary (21.5). — Let R be a Noetherian ring, M a nonzero Noetherian semilocal module, x ∈ rad(M ). Then dim(M/xM ) ≥ dim(M ) − 1, with equality if x ∈ /p for p ∈ Supp(M ) with dim(R/p) = dim(M ); equality holds if x ∈ / z.div(M ). Proof: By (21.3)(1), we have s(M/xM ) ≥ s(M )−1. So the asserted inequality holds by (21.4). If x ∈ / p ∈ Supp(M ) when dim(R/p) = dim(M ), then (21.3)(2) yields the opposite inequality, so equality. Finally, if x ∈ / z.div(M ), then x ∈ / p for any p ∈ Supp(M ) with dim(R/p) = dim(M ) owing to (17.17) and (17.14). □ (21.6) (Height). — Let R be a ring, and p a prime. The height of p, denoted ht(p), is defined by this formula: ht(p) := sup{ r | there’s a chain of primes p0 ⫋ · · · ⫋ pr = p }. The bijective correspondence p 7→ pRp of (11.18)(2) yields this formula: ht(p) = dim(Rp ). (21.6.1) Corollary (21.7). — Let R be a Noetherian ring, p a prime. Then ht(p) ≤ r if and only if p is minimal containing an ideal generated by r elements. Proof: Assume p is minimal containing an ideal a generated by r elements. Now, any prime of Rp containing aRp is of the form √ qRp where q is a prime of R with a ⊂ q ⊂ p by (11.18). So q = p. Hence pRp = aRp by the Scheinnullstellensatz. Hence r ≥ s(Rp ) by (21.2). But s(Rp ) = dim(Rp ) by (21.4), and dim(Rp ) = ht(p) by (21.6.1). Thus ht(p) ≤ r. Conversely, assume ht(p) ≤ r. Then Rp has a parameter ideal b generated by r elements, say y1 , . . . , yr by (21.6.1) and (21.4). Say yi = xi /si with si ∈ / p. Set a := ⟨x1 , . . . , xr ⟩. Then aRp = b. Suppose there is a prime q with √ a ⊂ q ⊂ p. Then b = aRp ⊂ qRp ⊂ pRp , and qRp is prime by (11.18)(2). But b = pRp . So qRp = pRp . Hence q = p by (11.18)(2). Thus p is minimal containing a, which is generated by r elements. □ September 3, 2012 11Nts.tex 112 21. Dimension Exercise (21.8). — Let R be a Noetherian ring, and p be a prime minimal containing x1 , . . . , xr . Given r′ with 1 ≤ r′ ≤ r, set R′ := R/⟨x1 , . . . , xr′ ⟩ and p′ := p/⟨x1 , . . . , xr′ ⟩. Assume ht(p) = r. Prove ht(p′ ) = r − r′ . Theorem (21.9) (Krull Principal Ideal). — Let R be a Noetherian ring, x ∈ R, and p a prime minimal containing x. If x ∈ / z.div(R), then ht(p) = 1. Proof: We have ht(p) ≤ 1 by (21.7). But if ht(p) = 0, then p ∈ Ass(R) by (17.17), and so x ∈ z.div(R) by (17.14). □ Exercise (21.10). — Let R be a Noetherian ring, p a prime ideal with ht(p) ≥ 2. Prove p is the union of infinitely many distinct prime ideals q with ht(q) = 1. Exercise (21.11). — Let R be a Noetherian ring with only finitely many prime ideals. Show dim(R) ≤ 1. Exercise (21.12). — Let R be a domain. Prove that, if R is a UFD, then every height-1 prime is principal, and that the converse holds if R is Noetherian. Exercise (21.13). — (1) Let A be a Noetherian local ring, and p a principal prime of height at least 1. Prove that A is a domain. (2) Let k be a field, P := k[[X]] the formal power series ring in one variable. Set R := P × P . Prove that P is Noetherian and semilocal, and that P contains a principal prime p of height 1, but that P is not a domain. Exercise (21.14). — Let R be a finitely generated algebra over a field. Assume R is a domain of dimension r. Let x ∈ R be neither 0 nor a unit. Set R′ := R/⟨x⟩. Prove that r − 1 is the length of any chain of primes in R′ of maximal length. Corollary (21.15). — Let A and B be Noetherian local rings, m and n their maximal ideals. Let φ : A → B be a local homomorphism. Then dim(B) ≤ dim(A) + dim(B/mB), with equality if B is flat over A. Proof: Set s := dim(A). By (21.4), there is a parameter ideal q generated by s elements. Then m/q is nilpotent by (21.2.1). Hence mB/qB is nilpotent. It follows that dim(B/mB) = dim(B/qB). But (21.5) yields dim(B/qB) ≥ dim(B)−s. Thus the inequality holds. Assume B is flat over A. Let p ⊃ mB be a prime with dim(B/p) = dim(B/mB). Then dim(B) ≥ dim(B/p) + ht(p) because the concatenation of a chain of primes containing p of length dim(B/p) with a chain of primes contained in p of length ht(p) is a chain of primes of B of length ht(p) + dim(B/p). Hence it suffices to show that ht(p) ≥ dim(A). As n ⊃ p ⊃ mB and as φ is local, φ−1 (p) = m. Since B is flat over A, (14.11) and induction yield a chain of primes of B descending from p and lying over any given chain in A. Thus ht(p) ≥ dim(A), as desired. □ Exercise (21.16). — Let R be a Noetherian ring. Prove that dim(R[X]) = dim(R) + 1. September 3, 2012 11Nts.tex 21. Dimension 113 Exercise (21.17). — Let A be a Noetherian local ring of dimension r. Let m be the maximal ideal, and k := A/m the residue class field. Prove that r ≤ dimk (m/m2 ), with equality if and only if m is generated by r elements. (21.18) (Regular local rings). — Let A be a Noetherian local ring of dimension r. We say A is regular if its maximal ideal is generated by r elements. Then any r generators are said to form a regular system of parameters. By (21.17), A is regular if and only if r = dimk (m/m2 ). For example, a field is a regular local ring of dimension 0, and is the only one. Lemma (21.19). — Let A be a Noetherian semilocal ring of dimension r, and q a parameter ideal. Then deg h(Gq (R), n) = r − 1. Proof: By (20.8), deg h(Gq (A), r) is equal to 1 less than the order of pole at 1 of the Hilbert series H(Gq (A), t). But that order is equal to d(A) by (21.2). Further, d(A) = r by the Dimension Theorem, (21.4). Thus the assertion holds. □ Proposition (21.20). — Let A be a Noetherian local ring of dimension r, and m its maximal ideal. Then A is regular if and only if its associated graded ring Gm (A) is a polynomial ring; if so, then the number of variables is r. Proof: Assume G(A) is a polynomial ring in s variables. Then dim(m/m2 ) = s. By (20.4), deg h(Gm (A), n) = s − 1. So s = r by (21.19). So A is regular by (21.18). Conversely, assume A is regular. Let x1 , . . . , xr be a regular sop, and x′i ∈ m/m2 the residue of xi . Set k := A/m, and let P := k[X1 , . . . , Xr ] be the polynomial ring. Form the k-algebra homomorphism φ : P → G(A) with φ(Xi ) = x′i . ⊕ Then φ is surjective as the xi generate G(A). Set a := Ker φ. Let P = Pn be the grading by total degree. Then φ preserves the gradings of P and G(A). So a ⊕ inherits a grading: a = an . So for n ≥ 0, there’s this canonical exact sequence: 0 → an → Pn → mn /mn+1 → 0. (21.20.1) Suppose a ̸= 0. Then there’s a nonzero f ∈ am for some m. Take n ≥ m. Then ∼ P Pn−m f ⊂ an . Since P is a domain, Pn−m −→ n−m f . Therefore, (21.20.1) yields dimk (mn /mn+1 ) = dimk (Pn ) − dimk (an ) ≤ dimk (Pn ) − dimk (Pn−m ) = (r−1+n) r−1 − (r−1+n−m) r−1 . The expression on the right is a polynomial in n of degree r − 2. On the other hand, dimk (mn /mn+1 ) = h(G(A), n) for n ≫ 0 by (20.8). Further, deg h(G(A), n) = r − 1 by (21.19). However, it follows from the conclusion of the preceding paragraph that deg h(G(A), n) ≤ r − 2. We have a contradiction! Hence a = 0. Thus φ is injective, so bijective, as desired. □ Exercise (21.21). — Let A be a Noetherian local ring of dimension r, and let x1 , . . . , xs ∈ A with s ≤ r. Set a := ⟨x1 , . . . , xs ⟩ and B := A/a. Prove equivalent: (1) A is regular, and there are xs+1 , . . . , xr ∈ A with x1 , . . . , xr a regular sop. (2) B is regular of dimension r − s. Theorem (21.22). — A regular local ring A is a domain. September 3, 2012 11Nts.tex 114 21. Dimension Proof: Use induction on r := dim(A). If r = 0, then A is a field, so a domain. Assume r ≥ 1. Let x be a member of a regular sop. Then A/⟨x⟩ is regular of dimension r − 1 by (21.21). By induction, A/⟨x⟩ is a domain. So ⟨x⟩ is prime. Thus A is a domain by (21.13). □ Lemma (21.23). — Let A be a local ring, m its maximal ideal, a a proper ideal. Set n := m/a and k := A/m. Then this sequence of k-vector spaces is exact: 0 → (m2 + a)/m2 → m/m2 → n/n2 → 0. Proof: The assertion is very easy to check. □ Proposition (21.24). — Let A be a regular local ring of dimension r, and a an ideal. Set B := A/a, and assume B is regular of dimension r − s. Then a is generated by s elements, and any such s elements form part of a regular sop. Proof: In its notation, (21.23) yields dim((m2 + a)/m2 ) = s. Hence, any set of generators of a includes s members of a regular sop of A. Let b be the ideal the s generate. Then A/b is regular of dimension r − s by (21.21). By (21.22), both A/b and B are domains of dimension r − s; whence, (15.10) implies a = b. □ September 3, 2012 11Nts.tex 22. Completion Completion is used to simplify a ring and its modules beyond localization. First, we discuss the topology of a filtration, and use Cauchy sequences to construct the completion. Then we discuss the inverse limit, the dual notion of the direct limit; thus we obtain an alternative construction. We conclude that, if we use the adic filtration of an ideal, then the functor of completion is exact on finitely generated modules over a Noetherian ring. Further, then the completion of a Noetherian ring is Noetherian; if the ideal is maximal, then the completion is local. We end with a useful version of the Cohen Structure Theorem for complete Noetherian local rings. (22.1) (Topology and completion). — Let R be a ring, M a module equipped with a filtration M = M0 ⊃ M1 ⊃ · · · . Then M has a topology: the open sets are the arbitrary unions of the sets m + Mn . Indeed, the intersection of two open sets is open, because the intersection of two unions is the union of the pairwise intersections; further, if the intersection U of m + Mn and m′ + Mn′ is nonempty and if n ≥ n′ , then U = m + Mn , because, if say m′′ ∈ U , then m + Mn = m′′ + Mn ⊂ m′′ + Mn′ = m′ + Mn′ . (22.1.1) The addition map M × M → M , given by (m, m′ ) 7→ m + m′ , is continuous, as (m + Mn ) + (m′ + Mn ) ⊂ (m + m′ ) + Mn . So, with m′ fixed, the translation m 7→ m + m′ is a homeomorphism M → M . (Similarly, inversion m 7→ −m is a homeomorphism; so M is a topological group.) Let a be an ideal, and give R the a-adic filtration. If the filtration on M is an a-filtration, then scalar multiplication (x, m) 7→ xm too is continuous, because (x + an )(m + Mn ) ⊂ xm + Mn . Further, if the filtration is a-stable, then it yields the same topology as the a-adic filtration, because Mn ⊃ an M ⊃ an Mn′ = Mn+n′ . Thus any two stable a-filtrations give the same topology, called the a-adic topology. When a is given, it is conventional to use the a-adic filtration and a-adic topology unless there’s explicit mention to the contrary. Further, if R is semi-local, then it is conventional to take a := rad(R). ∩ Let N be a submodule of M . Then the closure N of N is equal to n≥0 (N +Mn ), / (N + Mn ). because m ∈ / N means there’s n ≥ 0 with (m + Mn ) ∩ N = ∅, or ∩m ∈ In particular, each Mn is closed, and {0} is closed if and only if Mn = {0}. Further, M is separated — that is, Hausdorff — if and only if {0} is closed. For, if {0} is closed, then so is each {m}. Hence, given m′ ̸= m, there’s n′ so that m∈ / (m′ + Mn′ ). Take n ≥ n′ . Then (m + Mn ) ∩ (m′ + Mn′ ) = ∅ owing to (22.1.1). Finally, M is discrete — that is, every {m} is both open and closed — if and only if {0} is open. A sequence (mn )n≥0 in M is called Cauchy if, given n0 , there’s n1 with mn − mn′ ∈ Mn0 , or simply mn − mn+1 ∈ Mn0 , September 3, 2012 11Nts.tex for all n, n′ ≥ n1 ; 116 22. Completion the two conditions are equivalent because Mn0 is a subgroup and mn − mn′ = (mn − mn+1 ) + (mn+1 − mn+2 ) + · · · + (mn′ −1 − mn′ ). An m ∈ M is called a limit of (mn ) if, given n0 , there’s n1 with m − mn ∈ Mn0 for all n ≥ n1 . If every Cauchy sequence has a limit, then M is called complete. The Cauchy sequences form a module under termwise addition and termwise scalar multiplication. The sequences with 0 as a limit form a submodule. The c and is called the completion. There is a canonical quotient module is denoted M homomorphism, which carries m ∈ M to the class of the constant sequence (m): c by κ(m) := (m). κ: M → M It is straightforward to check that the notions of Cauchy sequence and limit c is separated and complete with respect to depend only on the topology. Similarly, M c=M c0 ⊃ M c1 ⊃ · · · , and κ is the universal example of a continuous the filtration M homomorphism from M into a separated and complete, filtered module. b Again, let a be an ideal. Under termwise multiplication of Cauchy sequences, R b c b is a ring, κ : R → R is a ring homomorphism, and M is an R-module. Further, c is a linear functor from ((R-mod)) to ((R-mod)). b M 7→ M For example, let k be a ring, and R := k[X1 , . . . , Xr ] the polynomial ring in r variables. Set a := ⟨X1 , . . . , Xr ⟩. Then a sequence (mn )n≥0 of polynomials is Cauchy if and only if, given n0 , there’s n1 such that, for all n ≥ n1 , the mn agree b is just the power series ring k[[X1 , . . . , Xr ]]. in degree less than n0 . Thus R For another example, take a prime integer p, and set a := ⟨p⟩. Then a sequence (mn )n≥0 of integers is Cauchy if and only if, given n0 , there’s n1 such that, for all n, n′ ≥ n1 , the difference mn − mn′ is a multiple of ∑ pn0 . The completion of Z is ∞ called the p-adic integers, and consists of the sums i=0 zi pi with 0 ≤ zi < p. b ). Proposition (22.2). — Let R be a ring, and a an ideal. Then b a ⊂ rad(R b is complete in the b Proof: Recall from (22.1) that R a-adic topology. Hence for b Thus b b ) by (3.2). □ x∈b a, we have 1/(1 − x) = 1 + x + x2 + · · · in R. a ⊂ rad(R Exercise (22.3). — In the 2-adic integers, evaluate the sum 1 + 2 + 4 + 8 + · · · . Exercise (22.4). — Let R be a ring, a an ideal, and M a module. Prove the following three conditions are equivalent: ∩ c is injective; (1) κ : M → M (2) an M = ⟨0⟩; (3) M is separated. Corollary (22.5). — Let R be a Noetherian ring, a ⊂ rad(R) an ideal, and M c. a finitely generated module. Then M ⊂ M Proof: The assertion results from (22.4), (18.28) or (20.19), and (3.2). □ (22.6) (Inverse limits). — Let R be a ring. Given modules Qn equipped with homomorphisms αnn+1 : Qn+1 → Qn for n ≥ 0, their inverse limit lim Qn is the ←− ∏ submodule of Qn of all vectors (qn ) with αnn+1 (qn+1 ) = qn for all n. Note that lim Qn = Ker(θ) (22.6.1) ←− n+1 where θ : Qn → Qn is the map defined by θ(qn ) := (qn − αn qn+1 ). Clearly, lim Qn has this UMP: given maps βn : P → Qn with αnn+1 βn+1 = βn , ←− there’s a unique map β : P → lim Qn with πn β = βn for all n. ←− September 3, 2012 11Nts.tex ∏ ∏ 22. Completion 117 Further, the UMP yields these two natural module isomorphisms: lim Hom(P, Qn ) = Hom(P, lim Qn ), ←− ←− lim Hom(Qn , N ) = Hom(lim Qn , N ). ←− −→ (The notion of inverse limit is formally dual to that of direct limit.) For example, let k be a ring, and R := k[X1 , . . . , Xr ] the polynomial ring in r variables. Set m := ⟨X1 , . . . , Xr ⟩ and Rn := R/mn+1 . Then Rn is just the R-algebra of polynomials of degree at most n, and the canonical map αnn+1 : Rn+1 → Rn is just truncation. Thus lim Rn is equal to the power series ring k[[X1 , . . . , Xr ]]. ←− For another example, take∑ a prime integer p, and set Zn := Z/⟨pn+1 ⟩. Then n i Zn is just the ring of sums i=0 zi p with 0 ≤ zi < p, and the canonical map αnn+1 : Zn+1 → Zn is just truncation. Thus lim Zn is just the ring of p-adic integers. ←− In general, consider exact sequences of modules βn γn 0 → Q′n −−→ Qn −→ Q′′n → 0 and commutative diagrams βn+1 γn+1 βn γn →0 0− → Q′n+1 −−−→ Qn+1 −−−→ Q′′n+1 −       ′′n+1 α′n+1 αn+1 αn y y y n n 0 −−→ Q′n −−−−−→ Qn −−−−−→ Q′′n −−→ 0 Then the induced sequence b β γ b → lim Q′′n 0 → lim Q′n − → lim Qn − ←− ←− ←− ′n+1 is exact; further, γ b is surjective if all the αn are surjective. Indeed, the above commutative diagrams yield the following one: ∏ ∏ ′ ∏ βn ∏ γn ∏ ′′ 0− → Qn −−−→ →0 Q − Qn −−−→  n      θy θ ′′ y θ′ y ∏ ∏ ′ ∏ βn ∏ γn ∏ ′′ Qn − →0 0− → Qn −−−→ Qn −−−→ (22.6.2) Owing to (22.6.1), the Snake Lemma (5.12) yields the exact sequence (22.6.2) and an injection Coker(b γ) ∏ ֒→ Coker(θ′ ). Also, Coker(θ′ ) = 0 if the αn′n+1 are surjec′ tive, because given (qn ) ∈∏ Q′n , we can solve the equations p′n − αn′n+1 (p′n+1 ) = qn′ recursively to get (p′n ) ∈ Q′n with θ′ (p′n ) = (qn′ ). Thus γ b is surjective. Proposition (22.7). — Let R be a ring, M a module, M = M0 ⊃ M1 ⊃ · · · a ∼ lim(M/M ). c −→ filtration. Then M n ←− c → lim(M/Mn ). Given a Cauchy sequence Proof: First, let’s define a map α : M ←− (mν ), let qn be the image of mν in M/Mn for ν ≫ 0. Then qn is independent of ν, because the sequence is Cauchy. Clearly, qn is the residue of qn+1 in M/Mn . Also, (mν ) has 0 as a limit if and only if qn = 0 for all n. Define α by α(mν ) := (qn ). Clearly, α is well defined, linear, and injective. As to surjectivity, given (qn ) ∈ lim(M/Mn ), let mν ∈ M represent qν ∈ M/Mν ←− for each ν. Then mµ − mν ∈ Mν for µ ≥ ν because the residue of qn in M/Mν is qν . Hence (mν ) is Cauchy. Thus α is surjective, so an isomorphism. □ September 3, 2012 11Nts.tex 118 22. Completion Exercise (22.8). — Let A be a Noetherian semilocal ring, and m1 , . . . , mm all its b = ∏A bm . maximal ideals. Prove that A i Exercise (22.9). — Let R be a ring, M a module, M = M0 ⊃ M1 ⊃ · · · a filtration, and N ⊂ M a submodule. Filter N by Nn := N ∩ Mn . Assume N ⊃ Mn b ⊂M c and M c/N b = M/N and G(M c ) = G(M ). for n ≥ n0 for some n0 . Prove N Exercise (22.10). — (1) Let R be a ring, a an ideal. If Ga (R) is a domain, show n b is an domain. If also ∩ R n≥0 a = 0, show R is a domain. (2) Use (1) to give an alternative proof that a regular local ring is a domain. b is a local Proposition (22.11). — Let A be a ring, m a maximal ideal. Then A b ring with maximal ideal m. b m b ⊃m b = A/m by (22.9); so m b is maximal. Next, rad(A) b by Proof: First, A/ ′ ′ b b (22.2). Finally, let m be any maximal ideal of A. Then m ⊃ rad(A ). Hence b Thus m b is the only maximal ideal. m′ = m. □ Exercise (22.12). — Let A be a semilocal ring, m1 , . . . , mm all its maximal ideals, b is a semilocal ring, that m b 1, . . . , m b m are all its and set m := rad(A). Prove that A b ). b = rad(A maximal ideals, and that m (22.13) (Completion, units, and localization). — Let R be a ring, a an ideal, and b the canonical map. Given t ∈ R, for each n denote by tn ∈ R/an the κ: R → R residue of t. Let’s show that κ(t) is a unit if and only if each tn is. b as a submodule of ∏ R/an . Then each tn Indeed, by (22.7), we may regard R is equal to the projection of κ(t). Hence tn is a unit if κ(t) is. Conversely, assume tn is a unit for each n. Then there are un ∈ R with un t ≡ 1 (mod an ).∏By the uniqueness of inverses, un+1 ≡ un in R/an for each n. Set u := (un ) ∈ R/an . b and uκ(t) = 1. Thus κ(t) is a unit. Then u ∈ R, b× ). Then by the above, T consists of the t ∈ R whose residue Set T := κ−1 (R n tn ∈ R/a is a unit for each n. So (2.29) and (1.8) yield T = { t ∈ R | t lies in no maximal ideal containing a }. (22.13.1) Set S := 1 + a. Then S ⊂ T owing to (22.13.1) as no maximal ideal can contain both x and 1 + x. Hence the UMP of localization (11.6) yields this diagram: ❚❚❚❚ R ❏ ❏❏❏ ❚❚❚ ❏❏❏ ❚❚❚κ❚❚ φS ❚❚❚❚ φT ❏$$  ❚ σ // τ ❚❚//** −1 −1 b S R T R R Further, S and T map into (R/an )× ; hence, (11.7), (11.22), and (12.18) yield: R/an = S −1 R/an S −1 R = T −1 R/an T −1 R. b is, by (22.7), equal to the completion of each of S −1 R and T −1 R in Therefore, R −1 their aS R-adic and aT −1 R-adic topologies. For example, take a to be a maximal ideal m. Then T = R − m by (22.13.1). b is equal to the completion of the localization Rm . Thus R Finally, assume R is Noetherian. Let’s prove that σ and τ are ∩ injective. Indeed, say τ σ(x/s) = 0. Then κ(x) = 0 as κ(s) is a unit. So x ∈ an . Hence the Krull Intersection Theorem, (18.28) or (20.19), yields an s′ ∈ S with s′ x = 0. So x/s = 0 in S −1 R. Thus σ is injective. Similarly, τ is injective. September 3, 2012 11Nts.tex 22. Completion 119 Theorem (22.14) (Exactness of completion). — Let R be a Noetherian ring, a c is exact. an ideal. Then on the finitely generated modules M , the functor M 7→ M Proof: Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of modules. Set Mn′ := M ′ ∩ an M . Then we obtain these exact sequences: 0 → M ′ /Mn′ → M/an M → M ′′ /an M ′′ → 0. ′ The maps M ′ /Mn+1 → M ′ /Mn′ are surjective. So (22.6) yields this exact sequence: 0 → lim M ′ /Mn′ → lim M/an M → lim M ′′ /an M ′′ → 0. ←− ←− ←− Assume R is Noetherian and M is finitely generated. Then M ′ = M0′ ⊃ M1′ ⊃ · · · is an a-stable filtration by the Artin–Rees Lemma (20.18). Hence, (22.1) and c′ → M c→M d′′ → 0. □ (22.7) yield the desired exactness of the sequencce 0 → M Exercise (22.15). — Let A be a Noetherian semilocal ring. Prove that an element b is also. x ∈ A is a nonzerodivisor if and only if its image x b∈A Exercise (22.16). — Let p ∈ Z be prime. For n > 0, define a Z-linear map αn : Z/⟨p⟩ → Z/⟨pn ⟩ by αn (1) = pn−1 . ⊕ ⊕ Set A := n≥1 Z/⟨p⟩ and B := n≥1 Z/⟨pn ⟩. Set α := αn ; so α : A → B. b is just A. (1) Show that the p-adic completion A (2) Show that, in the topology on A induced by the p-adic topology on B, the ∏∞ completion A is equal to n=1 Z/⟨p⟩. (3) Show that the natural sequence of p-adic completions ⊕ α b κ b b− b− 0→A →B → (B/A) b b (Thus p-adic completion is not right-exact on ((Z-mod)).) is not exact at B. Corollary (22.17). — Let R be a Noetherian ring, a an ideal, and M a finitely generated module. Then the natural map is an isomorphism: ∼ M b ⊗ M −→ c. R c is exact on the category of finitely Proof: By (22.14), the functor M 7→ M generated modules, and so (8.16) yields the conclusion. □ c preserves Exercise (22.18). — Let R be a ring, a an ideal. Show that M 7→ M b c surjections, and that R ⊗ M → M is surjective if M is finitely generated. Corollary (22.19). — Let R be a Noetherian ring, a and b ideals, M a finitely generated module. Then, using the a-adic topology, we have b = (bR b )n = (b c=b c and (2) (bn )b = bn R (1) (bM )b = bM bM b )n for any n ≥ 0. Proof: In general, the inclusion bM → M induces a commutative square b ⊗ (bM ) − b⊗M R →R     y y c (bM )b −−−−→ M b ⊗ M ). It is not hard to see that top map’s image is b(R In the present case, the two vertical maps are isomorphisms by (22.17), and the September 3, 2012 11Nts.tex 120 22. Completion c. bottom map is injective by (22.14). Thus (bM )b = bM b Hence bM c = bR bM c=b c. Thus (1) holds. Taking R for M yields b b = bR. bM n n b nb b In (1), taking b for b and R for M yields (b ) = b R. In particular, b b = bR; n n n ′ ′ n ′ b b so (bR ) = (b ) . But b R = (bR ) for any R-algebra R . Thus (2) holds. □ b is flat. Corollary (22.20). — Let R be a Noetherian ring, a an ideal. Then R b⊗b = b b by Proof: Let b be any ideal. Then R b by (22.17), and b b = bR b (22.19)(2). Thus R is flat by the Ideal Criterion (9.20). □ Exercise (22.21). — Let R be a Noetherian ring, and a and b ideals. Assume b is. a ⊂ rad(R), and use the a-adic topology. Prove b is principal if bR Lemma (22.22). — Let R be a ring, β : M → N a map of filtered modules (so β b preserves the filtration). If G(β) is injective or surjective, then so is β. Proof: Consider the following commutative diagram of exact sequences: 0− → Mn /Mn+1 − → M/Mn+1 − → M/Mn − →0       βn+1 y βn y Gn (β)y 0 −→ Nn /Nn+1 −→ N/Nn+1 −→ N/Nn −→ 0 Apply the Snake Lemma (5.12). It yields the following exact sequence: Ker Gn (β) → Ker βn+1 → Ker βn → Coker Gn (β) → Coker βn+1 → Coker βn . Assume G(β) is injective. Then Ker Gn (β) = 0. Hence induction on n yields Ker βn = 0 for all n. Thus βb is injective by (22.6). Assume G(β) is surjective. Then Coker Gn (β) = 0. So Ker βn+1 → Ker βn is surjective for all n. Also, induction on n yields Coker βn = 0 for all n; that is, βn 0− → Ker βn − → M/Mn −−→ N/Nn − →0 is exact. Thus βb is surjective by (22.6). □ Lemma (22.23). — Let R be a complete ring, M a separated module. If G(M ) is finitely generated over G(R), then M is finitely generated over R and complete. Proof: Take finitely many homogeneous generators of G(M ). Lift them to M . The lifts define a map α : Rn → M , and G(α) is surjective. So α b is surjective by (22.22). Now, form this canonical commutative diagram: κ n bn Rn −−R−→ R    αy α by κM c M −−− →M Since R is complete, κRn is surjective by (22.1). Since M is separated, κM is injective by (22.4). Hence κM is an isomorphism and α is surjective, as desired. □ Exercise (22.24) (Nakayama’s Lemma for a complete ring). — Let R be a ring, a an ideal, and M a module. Assume R is complete, and M separated. Show m1 , . . . , mn ∈ M generate if their images in M/aM generate. Proposition (22.25). — Let R be a complete ring, M a separated module. If G(M ) is a Noetherian G(R)-module, then M is a complete Noetherian R-module. September 3, 2012 11Nts.tex 22. Completion 121 Proof: Given a submodule N ⊂ M , set Nn := Mn ∩ N . Then G(N ) ⊂ G(M ). As G(M ) is Noetherian, G(N ) is finitely generated. Hence N is finitely generated and complete by (22.23). Thus M is Noetherian and complete. □ b Theorem (22.26). — Let R be a ring, a an ideal. If R is Noetherian, so is R. Proof: Assume R is Noetherian. Then G(R) is finitely generated as an (R/a)algebra by (20.12). So G(R) is Noetherian by the Hilbert Basis Theorem, (16.11). b by (22.9). Hence R b is Noetherian by (22.25) with R b for R and But G(R) = G(R) for M . □ Example (22.27). — Let k be a Noetherian ring, P := k[X1 , . . . , Xr ] the polynomial ring, and A := k[[X1 , . . . , Xr ]] the formal power series ring. Then A is the completion of P in the ⟨X1 , . . . , Xr ⟩-adic topology by (22.1). Further, P is Noetherian by the Hilbert Basis Theorem, (20.12). Thus A is Noetherian by (22.26). Assume k is a domain. Then A is a domain. Indeed, A is one if r = 1, because (am X1m + · · · )(bn X1n + · · · ) = am bn X1m+n + · · · . If r > 1, then A = k[[X1 , . . . , Xi ]] [[Xi+1 , . . . , Xr ]]; so A is a domain by induction. Set pi := ⟨Xi+1 , . . . , Xr ⟩. Then A/pi = k[[X1 , . . . , Xi ]] by (3.7). Hence pi is prime. So 0 = pr ⫋ · · · ⫋ p0 is a chain of primes of length r. Thus dim(A) ≥ r. Assume k is a field. Then A is local with maximal ideal ⟨X1 , . . . , Xr ⟩ and with residue field k by the above and either by (22.11) or again by (3.7). Therefore, dim(A) ≤ r by (21.17). Thus A is regular of dimension r. Exercise (22.28). — Let A be a Noetherian local ring, m the maximal ideal. b is a Noetherian local ring with m b as maximal ideal, (2) that Prove (1) that A b b is regular. dim(A) = dim(A ), and (3) that A is regular if and only if A Theorem (22.29) (UMP of Formal Power Series). — Let R be a ring, R′ an R-algebra, b an ideal of R′ , and x1 , . . . , xn ∈ b. Let P := R[[X1 , . . . , Xn ]] be the formal power series ring. If R′ is separated and complete, then there is a unique R-algebra map π : P → R′ with π(Xi ) = xi for 1 ≤ i ≤ n. Proof: For each m, there’s a unique R-algebra map R[X1 , . . . , Xn ] → R′ /bm sending Xi to the residue of xi . This map induces a map P/⟨X1 , . . . Xn ⟩m = R[X1 , . . . , Xn ]/⟨X1 , . . . Xn ⟩m −→ R′ /bm . Taking inverse limits yields π owing to (22.6) and (22.7). □ Theorem (22.30) (Cohen Structure). — Let A be a complete Noetherian local ring with maximal ideal m. Assume that A contains a coefficient field k; that ∼ A/m. Then A ≃ k[[X , . . . , X ]]/a for some variables X and ideal a. is, k −→ 1 n i Further, if A is regular of dimension r, then A ≃ k[[X1 , . . . , Xr ]]. Proof: Take generators x1 , . . . , xn ∈ m. Let π : k[[X1 , . . . , Xn ]] → A be the map with π(Xi ) = xi of (22.29). Then G(π) is surjective. Hence, π is surjective ∼ A. by (22.22). Set a := Ker(π). Then k[[X1 , . . . , Xn ]]/a −→ Assume A is regular of dimension r. Take n( = r. Then G(A) ) is a polynomial ring in r variables over k by (21.20). And G k[[X1 , . . . , Xr ]] is too by (22.6). Since G(π) is surjective, its kernel is a minimal prime, so equal to ⟨0⟩. Hence G(π) ∼ A. is bijective. So π is bijective by (22.22). Thus k[[X1 , . . . , Xr ]] −→ □ September 3, 2012 11Nts.tex 122 23. Discrete Valuation Rings 23. Discrete Valuation Rings A discrete valuation is a homomorphism from the multiplicative group of a field to the additive group integers such that the value of a sum is at least the minimum value of the summands. The corresponding discrete valuation ring consists of the elements whose values are nonnegative, plus 0. We characterize these rings in various ways; notably, we prove they are the normal Noetherian local domains of dimension 1. Then we prove that any normal Noetherian domain is the intersection of all the discrete valuation rings obtained by localizing at its height-1 primes. Finally, we prove Serre’s Criterion for normality of Noetherian domains. (23.1) (Discrete Valuations). — Let K be a field. We define a discrete valuation of K to be a surjective function v : K × → Z such that, for every x, y ∈ K × , (1) v(x · y) = v(x) + v(y), (2) v(x + y) ≥ min{v(x), v(y)} if x ̸= −y. (23.1.1) Condition (1) just means v is a group homomorphism. Hence, for any x ∈ K × , (1) v(1) = 0 and (2) v(x−1 ) = −v(x). (23.1.2) As a convention, we define v(0) := ∞. Consider the sets A := {x ∈ K | v(x) ≥ 0} and m := {x ∈ K | v(x) > 0}. Clearly, A is a subring, so a domain, and m is an ideal. Further, m is nonzero as v is surjective. We call A the discrete valuation ring (DVR) of v. Notice that, if x ∈ K, but x ∈ / A, then x−1 ∈ m; indeed, v(x) < 0, and so −1 v(x ) = −v(x) > 0. Hence, Frac(A) = K. Further, A× = {x ∈ K | v(x) = 0} = A − m. Indeed, if x ∈ A× , then v(x) ≥ 0 and −v(x) = v(x−1 ) ≥ 0; so v(x) = 0. Conversely, if v(x) = 0, then v(x−1 ) = −v(x) = 0; so x−1 ∈ A, and so x ∈ A× . Therefore, by the nonunit criterion, A is a local domain, not a field, and m is its maximal ideal. An element t ∈ m with v(t) = 1 is called a (local) uniformizing parameter. Such a t is irreducible, as t = ab with v(a) ≥ 0 and v(b) ≥ 0 implies v(a) = 0 or v(b) = 0 since 1 = v(a) + v(b). Further, any x ∈ K × has the unique factorization x = utn where u ∈ A× and n := v(x); indeed, v(u) = 0 as u = xt−n . In particular, t1 is uniformizing parameter if and only if t1 = ut with u ∈ A× ; also, A is a UFD. Moreover, A is a PID; in fact, any nonzero ideal a of A has the form a = ⟨tm ⟩ where m := min{ v(x) | x ∈ a }. n × (23.1.3) n Indeed, given a nonzero x ∈ a, say x = ut where u ∈ A . Then t ∈ a. So n ≥ m. Set y := utn−m . Then y ∈ A and x = ytm , as desired. In particular, m = ⟨t⟩ and dim(A) = 1. Thus A is regular local of dimension 1. Example (23.2). — The prototype is this example. Let ∑k be a field, t a variable, and K := k((t)) the field of formal Laurent series x := i≥n ai ti with n ∈ Z and with ai ∈ k and an ̸= 0. Set v(x) := n, the “order of vanishing” of x. Clearly, v is a discrete valuation, the formal power series ring k[[t]] is its DVR, and m := ⟨t⟩ is its maximal ideal. The preceding example can be extended to cover any DVR A that contains a ∼ A/⟨t⟩ where t is a uniformizing power. Indeed, A is a subring field k with k −→ September 3, 2012 11Nts.tex 23. Discrete Valuation Rings 123 b by (22.5), and A b = k[[t]] by the proof of the Cohen Structure of its completion A b restricts to that on A. Theorem (22.30). Further, clearly, the valuation on A A second old example is this. Let p ∈ Z be prime. Given x ∈ Q, write x = apn /b with a, b ∈ Z relatively prime and prime to p. Set v(x) := n. Clearly, v is a discrete valuation, the localization Z⟨p⟩ is its DVR, and pZ⟨p⟩ is its maximal ideal. We call v the p-adic valuation of Q. Lemma (23.3). — Let A be a local ∩ domain, m its maximal ideal. Assume that m is nonzero and principal and that n≥0 mn = 0. Then A is a DVR. Proof: Given a nonzero x ∈ A, there is an n ≥ 0 such that x ∈ mn − mn+1 . Say m = ⟨t⟩. Then x = utn , and u ∈ / m, so u ∈ A× . Set K := Frac(A). Given × m x ∈ K , write x = y/z where y = bt and z = ctk with b, c ∈ A× . Then x = utn with u := b/c ∈ A× and n := m − k ∈ Z. Define v : K × → Z by v(x) := n. If utn = wth with n ≥ h, then (u/w)tn−h = 1, and so n = h. Thus v is well defined. Since v(t) = 1, clearly v is surjective. To verify (23.1.1), take x = utn and y = wth with u, w ∈ A× . Then xy = (uw)tn+h . Thus (1) holds. To verify (2), we may assume n ≥ h. Then x + y = th (utn−h + w). Hence v(x + y) ≥ h = min{n, h} = min{v(x), v(y)}. Thus (2) holds. So v : K × → Z is a valuation. Clearly, A is the DVR of v. □ (23.4) (Depth). — Let R be a ring, M a nonzero module, and x1 , . . . , xn ∈ R. Set Mi := M/⟨x1 , . . . , xi ⟩. We say the sequence x1 , . . . , xn is regular on M , or is an M -sequence, and call n its length if Mn ̸= 0 and xi ∈ / z.div(Mi−1 ) for all i. We call the supremum of the lengths n of the M -sequences found in an ideal a the depth of a on M , and denote it by depth(a, M ). By convention, depth(a, M ) = 0 means a contains no nonzerodivisor on M . When M is semilocal, we call the depth of rad(M ) on M simply the depth of M and denote it by depth(M ). If depth(M ) = dim(M ), we call M Cohen– Macaulay. Lemma (23.5). — Let A be a Noetherian local ring, m its maximal ideal, and M a nonzero finitely generated module. (1) Then depth(M ) = 0 if and only if m ∈ Ass(M ). (2) Then depth(M ) = 1 if and only if there is an x ∈ m with x ∈ / z.div(M ) and m ∈ Ass(M/xM ). (3) Then depth(M ) ≤ dim(M ). Proof: Consider (1). If m ∈ Ass(M ), then it is immediate from the definitions that m ⊂ z.div(M ) and so depth(M ) = 0. ‘1Conversely, ∪ assume depth(M ) = 0. Then m ⊂ z.div(M ). Since A is Noetherian, z.div(M ) = p∈Ass(M ) p by (17.14). Since M is also finitely generated, Ass(M ) is finite by (17.20). Hence m = p for some p ∈ Ass(M ) by Prime Avoidance, (3.15). Consider (2). Assume depth(M ) = 1. Then there is an M -sequence of length 1, but none longer. So there is an x ∈ m with x ∈ / z.div(M ) and depth(M/xM ) = 0. Then m ∈ Ass(M/xM ) by (1). Conversely, assume there is x ∈ m with x ∈ / z.div(M ). Then depth(M ) ≥ 1 by definition. Assume m ∈ Ass(M/xM ). Then given any y ∈ m with y ∈ / z.div(M ), also m ∈ Ass(M/yM ) by (17.25). So depth(M/yM ) = 0 by (1). So there is no z ∈ m such that y, z is an M -sequence. Thus depth(M ) ≤ 1. Thus depth(M ) = 1. September 3, 2012 11Nts.tex 124 23. Discrete Valuation Rings Consider (3). Given any M -sequence x1 , . . . , xn , set Mi := M/⟨x1 , . . . , xi ⟩. Then dim(Mi+1 ) = dim(Mi ) − 1 by (21.5). Hence dim(M ) − n = dim(Mn ) ≥ 0. But depth(M ) := sup{n}. Thus (3) holds. □ Exercise (23.6). — Let R be a ring, M a module, and x, y ∈ R. (1) Prove that, if x, y form an M -sequence, then, given any m, n ∈ M such that xm = yn, there exists p ∈ M such that m = yp and n = xp. (2) Prove the converse of (1) if R is local, and x, y lie in its maximal ideal m, and M is Noetherian. Exercise (23.7). — Let R be a local ring, m its maximal ideal, M a Noetherian module, x1 , . . . , xn ∈ m, and σ a permutation of 1, . . . , n. Assume x1 , . . . , xn form an M -sequence, and prove xσ1 , . . . , xσn do too; first, say σ transposes i and i + 1. Exercise (23.8). — Prove that a Noetherian local ring A of dimension r ≥ 1 is regular if and only if its maximal ideal m is generated by an A-sequence. Theorem (23.9) (Characterization of DVRs). — Let A be a local ring, m its maximal ideal. Assume A is Noetherian. Then these five conditions are equivalent: (1) (2) (3) (4) (5) A A A A m is is is is is a DVR. a normal domain of dimension 1. a normal domain of depth 1. a regular local ring of dimension 1. principal and of height at least 1. Proof: Assume (1). Then A is UFD by (23.1); so A is normal by (10.28). Further, A has just two primes, ⟨0⟩ and m; so dim(A) = 1. Thus (2) holds. Further, (4) holds by (23.1). Clearly, (4) implies (5). ( ) Assume (2). Take a nonzero x ∈ m. Then A/⟨x⟩ ̸= 0, so Ass A/⟨x⟩ ̸= ∅ by (17.12). Now, A is a local domain of dimension 1, so A has just two primes, ⟨0⟩ and m. Clearly, ⟨0⟩ ∈ / Ass(A/⟨x⟩). Hence, m ∈ Ass(A/⟨x⟩). Thus (3) holds. Assume (3). By (23.5)(2), there are x, y ∈ m such that x is nonzero and y has residue y ∈ A/⟨x⟩ with m = Ann(y). So ym ⊂ ⟨x⟩. Set z := y/x ∈ Frac(A). Then zm = (ym)/x ⊂ A. Suppose zm ⊂ m. Then z is integral over A by (10.18). But A is normal, so z ∈ A. So y = zx ∈ ⟨x⟩, a contradiction. Hence, 1 ∈ zm; so there is t ∈ m with zt = 1. Given w ∈ m, therefore w = (wz)t with wz ∈ A. Thus m is principal. Finally, ht(m) ≥ 1 because x ∈ m and x ̸= 0. Thus (5) holds. Assume (18.28) yields an x ∈ m with ∩ (5). The Krull Intersection Theorem ∩ (1 + x) mn = 0. Then 1 + x ∈ A× . So mn = 0. Further, A is a domain by (21.13)(1). Hence (1) holds by (23.3). □ Exercise (23.10). — Let A be a DVR with fraction field K, and f ∈ A a nonzero nonunit. Prove A is a maximal proper subring of K. Prove dim(A) ̸= dim(Af ). Exercise (23.11). — Let k be a field, P := k[X, Y ] the polynomial ring in two variables, f ∈ P an irreducible polynomial. Say f = ℓ(X, Y ) + g(X, Y ) with ℓ(X, Y ) = aX + bY for a, b ∈ k and with g ∈ ⟨X, Y ⟩2 . Set R := P/⟨f ⟩ and p := ⟨X, Y ⟩/⟨f ⟩. Prove that Rp is a DVR if and only if ℓ ̸= 0. (Thus Rp is a DVR if and only if the plane curve C : f = 0 ⊂ k 2 is nonsingular at (0, 0).) September 3, 2012 11Nts.tex 23. Discrete Valuation Rings 125 Exercise (23.12). — Let k be a field, A a ring intermediate between the polynomial ring and the formal power series ring in one variable: k[X] ⊂ A ⊂ k[[X]]. Suppose that A is local with maximal ideal ⟨X⟩. Prove that A is a DVR. (Such local rings arise as rings of power series with curious convergence conditions.) Exercise (23.13). — Let L/K be an algebraic extension of fields, X1 , . . . , Xn variables, P and Q the polynomial rings over K and L in X1 , . . . , Xn . (1) Let q be a prime of Q, and p its contraction in P . Prove ht(p) = ht(q). (2) Let f, g ∈ P be two polynomials with no common prime factor in P . Prove that f and g have no common prime factor q ∈ Q. (23.14) (Serre’s Conditions). — Let R be a Noetherian ring. We say Serre’s Condition (Rn ) holds if, for any prime p of height m ≤ n, the localization Rp is regular of dimension m. We say Serre’s Condition (Sn ) holds if, for any prime p of any height m, the depth of p on Rp is at least min{m, n}, or equivalently, if depth(Rp ) ≥ min{dim(Rp ), n} as x1 , . . . , xr ∈ p is an Rp -sequence if and only if x1 /ti , . . . , xr /tr is for any ti ∈ / p. For example, (R0 ) holds if and only if Rp is a field for any minimal prime p. Also, (R1 ) holds if and only if (R0 ) does and Rp is a DVR for any p of height-1. Note depth(Rp ) ≤ dim(Rp ) by (23.5)(3). Hence (Sn ) holds if and only if Rp is Cohen–Macaulay when depth(Rp ) < n. In particular, (S1 ) holds if and only if p is minimal when p ∈ Ass(R) by (17.14); that is, there are no embedded primes. Exercise (23.15). — Let R be a Noetherian ring. Show that R is reduced if and only if (R0 ) and (S1 ) hold. Lemma (23.16). — Let R be a Noetherian domain. Set Φ := { p prime | ht(p) = 1 } Σ := { p prime | depth(Rp ) = 1 }. ∩ Then Φ ⊂ Σ, and Φ = Σ if and only if (S2 ) holds. Further, R = p∈Σ Rp . and Proof: Given p ∈ Φ, set q := pRp . Take 0 ̸= x ∈ q. Then q is minimal over ⟨x⟩. So q ∈ Ass(Rp /⟨x⟩) by (17.17). Hence depth(Rp ) = 1 by (23.5)(2). Thus Φ ⊂ Σ. However, (S1 ) holds by (23.15). Hence (S2 ) holds if and only if Φ ⊃ Σ. Thus Φ = Σ if and only if R satisfies (S2 ). ∩ Further, R ⊂ Rp for any prime p by (11.4); so R ⊂ p∈Σ Rp . As to the opposite ∩ inclusion, take an x ∈ p∈Σ Rp . Say x = a/b with a, b ∈ R and b ̸= 0. Then a ∈ bRp for all p ∈ Σ. But p ∈ Σ if p ∈ Ass(Rp /bRp ) by (23.5)(2). So a ∈ bR by (18.25). Thus x ∈ R, as desired. □ Theorem (23.17). — Let R be a normal Noetherian domain. Then ∩ R = p∈Φ Rp where Φ := { p prime | ht(p) = 1 }. Proof: As R is normal, so is Rp for any prime p by (11.31). So depth(Rp ) = 1 if and only if dim(Rp ) = 1 by (23.9). Thus (23.16) yields the assertion. □ Theorem (23.18) (Serre’s Criterion). — Let R be a Noetherian domain. Then R is normal if and only if (R1 ) and (S2 ) hold. September 3, 2012 11Nts.tex 126 23. Discrete Valuation Rings Proof: As R is a domain, (R0 ) and (S1 ) hold by (23.15). If R is normal, then so is Rp for any prime p by (11.31); whence, (R1 ) and (S2 ) hold by (23.9). Conversely, assume R satisfies (R1 ) and (S2 ). Let x be integral over R. Then x is integral over Rp for any prime p. Now, Rp is a DVR for all p of height 1 as R satisfies (R1 ). Hence, x ∈ Rp for all p of height 1, so for all p of depth 1 as R satisfies (S2 ). So x ∈ R owing to (23.16). Thus R is normal. □ Example (23.19). — Let k be an algebraically closed field, P := k[X, Y ] the polynomial ring in two variables, f ∈ P irreducible. Then dim(P ) = 2 by (15.12). Set R := P/⟨f ⟩. Then R is a domain. Let p ⊂ R be a nonzero prime. Say p = m/⟨f ⟩. Then 0 ⫋ ⟨f ⟩ ⫋ m is a chain of primes of length 2, the maximum. Thus m is maximal, and dim(R) = 1. Hence m = ⟨X − a, Y − b⟩ for some a, b ∈ k by (15.5). Write f (X, Y ) = ∂f /∂X(a, b)(X − a) + ∂f /∂Y (a, b)(Y − b) + g where g ∈ m2 . Then Rp is a DVR if and only if ∂f /∂X(a, b) and ∂f /∂Y (a, b) are not both equal to zero owing to (23.11) applied after making the change of variables X ′ := X − a and Y ′ := Y − b. Clearly, R satisfies (S2 ). Further, R satisfies (R1 ) if and only if Rp is a DVR for every nonzero prime p. Hence, by Serre’s Criterion, R is normal if and only if ∂f /∂X and ∂f /∂Y do not both belong to any maximal ideal m of P containing f . (Put geometrically, R is normal if and only if the plane curve C : f = 0 ⊂ k 2 is nonsingular everywhere.) Thus R is normal if and only if ⟨f, ∂f /∂X, ∂f /∂Y ⟩ = 1. Exercise (23.20). — Prove that a Noetherian domain R is normal if and only if, given any prime p associated to a principal ideal, pRp is principal. Exercise (23.21). — Let R be a Noetherian ring, K its total quotient ring, Φ := { p prime | ht(p) = 1 } and Σ := { p prime | depth(Rp ) = 1 }. Assuming (S1 ) holds in R, prove Φ ⊂ Σ, and prove Φ = Σ if and only if (S2 ) holds. Further, without assuming (S1 ) holds, prove this canonical sequence is exact: ∏ (23.21.1) R → K → p∈Σ Kp /Rp . Exercise (23.22). — Let R be a Noetherian ring, and K its total quotient ring. Set Φ := { p prime | ht(p) = 1 }. Prove these three conditions are equivalent: (1) R is normal. (2) (R1 ) and (S2 ) hold. ∏ (3) (R1 ) and (S1 ) hold, and R → K → p∈Φ Kp /Rp is exact. September 3, 2012 11Nts.tex 24. Dedekind Domains Dedekind domains are defined as the normal Noetherian domains of dimension 1. We prove they are the Noetherian domains whose localizations at nonzero primes are discrete valuation rings. Next we prove the Main Theorem of Classical Ideal Theory: in a Dedekind domain, every nonzero ideal factors uniquely into primes. Then we prove that a normal domain has a module-finite integral closure in any finite separable extension of its fraction field by means of Artin’s Character Theorem and the trace pairing of a separable extension. We conclude that a ring of algebraic integers is a Dedekind domain and that, if a domain is a finitely generated algebra over a field of characteristic 0, then in any algebraic extension of its fraction field — in particular, in the fraction field itself — the integral closure is a finitely generated module over the domain and is a finitely generated algebra over the field. Definition (24.1). — A domain R is said to be Dedekind if it is Noetherian, normal, and of dimension 1. Example (24.2). — of Dedekind domains include the integers Z, the [ √Examples ] Gaussian integers Z −1 , the polynomial ring k[X] in one variable over a field, and any DVR. Indeed, those rings are PIDs, and every PID R is a Dedekind domain: R is Noetherian by definition; R is a UFD, so normal by Gauss’s Theorem, (10.28); and R is of dimension 1 since every nonzero prime is maximal by (2.23). On the other hand, any local Dedekind domain is a DVR by (23.9). Example (24.3). — Let d ∈ Z be a square-free integer. Set R := Z + Zη where { √ (1 + d)/2 if d ≡ 1 (mod 4); η := √ d if not. √ Then R is the integral closure of Z in Q( d) by [1, Prp. (6.14), p. 412]; so R is normal. Also, dim(R) = dim(Z) by (15.11); so dim(R) = 1. Finally, R is Noetherian by (16.11) as Z is so and as R := Z + Zη. Thus R is Dedekind. Example (24.4). — Let k be an algebraically closed field, P := k[X, Y ] the polynomial ring in two variables, f ∈ P irreducible. By (23.19), R is a Noetherian domain of dimension 1, and R is Dedekind if and only if ⟨f, ∂f /∂X, ∂f /∂Y ⟩ = 1. Exercise (24.5). — Let R be a domain, S a multiplicative subset. (1) Assume dim(R) = 1. Prove dim(S −1 R) = 1 if and only if there is a nonzero prime p with p ∩ S = ∅. (2) Assume dim(R) ≥ 1. Prove dim(R) = 1 if and only if dim(Rp ) = 1 for every nonzero prime p. Exercise (24.6). — Let R be a Dedekind domain, S a multiplicative subset. Prove S −1 R is a Dedekind domain if and only if there’s a nonzero prime p with p ∩ S = ∅. Proposition (24.7). — Let R be a Noetherian domain, not a field. Then R is a Dedekind domain if and only if Rp is a DVR for every nonzero prime p. September 3, 2012 11Nts.tex 128 24. Dedekind Domains Proof: If R is Dedekind, then Rp is too by (24.6); so Rp is a DVR by (23.9). Conversely, suppose Rp is a DVR for every nonzero prime p. Then, trivially, R satisfies (R1 ) and (S2 ); so R is normal by Serre’s Criterion. Since R is not a field, dim(R) ≥ 1; whence, dim(R) = 1 by (24.5)(2). Thus R is Dedekind. □ Exercise (24.8). — Let R be a Dedekind domain, and a, b, c ideals. By first reducing to the case that R is local, prove that a ∩ (b + c) = (a ∩ b) + (a ∩ c), a + (b ∩ c) = (a + b) ∩ (a + c). Proposition (24.9). — In a Noetherian domain R of dimension 1, every ideal a ̸= 0 has a unique factorization a = q1 · · · qr with the qi primary and their primes pi distinct; further, {p1 , . . . , pr } = Ass(R/a) and qi = aRpi ∩ R for each i. Proof: The Lasker–Noether Theorem, (18.20), yields an irredundant primary ∩ decomposition a = qi . Say qi is pi -primary. Then by (18.18) the pi are distinct and {pi } = Ass(R/a). The qi are pairwise comaximal for the following reason. Suppose qi + qj lies in √ a maximal ideal m. Now, pi := qi by (18.5); so pni i ⊂ qi for some ni by (3.25). ni Hence pi ⊂ m. So pi ⊂ m by (2.2). But 0 ̸= a ⊂ pi ; hence, pi is maximal since dim(R) = 1. Therefore, pi = m. Similarly, pj = m. Hence i = j. Thus the ∏ qi are pairwise comaximal. So the ∏ Chinese Remainder Theorem, (1.13), yields a = i qi . As to uniqueness, let a = qi be any factorization with the qi primary and their primes pi distinct. The pi are minimal containing a as dim(R) = 1; so the pi are associated primes by∩(17.17). By the ∏ ∩ above reasoning, the qi are pairwise comaximal and so qi = qi . Hence a = qi is an irredundant primary decomposition by (18.18). So the pi are unique by the First Uniqueness Theorem, (18.19), and qi = aRpi ∩R by the Second Uniqueness Theorem, (18.24), and by (12.15)(3). □ Theorem (24.10) (Main Theorem of Classical Ideal Theory). — Let R be a domain. Assume R is Dedekind. Then every nonzero ideal a has a unique factorization into primes p. In fact, if vp denotes the valuation of Rp , then ∏ a= pvp (a) where vp (a) := min{ vp (a) | a ∈ a }. ∏ Proof: Using (24.9), write a = qi with the qi primary, their primes pi distinct and unique, and qi = aRpi ∩ R. Then Rpi is a DVR by (24.7). So (23.1.3) i yields aRpi = pm i Rpi with mi := min{ vpi (a/s) | a ∈ a and s ∈ R − pi }. But i vpi (1/s) = 0. So vpi (a/s) = vpi (a). Hence mi := min{ vpi (a) | a ∈ a }. Now, pm i mi mi is primary by (18.10) as pi is maximal; so pi Rpi ∩ R = pi by (18.22). Thus i □ qi = pm i . Corollary (24.11). — A Noetherian domain R of dimension 1 is Dedekind if and only if every primary ideal is a power of its radical. Proof: If R is Dedekind, every primary ideal is a power of its radical by (24.10). Conversely, given a nonzero prime p, set m := pRp . Then m ̸= 0. So m ̸= m2 by Nakayama’s Lemma. Take t ∈ m − m2 . Then m is the only prime containing t, as dim(Rp ) = 1 by (24.5)(2). So tRp is m-primary by (18.10). Set q := tRp ∩ R. Then q is p-primary by (18.8). So q = pn for some n by hypothesis. But qRp = tRp by (11.17)(3)(b). So tRp = mn . But t ∈ / m2 . So n = 1. So Rp is a DVR by (23.9). Thus R is Dedekind by (24.7). □ September 3, 2012 11Nts.tex 24. Dedekind Domains 129 Exercise (24.12). — Prove that a semilocal Dedekind domain A is a PID. Begin by proving that each maximal ideal is principal. Exercise (24.13). — Let R be a Dedekind domain, a and b two nonzero ideals. Prove (1) every ideal in R/a is principal, and (2) b is generated by two elements. Lemma (24.14) (E. Artin). — Let L be a field, G a group, σi : G → L× distinct homomorphisms. Then the σi are linearly independent over L in the vector space of set maps σ : G → L under valuewise addition and scalar multiplication. ∑m Proof: Suppose there’s an equation i=1 ai σi = 0 with nonzero ai ∈ L. Take × m ≥ 1 minimal. Now, σi ̸= 0 as σ∑ Since σ1 ̸= σ2 , there’s an i : G → L ; so m ≥ 2. ∑m m x ∈ G with σ1 (x) ̸= σ2 (x). Then i=1 ai σi (x)σi (y) = i=1 ai σi (xy) = 0 for every y ∈ G since σi is a homomorphism. Set τi (x) := 1 − σi (x)/σ1 (x). Then m ∑ i=1 ai τi (x)σi = m ∑ i=1 m ai σi − 1 ∑ ai σi (x)σi = 0. σ1 (x) i=1 But τ1 (x) = 0 and τ2 (x) ̸= 0, contradicting the minimality of m. □ (24.15) (Trace). — Let L/K be a finite Galois field extension. Its trace is this: ∑ tr : L → K by tr(x) := σ(x). σ∈Gal(L/K) Clearly, tr is K-linear. It is nonzero by (24.14) applied with G := L× . Consider the symmetric K-bilinear Trace Pairing: L×L→K by (x, y) 7→ tr(xy). (24.15.1) It is nondegenerate for this reason. Since tr is nonzero, there is a z ∈ L with tr(z) ̸= 0. Now, given x ∈ L× , set y := z/x. Then tr(xy) ̸= 0, as desired. Lemma (24.16). — Let R be a normal domain, K its fraction field, L/K a finite Galois field extension, and x ∈ L integral over R. Then tr(x) ∈ R. Proof: Let xn + a1 xn−1 + · · · + an = 0 be an equation of integral dependence for x over R. Let σ ∈ Gal(L/K). Then (σx)n + a1 (σx)n−1 + · · · + an = 0; so σx is integral over R. Hence tr(x) is integral over R, and lies in K. Thus tr(x) ∈ R since R is normal. □ Theorem (24.17) (Finiteness of integral closure). — Let R be a normal Noetherian domain, K its fraction field, L/K a finite separable field extension, and R′ the integral closure of R in L. Then R′ is module finite over R. Proof: Let L1 be the Galois closure of L/K, and R1′ the integral closure of R in L1 . Let z1 , . . . , zn ∈ L1 form a K-basis. Using (11.24), write zi = yi /ai with yi ∈ R1′ and ai ∈ R. Clearly, y1 , . . . , yn form a basis of L1 /K contained in R1′ . Let x1 , . . . , xn form the dual basis with respect ∑ to the Trace Paring, (24.15.1), so that tr(xi yj ) = δij . Given b ∈ R′ , write b = ci xi with ci ∈ K. Fix j. Then ) ∑ (∑ c i xi y j = ci tr(xi yj ) = cj for each j. tr(byj ) = tr September 3, 2012 11Nts.tex 130 24. Dedekind Domains But byj ∈ R1′ . So cj ∈ R by (24.16). Thus R′ ⊂ R′ is a finitely generated R-module, as desired. ∑ Rxi . Since R is Noetherian, □ Corollary (24.18). — Let R be a Dedekind domain, K its fraction field, L/K a finite separable field extension. Then the integral closure R′ of R in L is Dedekind. Proof: First, R′ is module finite over R by (24.17); so R′ is Noetherian by (16.18). Second, R′ is normal by (10.27). Finally, dim(R′ ) = dim(R) by (15.11), and dim(R) = 1 as R is Dedekind. Thus R is Dedekind. □ Theorem (24.19). — A ring of algebraic integers is a Dedekind domain. Proof: By (24.2), Z is a Dedekind domain; whence, so is its integral closure in any field that is a finite extension of Q by (24.18). □ Theorem (24.20) (Noether). — Let k be a field of characteristic 0, and R a domain that is a finitely generated k-algebra. Set K := Frac(R). Let L/K be a finite field extension (possibly L = K), and let R′ be the integral closure of R in L. Then R′ is a finitely generated R-module and a finitely generated k-algebra. Proof: By the Noether Normalization Lemma, (15.1), R is a module-finite k-algebra over a polynomial subring P . Then P is normal by Gauss’s Theorem, (10.28), and Noetherian by the Hilbert Basis Theorem, (16.11); also, L/ Frac(P ) is a finite field extension, which is separable as k is of characteristic 0. Hence, R′ is module finite over P by (24.17). The assertion follows. □ (24.21) (Other cases). — In (24.18), even if L/K is inseparable, the integral closure R′ of R in L is still Dedekind, as is proved below in Lecture 26. However, Akizuki constructed an example of a DVR R and a finite inseparable extension L/ Frac(R) such that the integral closure of R is a DVR, but is not module finite over R. The construction is nicely explained in [7, Secs. 9.4(1) and 9.5]. Thus separability is a necessary hypothesis in (24.17). Noether’s Theorem, (24.20), remains valid in positive characteristic, but the proof is more involved. See [3, (13.13), p. 297]. September 3, 2012 11Nts.tex 25. Fractional Ideals A fractional ideal is defined to be a submodule of the fraction field of a domain. A fractional ideal is called invertible if its product with another fractional ideal is equal to the given domain. We characterize the invertible fractional ideals as those that are nonzero, finitely generated, and principal locally at every maximal ideal. We prove that, in a Dedekind domain, any two nonzero integral (that is, ordinary) ideals have an invertible fractional ideal as their quotient. We characterize Dedekind domains as those domains whose integral ideals are, equivalently, all invertible, all projective, or all finitely generated and flat. Further, we prove a Noetherian domain is Dedekind if and only if every torsion-free module is flat. Finally, we prove the ideal class group is equal to the Picard group; the former is the group of invertible fractional ideals modulo those that are principal, and the latter is the group, under tensor product, of isomorphism classes of modules local free of rank 1. Definition (25.1). — Let R be a domain, and set K := Frac(R). We call an R-submodule M of K a fractional ideal. We call M integral if M ⊂ R. We call M principal if there is an x ∈ K with M = Rx. Given another fractional ideal N , form these two new fractional ideals: } {∑ and (M : N ) := { z ∈ K | zN ⊂ M }. M N := xi yi xi ∈ M and yi ∈ N We call them the product of M and N and the quotient of M by N . Exercise (25.2). — Let R be a domain, M and N nonzero fractional ideals. Prove that M is principal if and only if there exists some isomorphism M ≃ R. Construct the following canonical surjection and canonical isomorphism: π: M ⊗ N → → MN and ∼ Hom(N, M ). φ : (M : N ) −→ Proposition (25.3). — Let R be a domain, and K := Frac(R). Consider these finiteness conditions on a fractional ideal M : (1) There exist integral ideals a and b with b ̸= 0 and (a : b) = M . (2) There exists an x ∈ K × with xM ⊂ R. (3) There exists a nonzero x ∈ R with xM ⊂ R. (4) M is finitely generated. Then (1), (2), and (3) are equivalent, and they are implied by (4). Further, all four conditions are equivalent for every M if and only if R is Noetherian. Proof: Assume (1) holds. Take any nonzero x ∈ b. Given m ∈ M , clearly xm ∈ a ⊂ R; so xM ⊂ R. Thus (2) holds. Assume (2) holds. Write x = a/b with a, b ∈ R and a, b ̸= 0. Then aM ⊂ bR ⊂ R. Thus (3) holds. If (3) holds, then xM and xR are integral, and M = (xM : xR); thus (1) holds. Assume (4) holds. Say y1 /x1 , . . . , yn /xn ∈ K × generate M with xi , yi ∈ R. Set ∏ x := xi . Then x ̸= 0 and xM ⊂ R. Thus (3) holds. Assume (3) holds and R is Noetherian. Then xM ⊂ R. So xM is finitely generated, say by y1 , . . . , yn . Then y1 /x, . . . , yn /x generate M . Thus (4) holds. Finally, assume all four conditions are equivalent for every M . If M is integral, then (3) holds with x := 1, and so (4) holds. Thus R is Noetherian. □ September 3, 2012 11Nts.tex 132 25. Fractional Ideals Lemma (25.4). — Let R be a domain, M and N fractional ideals. Let S be a multiplicative subset. Then S −1 (M N ) = (S −1 M )(S −1 N ) and with equality if N is finitely generated. S −1 (M : N ) ⊂ (S −1 M : S −1 N ), ∑ Proof: Given x ∈ S −1 (M N i ∈ M , with ni ∈ N , ∑), write x = ( mi ni )/s with m−1 and with s ∈ S. Then x = (mi /s)(ni /1), and so x ∈ (S M )(S −1 N ). Thus S −1 (M N ) ⊂ (S −1 M )(S −1 N ). ∑ Conversely, given x ∈ (S −1 M )(S −1 ∏ N ), say x =∏ (mi /si )(ni /ti ) with mi ∈ M and ni ∈ N and si , ti ∈ S. Set s := si and t := ti . Then ∑ ∑ ′ ′ x = (mi ni /si ti ) = mi ni /st ∈ S −1 (M N ) with m′i ∈ M and n′i ∈ N . Thus S −1 (M N ) ⊃ (S −1 M )(S −1 N ), so equality holds. Given z ∈ S −1 (M : N ), write z = x/s with x ∈ (M : N ) and s ∈ S. Given y ∈ S −1 N , write y = n/t with n ∈ N and t ∈ S. Then z · n/t = xn/st and xn ∈ M and st ∈ S. So z ∈ (S −1 M : S −1 N ). Thus S −1 (M : N ) ⊂ (S −1 M : S −1 N ). −1 −1 Conversely, say N is generated by n1 , . . . , nr . Given ∏z ∈ (S M : S N ), write zni /1 = mi /si with mi ∈ M and si ∈ S. Set s := si . Then sz · ni ∈ M . So sz ∈ (M : N ). Hence z ∈ S −1 (M : N ), as desired. □ Definition (25.5). — Let R be a domain. We call a fractional ideal M locally principal if, for every maximal ideal m, the localization Mm is principal over Rm . Exercise (25.6). — Let R be a domain, M and N fractional ideals. Prove that the map π : M ⊗ N → M N is an isomorphism if M is locally principal. (25.7) (Invertible fractional ideals). — Let R be a domain. A fractional ideal M is said to be invertible if there is some fractional ideal M −1 with M M −1 = R. For example, a nonzero principal ideal Rx is invertible, as (Rx)(R · 1/x) = R. Proposition (25.8). — Let R be a domain, M an invertible fractional ideal. Then M −1 is unique; in fact, M −1 = (R : M ). Proof: Clearly M −1 ⊂ (R : M ) as M M −1 = R. But, if x ∈ (R : M ), then x · 1 ∈ (R : M )M M −1 ⊂ M −1 , so x ∈ M −1 . Thus (R : M ) ⊂ M −1 , as desired. □ Lemma (25.9). — An invertible ideal is finitely generated and nonzero. ∑ Proof: Let R be the domain, M the∑ideal. Say 1 = mi ni with mi ∈ M and ni ∈ M −1 . Let m ∈ M . Then m = mi mni . But mni ∈ R as m ∈ M and ni ∈ M −1 . So the mi generate M . Trivially, M ̸= 0. □ Lemma (25.10). — Let A be a local domain. Then a fractional ideal M is invertible if and only if M is principal and nonzero. ∑ Proof: Assume M is invertible. Say 1 = mi ni with mi ∈ M and ni ∈ M −1 . × As A is local, A − A is an ideal. So there’s a j with mj nj ∈ A× . Let m ∈ M . Then mnj ∈ A. Set a := (mnj )(mj nj )−1 ∈ A. Then m = amj . Thus M = Amj . Conversely, if M is principal and nonzero, then it’s always invertible by (25.7). □ Exercise (25.11). — Let R be a UFD. Show that a fractional ideal M is invertible if and only if M is principal and nonzero. September 3, 2012 11Nts.tex 25. Fractional Ideals 133 Theorem (25.12). — Let R be a domain, M a fractional ideal. Then M is invertible if and only if M is finitely generated and locally principal. Proof: Say M N = R. Then M is finitely generated and nonzero by (25.9). Let S be a multiplicative subset. Then (S −1 M )(S −1 N ) = S −1 R by (25.4). Let m be a maximal ideal. Then, therefore, Mm is an invertible fractional ideal over Rm . Thus Mm is principal by (25.10), as desired. Conversely, set a := M (R : M ) ⊂ R. Assume M is finitely generated. Then (25.4) yields am = Mm (Rm : Mm ). In addition, assume Mm is principal and nonzero. Then (25.7) and (25.8) yield am = Rm . Hence (13.16) yields a = R, as desired. □ Theorem (25.13). — Let R be a Dedekind domain, a, b nonzero integral ideals. Set M := (a : b). Then M is invertible, and has a unique factorization into powers of primes p. In fact, if vp denotes the valuation of Rp , then ∏ M= pvp (M ) where vp (M ) := min{ vp (x) | x ∈ M }. Finally, vp (M ) = min{vp (xi )} if the xi generate M . Proof: First, R is Noetherian. So (25.2) yields that M is finitely generated and that there is a nonzero x ∈ R with xM ⊂ R. Hence, each localization xMp is principal by (23.1.3). Thus M is invertible by (25.12). ∏ Next, the∏ Main Theorem of Classical Ideal Theory, (24.10), yields ⟨x⟩ = pvp (x) and xM = pvp (xM ) . Since vp (xM ) = vp (x) + vp (M ), we can cancel the vp (x) to ∏ conclude M = pvp (M ) . ∑n Finally, given x ∈ M , say x = i=1 ai xi with ai ∈ R. Then (23.1.1) yields vp (x) ≥ min{vp (ai xi )} ≥ min{vp (xi )} by induction on n. Thus vp (M ) = min{vp (xi )}. □ Exercise (25.14). — Show that a ring is a PID if and only if it’s a Dedekind domain and a UFD. (25.15) (Invertible modules). — Let R be an arbitrary ring. We call a module M invertible if there is another module N with M ⊗ N ≃ R. For example, suppose R is a domain. Let M be an invertible fractional ideal; say N is a fractional ideal with M N = R. Then M is locally principal by (25.12). So M ⊗ N = M N by (25.6). Thus M is an invertible abstract module. Exercise (25.16). — Let R be an ring, M an invertible module. Prove that M is finitely generated, and that, if R is local, then M is free of rank 1. Exercise (25.17). — Show these conditions on an R-module M are equivalent: (1) M is invertible. (2) M is finitely generated, and Mm ≃ Rm at each maximal ideal m. (3) M is locally free of rank 1. Assuming these conditions hold, show that M ⊗ Hom(M, R) = R. Lemma (25.18). — Let R be a domain, M a fractional ideal. Then M is an invertible fractional ideal if and only if M is a projective abstract module. September 3, 2012 11Nts.tex 134 25. Fractional Ideals Proof: Assume M is an invertible fractional ideal. Then M is an invertible abstract module by (25.15). Hence M is locally free of rank 1 by (25.17). So M is projective by (13.27). Conversely, assume M is projective. Then by (5.22), there exists a module M ′ with M ⊕ M ′ ≃ R⊕Λ . Let ρ : R⊕Λ → M be the projection, and set xλ := ρ(eλ ). Define φλ : M ֒→ R⊕Λ → R as the composition of the injection ∑ with the projection φλ on the λth factor. Then for all x ∈ M , we have x = λ∈Λ φλ (x)xλ and φλ (x) = 0 for almost all λ. ∑ Rqλ . Fix a nonzero y ∈ M . For λ ∈ Λ, set qλ := y1 φλ (y) ∈ Frac(R). Set N := Then for any nonzero x ∈ M , let’s check that xqλ = φλ (x). Write x = a/b and y = c/d with a, b, c, d ∈ R. Then a, c ∈ M ; whence, ∑ adφ(y) = φ(ac) = bcφ(x). Thus xqλ = φλ (x) ∈ R. Hence M · N ⊂ R. But y = φλ (y)yλ , so 1 = yλ qλ . Thus M · N = R. □ Theorem (25.19). — Let R (1) R is Dedekind; (2) every integral ideal is (3) every integral ideal is (4) every integral ideal is be a domain. Then the following are equivalent: invertible; projective; finitely generated and flat. Proof: Let a be an integral ideal. Assume (1). Since a = (a : R), it is invertible by (25.13). Thus (2) holds. Conversely, assume (2). Then a is finitely generated by (25.9). Thus R is Noetherian. Let p be any nonzero prime of R. Then by hypothesis, p is invertible. So by (25.12), it is locally principal. So Rp is a DVR by (23.9). Hence R is Dedekind by (24.7). Thus (1) holds. Thus (1) and (2) are equivalent. Recall that (2) and (3) are equivalent by (25.18). But (2) implies that R is Noetherian by (25.9). Thus (3) and (4) are equivalent by (16.18) and (13.27). □ Theorem (25.20). — A Noetherian domain R is Dedekind if and only if every torsion-free module is flat. Proof: (Of course, as R is a domain, every flat module is torsion free by (9.22).) Assume R is Dedekind. Let M be a torsion-free module, m a maximal ideal. Let’s see that Mm is torsion free over Rm . Let z ∈ Rm be nonzero, and say z = x/s with x, s ∈ R and s ∈ / m. Then µx : M → M is injective as M is torsion free. So µx : Mm → Mm is injective by the Exactness of Localization. But µx/s = µx µ1/s and µ1/s is invertible. So µx/s is injective. Thus Mm is torsion free. Since R is Dedekind, Rm is a DVR by (24.7), so a PID by (24.1). Hence Mm is flat over Rm by (9.22). But m is arbitrary. Therefore, M is flat over R by (13.23). Conversely, assume every torsion-free module is flat. Then, in particular, every integral ideal is flat. But R is Noetherian. Thus R is Dedekind by (25.19). □ (25.21) (The Picard Group). — Let R be a ring. We denote the collection of isomorphism classes of invertible modules by Pic(R). By (25.16), every invertible module is finitely generated, so isomorphic to a quotient of Rn for some integer n. Hence, Pic(R) is a set. Further, Pic(R) is, clearly, a group under tensor product with the class of R as identity. We call Pic(R) the Picard Group of R. Assume R is a domain, and set K := Frac(R). Given an invertible module M , we can embed M into K as follows. Set S := R − 0, and form the canonical map M → S −1 M . It is injective owing to (12.15) if the multiplication map µx : M → M September 3, 2012 11Nts.tex 25. Fractional Ideals 135 is injective for any x ∈ S. Fix x, and let’s prove µx is injective. Let m be a maximal ideal. Clearly, Mm is an invertible Rm -module. So Mm ≃ Rm by (25.16). Hence µx : Mm → Mm is injective. Therefore, µx : M → M is injective by (13.20). Thus M embeds canonically into S −1 M . Now, S −1 M is a localization of Mm , so is a 1-dimensional K-vector space, again as Mm ≃ Rm . Choose an isomorphism S −1 M ≃ K. It yields the desired embedding of M into K. Since M is invertible, M is finitely generated by (25.16). Further, as noted, Mm ≃ Rm at each maximal ideal m. Say x ∈ Mm corresponds to 1 ∈ Rm . Then yx ∈ Mm corresponds to y ∈ Rm . Thus M is locally principal. So, by (25.12), M is also invertible as a fractional ideal. The invertible fractional ideals M , clearly, form a group F(R). Sending an M to its isomorphism class yields a map κ : F(R) → Pic(R) by (25.15). By the above, κ is surjective. Further, κ is a group homomorphism by (25.6). It’s not hard to check that its kernel is the group P(R) of principal ideals and that P(R) = K × /R× . We call F(R)/P(R) the Ideal Class Group of R. Thus F(R)/P(R) = Pic(R); in other words, the Ideal Class Group is canonically isomorphic to the Picard Group. Every invertible fractional ideal is, by (25.12), finitely generated and nonzero, so of the form (a : b) where a and b are integral and nonzero by (25.3). Conversely, by (25.13) and (25.19), every fractional ideal of this form is invertible if and only if R is Dedekind. In fact, then F(R) is the free abelian group on the prime ideals. Further, then Pic(R) = 0 if and only if R is UFD, or equivalently by (25.14), a PID. See [1, Ch. 11, Sects. 10–11, pp. 424–437] for a discussion of the case in which R is a ring of quadratic integers, including many examples where Pic(R) ̸= 0. September 3, 2012 11Nts.tex 136 26. Arbitrary Valuation Rings 26. Arbitrary Valuation Rings A valuation ring is, by definition, a subring of a field whose elements either lie in the subring or their reciprocals do. Valuation rings are normal local domains. They are maximal under domination, that is, inclusion of both the local rings and their maximal ideals. Given any subring, its normalization is equal to the intersection of all the valuation rings containing it. We end with the Krull–Akizuki Theorem: given a 1-dimensional Noetherian domain, a finite extension of its fraction field, and a proper subring of the extension containing the domain, that subring too is 1dimensional and Noetherian. We conclude that, if we normalize a Dedekind domain in any finite extension of its fraction field, we obtain another Dededind domain. Definition (26.1). — A subring V of a field K is said to be a valuation ring of K if, whenever z ∈ K − V , then 1/z ∈ V . Proposition (26.2). — Let V be a valuation ring of a field K, and set m := {1/z | z ∈ K − V } ∪ {0}. Then V is local, m is its maximal ideal, and K is its fraction field. Proof: Clearly m = V −V × . Let’s show m is an ideal. Take a nonzero a ∈ V and nonzero x, y ∈ m. Suppose ax ∈ / m. Then ax ∈ V × . So a(1/ax) ∈ V . So 1/x ∈ V . × So x ∈ V , a contradiction. Thus ax ∈ m. Now, by hypothesis, either x/y ∈ V or y/x ∈ V . Say y/x ∈ V . Then 1 + (y/x) ∈ V . So x + y = (1 + (y/x))x ∈ m. Thus m is an ideal. Hence V is local and m is its maximal ideal by (3.4). Finally, K is its fraction field, because whenever z ∈ K − V , then 1/z ∈ V . □ Exercise (26.3). — Let V be a domain. Show that V is a valuation ring if and only if, given any two ideals a and b, either a lies in b or b lies in a. Exercise (26.4). — Let V be a valuation ring, m its maximal ideal, and p ⊂ m another prime ideal. Prove that Vp is a valuation ring, that its maximal ideal pVp is equal to p, and that V /p is a valuation ring of the field Vp /p. Exercise (26.5). — Prove that a valuation ring V is normal. Lemma (26.6). — Let R be a domain, a an ideal, K := Frac(R), and x ∈ K × . Then either 1 ∈ / aR[x] or 1 ∈ / aR[1/x]. Proof: Assume 1 ∈ aR[x] and 1 ∈ aR[1/x]. Then there are equations 1 = a0 + · · · + an xn and 1 = b0 + · · · + bm /xm with all ai , bj ∈ a. Assume n, m minimal and m ≤ n. Multiply through by 1 − b0 and an xn , getting 1 − b0 = (1 − b0 )a0 + · · · + (1 − b0 )an xn n (1 − b0 )an x = an b1 x n−1 Combine the latter equations, getting + · · · + an bm x and n−m . 1 − b0 = (1 − b0 )a0 + · · · + (1 − b0 )an−1 xn−1 + an b1 xn−1 + · · · + an bm xn−m . Simplify, getting an equation of the form 1 = c0 + · · · + cn−1 xn−1 with ci ∈ a, which contradicts the minimality of n. □ September 3, 2012 11Nts.tex 26. Arbitrary Valuation Rings 137 Lemma (26.7). — Let A, B be local rings, and m, n their maximal ideals. If B ⊃ A, then these conditions are equivalent: (1) n ∩ A = m; (2) 1 ∈ / mB; (3) m ⊂ n. Proof: Assume B ⊃ A. If (1) holds, then mB ⊂ n, so (2) holds. If (2) holds, then mB ⊂ n, so (3) holds. If (3) holds, then m ⊂ n ∩ A ⊊ A, so (1) holds. □ (26.8) (Domination). — Let A, B be local rings, and m, n their maximal ideals. We say B dominates A if B ⊃ A and n ∩ A = m. Proposition (26.9). — Let K be a field, A a local subring. Then A is dominated by a valuation ring V of K. Proof: Let m be the maximal ideal of A. Let S be the set of subrings R of K with R ⊃ A and 1 ∈ / mR. ∪ Then A ∈ S. Order S by inclusion. Let {Rλ } be a totally ordered subset. Set R := Rλ . If 1 ∈ mR, then 1 = a1 x1 + · · · + an xn with ai ∈ m and xi ∈ R. But then there is λ such that xi ∈ Rλ for all i; so 1 ∈ mRλ , a contradiction. Thus R ∈ S. Hence, by Zorn’s Lemma, S has a maximal element V . For any nonzero x ∈ K, set V ′ := V [x] and V ′′ := V [1/x]. By (26.6), either 1∈ / mV ′ or 1 ∈ / mV ′′ . Hence by maximality, either V = V ′ or V = V ′′ . So either x ∈ V or 1/x ∈ V . Thus V is a valuation ring. So V is local by (26.2), and dominates A by (26.8) as 1 ∈ / mV . □ Exercise (26.10). — Let K be a field, S the set of local subrings with fraction field K, ordered by domination. Show its maximal elements are the valuation rings. Theorem (26.11). — Let K be a field, and R a subring of K. Then the integral closure R′ of R in K is the intersection of all valuation rings V of K containing R. Further, if R is local, then the V dominating R suffice. Proof: Every valuation ring V is normal by (26.5). So if V ⊃ R, then V ⊃ R′ . ∩ Thus V ⊃R V ⊃ R′ . To prove the opposite inclusion, take any x ∈ K − R′ . To find a valuation ring V with V ⊃ R and x ∈ / V , set y := 1/x. If 1/y ∈ R[y], then for some n, 1/y = a0 y n + a1 y n−1 + · · · + an n n+1 with n aλ ∈ R. Multiplying by x yields x − an x − · · · − a0 = 0. So x ∈ R′ , a contradiction. Thus 1 ∈ / yR[y]. So there is a maximal ideal m of R[y] containing y. Then the composition R → R[y] → R[y]/m is surjective as y ∈ m. So m ∩ R is a maximal ideal of R. By (26.9), there is a valuation ring V that dominates R[y]m ; whence, if R is local, then V also dominates R. But y ∈ m; so x = 1/y ∈ / V , as desired. □ (26.12) (Valuations). — We call an additive abelian group Γ totally ordered if Γ has a subset Γ+ that is closed under addition and satisfies −Γ+ ⊔ {0} ⊔ Γ+ = Γ. Given x, y ∈ Γ, write x > y if x − y ∈ Γ+ . Note that either x > y or x = y or y > x. Note that, if x > y, then x + z > y + z for any z ∈ Γ. Let V be a domain, and set K := Frac(V ) and Γ := K × /V × . Write the group Γ additively, and let v : K × → Γ be the quotient map. It is a homomorphism: ( ) v(xy) = v(x) + v(y). (26.12.1) Set Γ+ := v V − 0 − 0. Then Γ+ is closed under addition. Clearly, V is a valuation ring if and only if −Γ+ ⊔ {0} ⊔ Γ+ = Γ, so if and only if Γ is totally ordered. September 3, 2012 11Nts.tex 138 26. Arbitrary Valuation Rings Assume V is a valuation ring. Let’s prove that, for all x, y ∈ K × , v(x + y) ≥ min{v(x), v(y)} if x ̸= −y. (26.12.2) Indeed, say v(x) ≥ v(y). Then z := x/y ∈ V . So v(z + 1) ≥ 0. Hence v(x + y) = v(z + 1) + v(y) ≥ v(y) = min{v(x), v(y)}, Note that (26.12.1) and (26.12.2) are the same as (1) and (2) of (23.1). Conversely, start with a field K, with a totally ordered additive abelian group Γ, and with a surjective homomorphism v : K × → Γ satisfying (26.12.2). Set V := {x ∈ K × | v(x) ≥ 0} ∪ {0}. Then V is a valuation ring, and Γ = K × /V × . We call such a v a valuation of K, and Γ the value group of v or of V . For example, a DVR V of K is just a valuation ring with value group Z, since any x ∈ K × has the form x = utn with u ∈ V × and n ∈ Z. Example (26.13). — Fix totally ordered additive abelian group Γ, and a field k. Form the k-vector space R with basis the symbols X a for a ≥ 0 in Γ. Define X a X b := X a+b , and extend this product to R by linearity. Then R is a k-algebra with X0 = 1. We call R the group algebra of Γ.Define v : (R − 0) → Γ by (∑ ) v ra X a := min{a | ra ̸= 0}. Then for x, y ∈ (R − 0), clearly v(xy) = v(x) + v(y) because k is a domain and Γ is ordered. Hence R is a domain. Moreover, if v(x + y) = a, then either v(x) ≤ a or v(y) ≤ a. Thus v(x + y) ≥ min{v(x), v(y)}. Set K := Frac(R), and extend v to a map v : K × → Γ by v(x/y) := v(x) − v(y) if y ̸= 0. Clearly v is well defined, surjective, and a homomorphism. Further, for x, y ∈ K × , clearly v(x + y) ≥ min{v(x), v(y)}. Thus v is a valuation with group Γ. Set R′ := {x ∈ R | v(x) ≥ 0} and p := {x ∈ R | v(x) > 0}. Clearly, R′ is a ring, and p is a prime of R′ . Further, Rp′ is the valuation ring of v. There are many choices for Γ other than Z. Examples include the additive rationals, the additive reals, its subgroup generated by two incommensurate reals, and the lexicographically ordered product of any two totally ordered abelian groups. Proposition (26.14). — Let v be a valuation of a field K, and x1 , . . . , xn ∈ K × with n ≥ 2. Set m := min{v(xi )}. (1) If n = 2 and if v(x1 ) ̸= v(x2 ), then v(x1 + x2 ) = m. (2) If x1 + · · · + xn = 0, then m = v(xi ) = v(xj ) for some i ̸= j. Proof: For (1), say v(x1 ) > v(x2 ); so v(x2 ) = m. Set z := x1 /x2 . Then v(z) > 0. Also v(−z) = v(z) + v(−1) > 0. Now, 0 = v(1) = v(z + 1 − z) ≥ min{v(z + 1), v(−z)} ≥ 0. Hence v(z + 1) = 0. Now, x1 + x2 = (z + 1)x2 . Therefore, v(x1 + x2 ) = v(x2 ) = m. Thus (1) holds. For (2), reorder the xi so v(xi ) = m for i ≤ k and v(xi ) > m for i > k. By induction, (26.12.2) yields v(xk+1 + · · · + xn ) ≥ mini>k {v(xi )}. Therefore, v(xk+1 + · · · + xn ) > m. If k = 1, then (1) yields v(0) = v(x1 + (x2 + · · · + xn )) = m, a contradiction. So k > 1, and v(x1 ) = v(x2 ) = m, as desired. □ September 3, 2012 11Nts.tex 26. Arbitrary Valuation Rings 139 Exercise (26.15). — Let V be a valuation ring, such as a DVR, whose value group Γ is Archimedean; that is, given any nonzero α, β ∈ Γ, there’s n ∈ Z such that nα > β. Show that V is a maximal proper subring of its fraction field K. Exercise (26.16). — Let V be a valuation ring. Show that (1) every finitely generated ideal a is principal, and (2) V is Noetherian if and only if V is a DVR. Lemma (26.17). — Let R be a 1-dimensional Noetherian domain, K its fraction field, M a torsion-free module, and x ∈ R nonzero. Then ℓ(R/xR) < ∞. Further, ℓ(M/xM ) ≤ dimK (M ⊗R K) ℓ(R/xR), (26.17.1) with equality if M is finitely generated. Proof: Set r := dimK (M ⊗R K). If r = ∞, then (26.17.1) is trivial; so we may assume r < ∞. Set S := R−{0}. Given any module N , set NK := S −1 N . Recall NK = N ⊗R K. First, assume M is finitely generated. Choose any K-basis m1 /s1 , . . . , mr /sr of MK with mi ∈ M and si ∈ S. Then m1 /1, . . . , mr /1 is also a basis. Define an R-map α : Rr → M by sending the standard basis elements to the mi . Then its localization αK is an K-isomorphism. But Ker(α) is a submodule of Rr , so torsion free. Further, S −1 Ker(α) = Ker(αK ) = 0. Hence Ker(α) = 0. Thus α is injective. Set N := Coker(α). Then NK = 0, and N is finitely generated. Hence, Supp(N ) is a proper closed subset of Spec(R). But dim(R) = 1 by hypothesis. Hence, Supp(N ) consists entirely of maximal ideals. So ℓ(N ) < ∞ by (19.4). Similarly, Supp(R/xR) is closed and proper in Spec(R). So ℓ(R/xR) < ∞. Consider the standard exact sequence: 0 → N ′ → N → N → N/xN → 0 where N ′ := Ker(µx ). ( ) Apply Additivity of Length, (19.9); it yields ℓ N ′ = ℓ(N/xN ). Since M is torsion free, µx : M → M is injective. Consider this commutative diagram with exact rows: α 0− → Rr − →M →N →0  −  −    µx y µx y µx y α →M − →N − →0 0− → Rr − Apply the snake lemma (5.12). It yields this exact sequence: 0 → N ′ → (R/xR)r → M/xM → N/xN → 0. ( ) ( ) Hence ℓ(M/xM ) = ℓ (R/xR)r by additivity. But ℓ (R/xR)r = r ℓ(R/xR) also by additivity. Thus equality holds in (26.17.1) when M is finitely generated. Second, assume M is arbitrary, but (26.17.1) fails. Then M possesses a finitely generated submodule M ′ whose image H in M/xM satisfies ℓ(H) > rℓ(R/xR). ′ ′ Now, MK ⊃ MK ; so r ≥ dimK (MK ). Therefore, ( ) ′ ′ ′ ) ℓ R/xR . ℓ(M /xM ) ≥ ℓ(H) > r ℓ(R/xR) ≥ dimK (MK However, together these inequalities contradict the first case with M ′ for M . □ Theorem (26.18) (Krull–Akizuki). — Let R be a 1-dimensional Noetherian domain, K its fraction field, K ′ a finite extension field, and R′ a proper subring of K ′ containing R. Then R′ is, like R, a 1-dimensional Noetherian domain. September 3, 2012 11Nts.tex 140 26. Arbitrary Valuation Rings Proof: Given a nonzero ideal a′ of R′ , take any nonzero x ∈ a′ . Since K ′ /K is finite, there is an equation an xn + · · · + a0 = 0 with ai ∈ R and a0 ̸= 0. Then a0 ∈ a′ ∩ R. Further, (26.17) yields ℓ(R/a0 R) < ∞. Clearly, R′ is a domain, so a torsion free R-module. Further, R′ ⊗R K ⊂ K ′ ; hence, dimK (R′ ⊗R K) < ∞. Therefore, (26.17) yields ℓR (R′ /a0 R′ ) < ∞. But a′ /a0 R′ ⊂ R′ /a0 R′ . So ℓR (a′ /a0 R′ ) < ∞. So a′ /a0 R′ is finitely generated over R by (19.2)(3). Hence a′ is finitely generated over R′ . Thus R′ is Noetherian. Set R′′ := R′ /a0 R′ . Clearly, ℓR′′ R′′ ≤ ℓR R′′ . So ℓR′′ R′′ < ∞. So, in R′′ , every prime is maximal by (19.4). So if a′ is prime, then a′ /a0 R′ is maximal, whence a′ maximal. So in R, every nonzero prime is maximal. Thus R′ is 1-dimensional. □ Corollary (26.19). — Let R be a 1-dimensional Noetherian domain, such as a Dedekind domain. Let K be its fraction field, K ′ a finite extension field, and R′ the normalization of R in K ′ . Then R′ is Dedekind. Proof: Since R is 1-dimensional, it’s not a field. But R′ is the normalization of R. So R′ is not a field by (14.1). Hence, R′ is Noetherian and 1-dimensional by Theorem (26.18). Thus R′ is Dedekind by Definition (24.1). □ Corollary (26.20). — Let K ′ /K be a field extension, and V ′ a valuation ring of K ′ not containing K. Set V := V ′ ∩ K. Then V is a DVR if and only if V ′ is. Proof: It follows easily from Definition (26.1) that V is a valuation ring, and from Subsection (26.12) that its value group is a subgroup of that of V ′ . Now, a nonzero subgroup of Z is a copy of Z. Thus V is a DVR if V ′ is. Conversely, assume V is a DVR, so Noetherian and 1-dimensional. Now, V ′ does not contain K, so is proper in K ′ .- Hence, V ′ is Noetherian by Theorem (26.18), so a DVR by Exercise (26.16)(2). □ September 3, 2012 11Nts.tex Solutions 1. Rings and Ideals Exercise (1.6). — Let R be a ring, a an ideal, and P := R[X1 , . . . , Xn ] the polynomial ring. Construct an isomorphism ψ from P/aP onto (R/a)[X1 , . . . , Xn ]. Solution: Let κ : R → R/a be the quotient map. Form the homomorphism φ : P → (R/a)[X1 , . . . , Xn ] such that φ|R = κ and φ(Xi ) = Xi . Then (∑ ) ∑ a(i1 ,...,in ) X1i1 · · · Xnin = φ κ(a(i1 ,...,in ) )X1i1 · · · Xnin . Since κ is surjective, so is φ. Since Ker(κ) = a, it follows that ∑ Ker(φ) = aX1i1 · · · Xnin = aP. Therefore, φ induces the desired isomorphism ψ by (1.5.1). □ Exercise (1.9). — Let R be ring, and P := R[X1 , . . . , Xn ] the polynomial ring. Let m ≤ n and a1 , . . . , am ∈ R. Set p := ⟨X1 − a1 , . . . , Xm − am ⟩. Prove that P/p = R[Xm+1 , . . . , Xn ]. Solution: First, assume m = n. Set P ′ := R[X1 , . . . , Xn−1 ] and p′ := ⟨X1 − a1 , . . . , Xn−1 − an−1 ⟩ ⊂ P ′ . By induction on n, we may assume P ′ /p′ = R. However, P = P ′ [Xn ]. Hence P/p′ P = (P ′ /p′ )[Xn ] by (1.6). Thus P/p′ P = R[Xn ]. / ′ We have P/p = (P/p P ) p(P/p′ P ) by (1.8). But p = p′ P + ⟨Xn − an ⟩P . Hence p(P/p′ P ) = ⟨Xn − an ⟩(P/p′ P ). So P/p = R[Xn ]/⟨Xn − an ⟩. So P/p = R by (1.7). In general, P = (R[X1 , . . . , Xm ])[Xm+1 , . . . , Xn ]. Thus P/p = R[Xm+1 , . . . , Xn ] by (1.6). □ Exercise (1.13) (Chinese Remainder Theorem). — Let R be a ring. (1) Let a and b be comaximal ideals; that is, a + b = R. Prove (a) ab = a ∩ b and (b) R/ab = (R/a) × (R/b). (2) Let a be comaximal to both b and b′ . Prove a is also comaximal to bb′ . (3) Let a, b be comaximal, and m, n ≥ 1. Prove am and bn are comaximal. (4) Let a1 , . . . , an be pairwise comaximal. Prove (a) a1 and a2 · · · an are comaximal; (b) a1 ∩ · · · ∩ an = a1∏ · · · an ; ∼ (c) R/(a1 · · · an ) −→ (R/ai ). Solution: To prove (1)(a), note that always ab ⊆ a ∩ b. Conversely, a + b = R implies x+y = 1 with x ∈ a and y ∈ b. So given z ∈ a∩b, we have z = xz +yz ∈ ab. To prove (1)(b), form the map R → R/a × R/b that carries an element to its pair of residues. The kernel is a ∩ b, which is ab by (1). So we have an injection φ : R/ab ֒→ R/a × R/b. To show that φ is surjective, take any element (x̄, ȳ) in R/a × R/b. Say x̄ and ȳ 141 142 Solutions: 1. Rings and Ideals are the residues of x and y. Since a + b = R, we can find a ∈ a and b ∈ b such that a + b = y − x. Then φ(x + a) = (x̄, ȳ), as desired. Thus (1) holds. To prove (2), note that R = (a + b)(a + b′ ) = (a2 + ba + ab′ ) + bb′ ⊆ a + bb′ ⊆ R. To prove (3), note that (2) implies a and bn are comaximal for any n ≥ 1 by induction on n. Hence, bn and am are comaximal for any m ≥ 1. To prove (4)(a), assume a1 and a2 · · · an−1 are comaximal by induction on n. By hypothesis, a1 and an are comaximal. Thus (2) yields (a). To prove (4)(b) and (4)(c), again proceed by induction on n. Thus (1) yields a1 ∩ (a2 ∩ · · · ∩ an ) = a1 ∩ (a2 · · · an ) = a1 a2 · · · an ; ∏ ∼ R/a × R/(a · · · a ) −→ ∼ R/(a1 · · · an ) −→ (R/ai ). 1 2 n □ Exercise (1.14). — First, given a prime number p and a k ≥ 1, find the idempotents in Z/⟨pk ⟩. Second, find the idempotents in Z/⟨12⟩. Third, find the number ∏N of idempotents in Z/⟨n⟩ where n = i=1 pni i with pi distinct prime numbers. Solution: First, let m ∈ Z be idempotent modulo pk . Then m(m−1) is divisible by pk . So either m or m − 1 is divisible by pk , as m and m − 1 have no common prime divisor. Hence 0 and 1 are the only idempotents in Z/⟨pk ⟩. Second, since −3 + 4 = 1, the Chinese Remainder Theorem (1.13) yields Z/⟨12⟩ = Z/⟨3⟩ × Z/⟨4⟩. Hence m is idempotent modulo 12 if and only if m is idempotent modulo 3 and modulo 4. By the previous case, we have the following possibilities: m ≡ 0 (mod 3) and m ≡ 1 (mod 3) and m ≡ 1 (mod 3) and m ≡ 0 (mod 3) and m ≡ 0 (mod 4); m ≡ 1 (mod 4); m ≡ 0 (mod 4); m ≡ 1 (mod 4). Therefore, m ≡ 0, 1, 4, 9 (mod 12). ni−1 and pni i have no common prime Third, for each i, the two numbers pn1 1 · · · pi−1 divisor. Hence some linear combination is equal to 1 by the Euclidean Algorithm. So the principal ideals they generate are comaximal. Hence by induction on N , the Chinese Remainder Theorem yields Z/⟨n⟩ = N ∏ i=1 Z/⟨pni i ⟩. So m is idempotent modulo n if and only if m is idempotent modulo pni for all i; hence, if and only if m is 0 or 1 modulo pni for all i by the first case. Thus there are 2N idempotents in Z/⟨n⟩. □ Exercise (1.15). — Let R := R′ × R′′ be a product of rings, a ⊂ R an ideal. Show a = a′ × a′′ with a′ ⊂ R′ and a′′ ⊂ R′′ ideals. Show R/a = (R′ /a′ ) × (R′′ /a′′ ). Solutions: 2. Prime Ideals 143 Solution: Set a′ := {x′ | (x′ , 0) ∈ a} and a′′ := {x′′ | (0, x′′ ) ∈ a}. Clearly a ⊂ R′ and a′′ ⊂ R′′ are ideals. Clearly, ′ a ⊃ a′ × 0 + 0 × a′′ = a′ × a′′ . The opposite inclusion holds, because if a ∋ (x′ , x′′ ), then a ∋ (x′ , x′′ ) · (1, 0) = (x′ , 0) and a ∋ (x′ , x′′ ) · (0, 1) = (0, x′′ ). Finally, the equation R/a = (R/a′ ) × (R/a′′ ) is now clear from the construction of the residue class ring. □ Exercise (1.16). — Let R be a ring, and e, e′ idempotents. (See (10.6) also.) (1) Set a := ⟨e⟩. Show a is idempotent; that is, a2 = a. (2) Let a be a principal idempotent ideal. Show a = ⟨f ⟩ with f idempotent. (3) Assume ⟨e⟩ = ⟨e′ ⟩. Show e = e′ . (4) Set e′′ := e + e′ − ee′ . Show ⟨e, e′ ⟩ = ⟨e′′ ⟩ and e′′ is idempotent. (5) Let e1 , . . . , er be idempotents. Show ⟨e1 , . . . , er ⟩ = ⟨f ⟩ with f idempotent. (6) Assume R is Boolean. Show every finitely generated ideal is principal. Solution: For (1), note a2 ⊂ a always. Conversely, xe = xe2 for any x ∈ R; so a ⊂ a2 . Thus (1) holds. For (2), say a = ⟨g⟩. Then a2 = ⟨g 2 ⟩. But a2 = a. So g = xg 2 for some x. Set f := xg. Then f ∈ a; so ⟨f ⟩ ⊂ a. And g = f g. So a ⊂ ⟨f ⟩. Thus (2) holds. For (3), say e′ = xe. So e′ = xe2 = e′ e. By symmetry, e = ee′ . Thus (3) holds. For (4), note ⟨e′′ ⟩ ⊂ ⟨e, e′ ⟩. Conversely, ee′′ = e2 + ee′ − e2 e′ = e + ee′ − ee′ = e. By symmetry, e′ e′′ = e′ . So ⟨e, e′ ⟩ ⊂ ⟨e′′ ⟩ and e′′2 = ee′′ + e′ e′′ − ee′ e′′ = e′′ . Thus (4) holds. For (5), induct on r. Thus (4) yields (5). For (6), recall that every element of R is idempotent. Thus (5) yields (6). □ 2. Prime Ideals Exercise (2.2). — Let a and b be ideals, and p a prime ideal. Prove that these conditions are equivalent: (1) a ⊂ p or b ⊂ p; and (2) a ∩ b ⊂ p; and (3) ab ⊂ p. Solution: Trivially, (1) implies (2). If (2) holds, then (3) follows as ab ⊂ a ∩ b. Finally, assume a ̸⊂ p and b ̸⊂ p. Then there are x ∈ a and y ∈ b with x, y ∈ / p. Hence, since p is prime, xy ∈ / p. However, xy ∈ ab. Thus (3) implies (1). □ Exercise (2.4). — Given a prime number p and an integer n ≥ 2, prove that the residue ring Z/⟨pn ⟩ does not contain a domain. Solution: Any subring of Z/⟨pn ⟩ must contain 1, and 1 generates Z/⟨pn ⟩ as an abelian group. So Z/⟨pn ⟩ contains no proper subrings. However, Z/⟨pn ⟩ is not a domain, because in it, p · pn−1 = 0 but neither p nor pn−1 is 0. □ Exercise (2.5). — Let R := R′ × R′′ be a product of two rings. Show that R is a domain if and only if either R′ or R′′ is a domain and the other is 0. Solution: Assume R is a domain. As (1, 0) · (0, 1) = (0, 0), either (1, 0) = (0, 0) or (0, 1) = (0, 0). Correspondingly, either R′ = 0 and R = R′′ , or R′′ = 0 and R = R′′ . The assertion is now obvious. □ 144 Solutions: 2. Prime Ideals Exercise (2.10). — Let R be a ring, p a prime ideal, R[X] the polynomial ring. Show that pR[X] and pR[X] + ⟨X⟩ are prime ideals of R[X]. Solution: Note R[X]/pR[X] = (R/p)[X] by (1.6). But R/p is a domain by (2.9). So R[X]/pR[X]/is a domain by (2.3). Thus pR[X] is prime by (2.9). Note (pR[X] + ⟨X⟩) pR[X] is/ equal to ⟨X⟩ ⊂ (R/p)[X]. But (R/p)[X]/⟨X⟩ is equal to R/p by (1.7). So R[X] (pR[X] + ⟨X⟩) is equal to R/p by (1.8). But R/p is a domain by (2.9). Thus pR[X] + ⟨X⟩ is prime again by (2.9). □ Exercise (2.11). — Let R be a domain, and R[X1 , . . . , Xn ] the polynomial ring in n variables. Let m ≤ n, and set p := ⟨X1 , . . . , Xm ⟩. Prove p is a prime ideal. Solution: Simply combine (2.9), (2.3), and (1.9) □ Exercise (2.12). — Let R := R′ × R′′ be a product of rings. Show every prime ideal of R has the form p′ × R′′ with p′ ⊂ R′ prime or R′ × p′′ with p′′ ⊂ R′′ prime. Solution: Simply combine (1.15), (2.9), and (2.5). □ Exercise (2.16). — Let k be a field, R a nonzero ring, φ : k → R a ring map. Prove φ is injective. Solution: By (1.1), 1 ̸= 0 in R. So Ker(φ) ̸= k. So Ker(φ) = 0 by (2.15). Thus φ is injective. □ Exercise (2.18). — Let B be a Boolean ring. Show that every prime p is maximal, and B/p = F2 . Solution: Take any z ∈ B/p. Then z(z − 1) = 0. But B/p is a domain. So z = 0 or z = 1. Thus B/p = F2 . Clearly, F2 is a field. Thus (2.17) yields (1). □ Exercise (2.20). — Prove the following statements or give a counterexample. (1) (2) (3) (4) The complement of a multiplicative subset is a prime ideal. Given two prime ideals, their intersection is prime. Given two prime ideals, their sum is prime. Given a ring map φ : R → R′ , the operation φ−1 carries maximal ideals of R′ to maximal ideals of R. (5) In (1.8), κ−1 takes maximal ideals of R/a to maximal ideals of R. Solution: (1) False. In the ring Z, consider the set S of powers of 2. The complement T of S contains 3 and 5, but not 8; so T is not an ideal. (2) False. In the ring Z, consider the prime ideals ⟨2⟩ and ⟨3⟩; their intersection ⟨2⟩ ∩ ⟨3⟩ is equal to ⟨6⟩, which is not prime. (3) False. Since 2 · 3 − 5 = 1, we have ⟨3⟩ + ⟨5⟩ = Z. (4) False. Let φ : Z → Q be the inclusion map. Then φ−1 ⟨0⟩ = ⟨0⟩. (5) True. The assertion is immediate from (1.8). □ Exercise (2.21). — Let k be a field, P := k[X1 , . . . , Xn ] the polynomial ring, f ∈ P nonzero. Let d be the highest power of any variable appearing in f . (1) Let S ⊂ k have at least dn + 1 elements. Proceeding by induction on n, find a1 , . . . , an ∈ S with f (a1 , . . . , an ) ̸= 0. (2) Using the algebraic closure K of k, find a maximal ideal m of P with f ∈ / m. Solutions: 3. Radicals 145 Solution: Consider (1). Assume n = 1. Then f has at most d roots by [Artin, (1.8), p. 392]. So f (a1 ) ̸= 0 for some a1 ∈ S. ∑ Assume n > 1. Say f = j gj X1j with gj ∈ k[X2 , . . . , Xn ]. But f ̸= 0. So gi ̸= 0 for some i. By induction, gi (a2 , . . . , an ) ̸= 0 for some a2 , . . . , an ∈ S. By n = 1, ∑ find a1 ∈ S such that f (a1 , . . . , an ) = j gj (a2 , . . . , an )aj1 ̸= 0. Thus (1) holds. Consider (2). As K is infinite, (1) yields a1 , . . . , an ∈ K with fi (a1 , . . . , an ) ̸= 0. Define φ : P → K by φ(Xi ) = ai . Then Im(φ) ⊂ K is the k-subalgebra generated by the ai . It is a field by [Artin, (2.6), p. 495]. Set m := Ker(φ). Hence m is a maximal ideal, and fi ∈ / m as φ(fi ) = fi (a1 , . . . , an ) ̸= 0. Thus (2) holds. □ Exercise (2.24). — Prove that, in a PID, elements x and y are relatively prime (share no prime factor) if and only if the ideals ⟨x⟩ and ⟨y⟩ are comaximal. Solution: Say ⟨x⟩ + ⟨y⟩ = ⟨d⟩. Then d = gcd(x, y), as is easy to check. The assertion is now obvious. □ Exercise (2.27). — Preserve the setup of (2.26). Let f := a0 X n + · · · + an be a polynomial of positive degree n. Assume that R has infinitely many prime elements p, or simply that there is a p such that p ∤ a0 . Show that ⟨f ⟩ is not maximal. Solution: Set a := ⟨p, f ⟩. Then a ⫌ ⟨f ⟩, because p is not a multiple of f . Set k := R/⟨p⟩. Since p is irreducible, k is a domain by (2.6) and (2.8). Let f ′ ∈ k[X] denote the image of f . By hypothesis, deg(f ′ ) = n ≥ 1. Hence f ′ is not a unit by ∼ k[X]/⟨f ′ ⟩ by (2.3) since k is a domain. Therefore, ⟨f ′ ⟩ is proper. But P/a −→ (1.6) and (1.8). So a is proper. Thus ⟨f ⟩ is not maximal. □ 3. Radicals Exercise (3.6). — Let A be a ring, m a maximal ideal such that 1 + m is a unit for every m ∈ m. Prove A is local. Is this assertion still true if m is not maximal? Solution: Take y ∈ A. Let’s prove that, if y ∈ / m, then y is a unit. Since m is maximal, ⟨y⟩ + m = A. Hence there exist x ∈ R and m ∈ m such that xy + m = 1, or in other words, xy = 1 − m. So xy is a unit by hypothesis; whence, y is a unit. Thus A is local by (3.4). The assertion is not true if m is not maximal. Indeed, take any ring that is not local, for example Z, and take m := ⟨0⟩. □ Exercise (3.10). — Let φ : R → R′ be a map of rings, p an ideal of R. Prove (1) there is an ideal q of R′ with φ−1 (q) = p if and only if φ−1 (pR′ ) = p; (2) if p is prime with φ−1 (pR′ ) = p, then there’s a prime q of R′ with φ−1 (q) = p. Solution: In (1), given q, note φ(p) ⊂ q, as always φ(φ−1 (q)) ⊂ q. So pR′ ⊂ q. Hence φ−1 (pR′ ) ⊂ φ−1 (q) = p. But, always p ⊂ φ−1 (pR′ ). Thus φ−1 (pR′ ) = p. The converse is trivial: take q := pR′ . In (2), set S := φ(R − p). Then S ∩ pR′ = ∅, as φ(x) ∈ pR′ implies x ∈ φ−1 (pR′ ) and φ−1 (pR′ ) = p. So there’s a prime q of R′ containing pR′ and disjoint from S by (3.9). So φ−1 (q) ⊃ φ−1 (pR′ ) = p and φ−1 (q) ∩ (R − p) = ∅. Thus φ−1 (q) = p. □ Exercise (3.11). — Use Zorn’s lemma to prove that any prime ideal p contains a prime ideal q that is minimal containing any given subset s ⊂ p. 146 Solutions: 3. Radicals Solution: Let S be the set of all prime ideals q such that s ⊂ q ⊂ p. Then p ∈ S, so S ̸= ∅. Order S by reverse inclusion. To apply Zorn’s Lemma, we must ∩ show that, for any decreasing chain {qλ } of prime ideals, the intersection q := qλ is a prime ideal. Plainly q is always an ideal. So take x, y ∈ / q. Then there exists λ such that x, y ∈ / qλ . Since qλ is prime, xy ∈ / qλ . So xy ∈ / q. Thus q is prime. □ Exercise (3.13). — Let R be a ring, S a subset. Show that S is saturated multiplicative if and only if R − S is a union of primes. Solution: First, take x ∈ R − S. Assume S is multiplicative. Then xy ∈ / S for all y ∈ R. So ⟨x⟩ ∩ S = ∅. Assume S is saturated too. Then (3.9) gives a prime p ⊃ ⟨x⟩ with p ∩ S = ∅. Thus R − S is a union of primes. Conversely, assume R − S is a union of primes p. Then 1 ∈ S as 1 lies in no p. Take x, y ∈ R. Then x, y ∈ S if and only if x, y lie in no p; if and only if xy lies in no p, as every p is prime; if and only if xy ∈ S. Thus S is saturated multiplicative. □ Exercise (3.14). — Let R be a ring, and S a multiplicative subset. Define its saturation to be the subset S := { x ∈ R | there is y ∈ R with xy ∈ S }. (1) Show (a) that S ⊃ S, and (b) that S is saturated multiplicative, and (c) that any saturated multiplicative subset T containing S also contains S. (2) Show that R − S is the union U of all the primes p with p ∩ S = ∅. ∪ (3) Let a be an ideal; assume S = 1 + a; set W := p∈V(a) p. Show R − S = W . Solution: Consider (1). Trivially, if x ∈ S, then x · 1 ∈ S. Thus (a) holds. Hence 1 ∈ S as 1 ∈ S. Now, take x, x′ ∈ S. Then there are y, y ′ ∈ R with xy, x′ y ′ ∈ S. But S is multiplicative. So (xx′′ )(yy ′ ) ∈ S. Hence xx′ ∈ S. Thus S is multiplicative. Conversely, take x, x′ ∈ R with xx′ ∈ S. Then there is y ∈ R with xx′ y ∈ S. So x, x′ ∈ S. Thus S is saturated. Thus (b) holds Finally, given x ∈ S, there is y ∈ R with xy ∈ S. So xy ∈ T . But T is saturated multiplicative. So x ∈ T . Thus T ⊃ S. Thus (c) holds. Consider (2). Plainly, R−U contains S. Further, R−U is saturated multiplicative by (3.13). So R − U ⊃ S by (1)(c). Thus U ⊂ R − S. Conversely, R − S is a union of primes p by (3.13). Plainly, p ∩ S = ∅ for all p. So U ⊃ R − S. Thus (2) holds. For (3), first take a prime p with p ∩ S = ∅. Then 1 ∈ / p + a. So p + a lies in a maximal ideal m by (3.9). Then a ⊂ m; so m ∈ V(a). Also, p ⊂ m. Thus U ⊂ W . Conversely, take p ∈ V(a). Then S ⊂ 1 + p. But p ∩ (1 + p) = ∅. So p ∩ S = ∅. Thus U ⊃ W . Thus U = W . Thus (2) yields (3). □ Exercise (3.16). — Let k be an infinite field. ∪ (1) Let V be a vector ∪ space, W1 , . . . , Wr proper subspaces. Show Wi ̸= V . (2) In (1), let W ⊂ Wi be a subspace. Show W∪⊂ Wi for some i. (3) Let R a k-algebra, a, a1 , . . . , ar ideals. If a ⊂ ai , show a ⊂ ai for some i. Solution: For (1), for all i, take vi ∈ V − Wi . Form their span V∪′ ⊂ V . Set n := dim V ′ and Wi′ := Wi ∩ V ′ . Then n < ∞, and it suffices to show Wi′ ̸= V ′ . Identify V ′ with k n . Form the polynomial ring P := k[X1 , . . . , Xn ]. For each i, take a linear form fi ∈ P that vanishes on Wi′ . Set f := f1 · · · fr . Then ∪ (2.21)(1) yields a1 , . . . , an ∈ k with f (a1 , . . . , an ) ̸= 0. ∪ Then (a1 , . . . , an ) ∈ V ′ − Wi′ . For (2), for all i, set Ui := W ∩ Wi . Then Ui = W . So (1) implies Ui = W for Solutions: 3. Radicals 147 some i. Thus W ⊂ Wi . Finally, as every ideal is a k-vector space, (3) is a special case of (2). □ Exercise (3.17). — Let k be a field, R := k[X, Y ] the polynomial ring in two variables, m := ⟨X, Y ⟩. Show m is a union of smaller primes. Solution: Since R is a UFD, and∪m is maximal, so prime, any nonzero f ∈ m has a prime factor p ∈ m. Thus m = p ⟨p⟩, but m ̸= ⟨p⟩ as m is not principal. □ Exercise (3.19). — Find the nilpotents in Z/⟨n⟩. In particular, take n = 12. Solution: An integer m is nilpotent modulo n if and only if some power mk is divisible by n. The latter holds if and only if every prime factor of n occurs in m. In particular, in Z/⟨12⟩, the nilpotents are 0 and 6. □ Exercise (3.20). — Let φ : R → R′ be a ring map, b ⊂ R′ a subset. Prove √ √ φ−1 b = φ−1 b. Solution: Below, (1) is clearly equivalent √ −1 (1) x ∈ φ√ b; (4) (5) (2) φx ∈ b; (3) (φx)n ∈ b for some n; (6) to (2); and (2), to (3); and so forth: φ(xn ) ∈ b for some n; xn ∈√φ−1 b for some n; x ∈ φ−1 b. □ √ Exercise (3.21). — Let R be a ring, a ⊂ ⟨0⟩ an ideal, and P := R[Y ] the polynomial ring in one variable. Let u ∈ R be a unit, and x ∈ R a nilpotent. (1) Prove (a) that u + x is a unit in R and (b) that u + xY is a unit in P . (2) Suppose w ∈ R maps to a unit of R/a. Prove that w is a unit in R. Solution: In (1), say xn = 0. Set y := −xu−1 . Then (a) holds as (u + x) · u−1 (1 + y + y 2 + · · · + y n−1 ) = 1. Now, u is also a unit in P , and (xY )n = 0; hence, (a) implies (b). In (2), say wy ∈ R maps to 1 ∈ R/a. Set z := wy − 1. Then z ∈ a, so z is nilpotent. Hence, 1+z is a unit by (1)(a). So wy is a unit. Then w·y(wy)−1 = 1. □ Exercise (3.23). — Let B be a Boolean ring. Show that rad(B) = nil(B) = ⟨0⟩. ∩ Solution: By (3.22), nil(B) = p where p runs through all the primes of B. Every p is maximal by (2.18); the converse holds by (2.22). Thus rad(B) = nil(B). Let f ∈ nil(B). Then f n = 0 for some n ≥ 1 by (3.18). But f 2 = f by (1.2). So f = 0. Thus nil(B) = ⟨0⟩. □ √ Exercise (3.25). (√—)nLet R be a ring, and a an ideal. Assume a is finitely generated. Show a ⊂ a for all large n. √ Solution: Let x∑ of a. For each∑i, there is ni such that 1 , . . . , xm be generators √ m xni i ∈ a. Let n > (ni − 1). Given a ∈ a, write a = i=1 yi xi with yi ∈ R. ∑m j Then an is a linear combination of terms of the form x11 · · · xjmm∑ with ∑ i=1 ji = n. Hence ji ≥ ni for some i, because if ji ≤ ni − 1 for all i, then ji ≤ (ni − 1). Thus an ∈ a, as desired. □ Exercise (3.26). — Let R be a ring, q an ideal, p a finitely generated prime. √ Prove that p = q if and only if there is n ≥ 1 such that p ⊃ q ⊃ pn . 148 Solutions: 4. Modules √ Solution: If p = q, then p ⊃ q ⊃ pn by (3.25). Conversely, if q ⊃ pn , then √ √ clearly q ⊃ p . Further, since p is prime, if p ⊃ q, then p ⊃ q. □ Exercise (3.28). — Let R be a ring. Assume R is reduced and has finitely many ∏ minimal prime ideals p1 , . . . , pn . Prove that φ : R → (R/pi ) is injective, and for each i, there is some (x1 , . . . , xn ) ∈ Im(φ) with xi ̸= 0 but xj = 0 for j ̸= i. ∩ Solution: Clearly Ker(φ) = pi . Now, R is reduced and the pi are its minimal primes; hence, (3.22) and (3.11) yield ∩ √ pi . ⟨0⟩ = ⟨0⟩ = Thus Ker(φ) = ⟨0⟩, and so φ is injective. Finally, fix i. Since pi is minimal, pi ̸⊃ pj for j ̸= i; say aj ∈ pj − pi . Set ∏ a := j̸=i aj . Then a ∈ pj − pi for all j ̸= i. Thus Im(φ) meets R/pi . □ 4. Modules Exercise (4.3). — Let R be a ring, M a module. Consider the set map θ : Hom(R, M ) → M defined by θ(ρ) := ρ(1). Show that θ is an isomorphism, and describe its inverse. Solution: First off, θ is R-linear, because θ(xρ + x′ ρ′ ) = (xρ + x′ ρ′ )(1) = xρ(1) + x′ ρ′ (1) = xθ(ρ) + x′ θ(ρ′ ). Set H := Hom(R, M ). Define η : M → H by η(m)(x) := xm. It is easy to check that ηθ = 1H and θη = 1M . Thus θ and η are inverse isomorphisms by (4.2). □ Exercise (4.12). — Let R be a domain, and x ∈ R nonzero. Let M be the submodule of Frac(R) generated by 1, x−1 , x−2 , . . . . Suppose that M is finitely generated. Prove that x−1 ∈ R, and conclude that M = R. ∑ni Solution: Suppose M is generated by m1 , . . . , mk . Say mi = j=0 aij x−j for some ni and aij ∈ R. Set n := max{ni }. Then 1, x−1 , . . . , x−n generate M . So x−(n+1) = an x−n + · · · + a1 x−1 + a0 for some ai ∈ R. Thus x−1 = an + · · · + a1 xn−1 + a0 xn ∈ R. Finally, as x−1 ∈ R and R is a ring, also 1, x−1 , x−2 , . . . ∈ R; so M ⊂ R. Conversely, M ⊃ R as 1 ∈ M . Thus M = R. □ ∏ Exercise (4.14). — Let Λ be an infinite set, Rλ a ring for λ ∈ Λ. Endow Rλ ⊕ ∏ and Rλ with componentwise addition and multiplication. Show that R has λ ⊕ a multiplicative identity (so is a ring), but that Rλ does not (so is not a ring). Solution: Consider the vector ∏ (1) whose every component is 1. Obviously, (1) is a multiplicative identity of Rλ⊕ . On the other hand, no restricted vector (eλ ) can be a multiplicative identity in Rλ ; indeed, because Λ is infinite, eµ must be zero for some µ. So (eλ ) · (xλ ) ̸= (xλ ) if xµ ̸= 0. □ Solutions: 5. Exact Sequences 149 Exercise (4.15). — Let L, M , and N be modules. Consider a diagram β α → → L− M− N ← − ← − ρ σ where α, β, ρ, and σ are homomorphisms. Prove that M =L⊕N and α = ι L , β = πN , σ = ι N , ρ = πL if and only if the following relations hold: βα = 0, βσ = 1, ρσ = 0, ρα = 1, and αρ + σβ = 1. Solution: If M = L ⊕ N and α = ιL , β = πN , σ = ιN , ρ = πL , then the definitions immediately yield αρ + σβ = 1 and βα = 0, βσ = 1, ρσ = 0, ρα = 1. Conversely, assume αρ + σβ = 1 and βα = 0, βσ = 1, ρσ = 0, ρα = 1. Consider the maps φ : M → L ⊕ N and θ : L ⊕ N → M given by φm := (ρm, βm) and θ(l, n) := αl + σn. They are inverse isomorphisms, because φθ(l, n) = (ραl + ρσn, βαl + βσn) = (l, n) and θφm = αρm + σβm = m. Lastly, β = πN φ and ρ = πL φ by definition of φ, and α = θιL and σ = θιN by definition of θ. □ Exercise (4.16). — Let N be a module,⊕ Λ a nonempty set, Mλ a module for λ ∈ Λ. Prove that the injections ικ : Mκ → Mλ induce an injection ⊕ ⊕ Hom(N, Mλ ) ֒→ Hom(N, Mλ ), and that it is an isomorphism if N is finitely generated. Solution: For λ ∈ Λ, let αλ : N → Mλ be maps, almost all 0. Then (∑ ) ( ) ⊕ ιλ αλ (n) = αλ (n) ∈ Mλ . ∑ So if ιλ αλ = 0, then αλ = 0 for all λ. Thus the ικ induce an injection. ⊕ Assume N is finitely generated, say by n1 , . . .⊕ , nk . Let α : N → Mλ be a map. Then each α(ni ) lies in a finite direct subsum of Mλ . So α(N ) lies in ⊕ one too. Set ακ := π α for all κ ∈ Λ. Then almost all α vanish. So (α ) lies in Hom(N, Mλ ), κ κ κ ∑ and ικ ακ = α. Thus the ικ induce a surjection, so an isomorphism. □ Exercise (4.17). — Let N be a module,⊕ Λ a nonempty set, Mλ a module for λ ∈ Λ. Prove that the injections ικ : Mκ → Mλ induce an injection ⊕ ⊕ Hom(N, Mλ ) ֒→ Hom(N, Mλ ), and that it is an isomorphism if N is finitely generated. ) (⊕ ⊕ aMλ because a · (mλ ) = (amλ ). Conversely, Solution: Mλ ⊂ ) ⊕First, a (⊕ ∑ aλ ιλ mλ since the sum is finite. a Mλ ⊃( aMλ) because (aλ mλ ) = ∏ ∏ Second, a Mλ ⊂( aM)λ as a(mλ ) = (amλ ). Conversely, say a is generated ∏ ∏ ∏ by f1 , . . . , fn . Then a Mλ ⊃ aMλ . Indeed, take (m′λ ) ∈ aMλ . Then for ∑ n λ each λ, there is nλ such that m′λ = j=1 aλj mλj with aλj ∈ a and mλj ∈ Mλ . ∑n Write aλj = i=1 xλji fi with the xλji scalars. Then ) (∑ ) ∑ (∑ nλ nλ ∑ n n ) (∏ Mλ . □ xλji mλj ∈ a fi fi xλji mλj = (m′λ ) = j=1 i=1 5. Exact Sequences i=1 j=1 150 Solutions: 5. Exact Sequences Exercise (5.5). — Let M ′ and M ′′ be modules, N ⊂ M ′ a submodule. Set M := M ′ ⊕ M ′′ . Using (5.2)(1) and (5.3) and (5.4), prove M/N = M ′ /N ⊕ M ′′ . Solution: By (5.2)(1) and (5.3), the two sequences 0 → M ′′ → M ′′ → 0 and 0 → N → M ′ → M ′ /N → 0 are exact. So by (5.4), the sequence 0 → N → M ′ ⊕ M ′′ → (M ′ /N ) ⊕ M ′′ → 0 is exact. Thus (5.3) yields the assertion. □ Exercise (5.6). — Let 0 → M ′ → M → M ′′ → 0 be a short exact sequence. Prove that, if M ′ and M ′′ are finitely generated, then so is M . Solution: Let m′′1 , . . . , m′′n ∈ M map to elements generating M ′′ . Let m ∈ M , and write its image in M ′′ as a linear combination of the images of the m′′i . Let m′′ ∈ M be the same combination of the m′′i . Set m′ := m − m′′ . Then m′ maps to 0 in M ′′ ; so m′ is the image of an element of M ′ . Let m′1 , . . . , m′l ∈ M be the images of elements generating M ′ . Then m′ is a linear combination of the m′j . So m is a linear combination of the m′′i and m′j . Thus the m′i and m′′j together generate M . □ Exercise (5.10). — Let M ′ , M ′′ be modules, and set M := M ′ ⊕ M ′′ . Let N be a submodule of M containing M ′ , and set N ′′ := N ∩ M ′′ . Prove N = M ′ ⊕ N ′′ . Solution: Form the sequence 0 → M ′ → N → πM ′′ N → 0. It splits by (5.9) as (πM ′ |N ) ◦ ιM ′ = 1M ′ . Finally, if (m′ , m′′ ) ∈ N , then (0, m′′ ) ∈ N as M ′ ⊂ N ; hence, πM ′′ N = N ′′ . □ Exercise (5.11). — Criticize the following misstatement of (5.9): given a short α β exact sequence 0 → M ′ − →M − → M ′′ → 0, there is an isomorphism M ≃ M ′ ⊕ M ′′ if and only if there is a section σ : M ′′ → M of β. Solution: We have α : M ′ → M , and ιM ′ : M ′ → M ′ ⊕ M ′′ , but (5.9) requires that they be compatible with the isomorphism M ≃ M ′ ⊕ M ′′ , and similarly for β : M → M ′′ and πM ′′ : M ′ ⊕ M ′′ → M ′′ . Let’s construct a counterexample (due to B. Noohi). For each integer ⊕ n ≥ 2, let Mn be the direct sum of countably many copies of Z/⟨n⟩. Set M := Mn . First, let us check these two statements: (1) For any finite abelian group G, we have G ⊕ M ≃ M . (2) For any finite subgroup G ⊂ M , we have M/G ≃ M . Statement (1) holds since G is isomorphic to a direct sum of copies of Z/⟨n⟩ for various n by the structure theorem for finite abelian groups [1, (6.4), p. 472], [4, Thm. 13.3, p. 200]. ⊕ To prove (2), write M = B M ′ , where B contains G and involves only finitely many components of M . Then M ′ ≃ M . Therefore, (5.10) and (1) yield M/G ≃ (B/G) ⊕ M ′ ≃ M. To construct the counterexample, let p be a prime number. Take one of the Z/⟨p2 ⟩ components of M , and let M ′ ⊂ Z/⟨p2 ⟩ be the cyclic subgroup of order p. There is no retraction Z/⟨p2 ⟩ → M ′ , so there is no retraction M → M ′ either, since the latter would induce the former. Finally, take M ′′ := M/M ′ . Then (1) and (2) yield M ≃ M ′ ⊕ M ′′ . □ Solutions: 5. Exact Sequences 151 Exercise (5.13). — Referring to (4.8), give an alternative proof that β is an isomorphism by applying the Snake Lemma to the diagram 0 −−→ M −−−→ N −−−−−→ N/M −−−−→ 0       κy βy y / λ 0− → M/L − → N/L − → (N/L) (M/L) − →0 Solution: The Snake Lemma yields an exact sequence, 1 L− → L → Ker(β) → 0; hence, Ker(β) = 0. Moreover, β is surjective because κ and λ are. □ Exercise (5.14) (Five Lemma). — Consider this commutative diagram: α α α α β4 β3 β2 β1 4 3 2 1 M 4 −−→ M 3 −−→ M 2 −−→ M 1 −−→ M 0      γ4 y γ3 y γ2 y γ1 y γ0 y N4 −−→ N3 −−→ N2 −−→ N1 −−→ N0 Assume it has exact rows. Via a chase, prove these two statements: (1) If γ3 and γ1 are surjective and if γ0 is injective, then γ2 is surjective. (2) If γ3 and γ1 are injective and if γ4 is surjective, then γ2 is injective. Solution: Let’s prove (1). Take n2 ∈ N2 . Since γ1 is surjective, there is m1 ∈ M1 such that γ1 (m1 ) = β2 (n2 ). Then γ0 α1 (m1 ) = β1 γ1 (m1 ) = β1 β2 (n2 ) = 0 by commutativity and exactness. Since γ0 is injective, α1 (m1 ) = 0. Hence exactness yields m2 ∈ M2 with α2 (m2 ) = m1 . So β2 (γ2 (m2 ) − n2 ) = γ1 α2 (m2 ) − β2 (n2 ) = 0. Hence exactness yields n3 ∈ N3 with β3 (n3 ) = γ2 (m2 )−n2 . Since γ3 is surjective, there is m3 ∈ M3 with γ3 (m3 ) = n3 . Then γ2 α3 (m3 ) = β3 γ3 (m3 ) = γ2 (m2 ) − n2 . Hence γ2 (m2 − α3 (m3 )) = n2 . Thus γ2 is surjective. The proof of (2) is similar. □ Exercise (5.15) (Nine Lemma). — Consider this commutative diagram: 0   y 0   y 0   y ′ ′′ 0 −→ L  −−→ L  −−→ L −→ 0    y y y ′ ′′ 0− →M →M →M →0  −  −  −    y y y ′ 0− →N  −→  y 0 N  −→  y 0 N′′ − →0  y 0 Assume all the columns are exact and the middle row is exact. Prove that the first row is exact if and only if the third is. Solution: The first row is exact if the third is owing to the Snake Lemma (5.12) applied to the bottom two rows. The converse is proved similarly. □ 152 Solutions: 6. Direct Limits Exercise (5.16). — Consider this commutative diagram with exact rows: β γ β′ γ′ ′ ′′ M →M →M  −  −     αy α′ y α′′ y N ′ −→ N −→ N ′′ Assume α′ and γ are surjective. Given n ∈ N and m′′ ∈ M ′′ with α′′ (m′′ ) = γ ′ (n), show that there is m ∈ M such that α(m) = n and γ(m) = m′′ . there is m1 ∈ M with γ(m1 ) = m′′ . Then (Solution:) Since γ ′′is surjective, ′′ ′ γ n−α(m1 ) = 0 as α (m ) = γ (n) and as the right-hand square is commutative. So by exactness of the bottom row, there is n′ ∈ N ′ with β ′ (n′ ) = n − α(m1 ). Since α′ is surjective, there is m′ ∈ M ′ with α′ (m′ ) = n′ . Set m := m1 + β(m′ ). Then γ(m) = m′′ as γβ = 0. Further, α(m) = α(m1 ) + β ′ (n′ ) = n as the left-hand square is commutative. Thus m works. □ ′ Exercise (5.21). — Show that a free module R⊕Λ is projective. Solution: Given β : M → → N and α : R⊕Λ → N , use the UMP of (4.10) to define γ : R⊕Λ → M by sending the standard basis vector eλ to any lift of α(eλ ), that is, any mλ ∈ M with β(mλ ) = α(eλ ). (The Axiom of Choice permits a simultaneous choice of all mλ if Λ is infinite.) Clearly α = βγ. Thus R⊕Λ is projective. □ Exercise (5.24). — Let R be a ring, and 0 → L → Rn → M → 0 an exact sequence. Prove M is finitely presented if and only if L is finitely generated. Solution: Assume M is finitely presented; say Rl → Rm → M → 0 is a finite presentation. Let L′ be the image of Rl . Then L′ ⊕ Rn ≃ L ⊕ Rm by Schanuel’s Lemma (5.23). Hence L is a quotient of Rl ⊕ Rn . Thus L is finitely generated. Conversely, assume L is generated by ℓ elements. They yield a surjection Rℓ → →L by (4.10)(1). It yields a sequence Rℓ → Rn → M → 0. The latter is, plainly, exact. Thus M is finitely presented. □ Exercise (5.25). — Let R be a ring, X1 , X2 , . . . infinitely many variables. Set P := R[X1 , X2 , . . . ] and M := P/⟨X1 , X2 , . . . ⟩. Is M finitely presented? Explain. Solution: No, otherwise by (5.24), the ideal ⟨X1 , X2 , . . . ⟩ would be generated by some f1 , . . . , fn ∈ P , so also by X1 , . . . , Xm for some m, but plainly it isn’t. □ β α Exercise (5.27). — Let 0 → L − →M − → N → 0 be a short exact sequence with M finitely generated and N finitely presented. Prove L is finitely generated. Solution: Let R be the ground ring. Say M is generated by m elements. They yield a surjection µ : Rm → → M by (4.10)(1). As in (5.26), µ induces the following commutative diagram, with λ surjective: 0− →K →  −  λy α Rm − →N →0  −  1  µy Ny β 0 −→ L −→ M −→ N − →0 By (5.24), K is finitely generated. Thus L is too, as λ is surjective. □ Solutions: 6. Direct Limits 153 6. Direct Limits Exercise (6.3). — (1) Show that the condition (6.2)(1) is equivalent to the commutativity of the corresponding diagram: ( ) HomC (B, C) − → HomC′ F (B), F (C)     y y ( ) HomC (A, C) − → HomC′ F (A), F (C) (2) Given γ : C → D, show (6.2)(1) yields the commutativity of this diagram: ( ) HomC (B, C) − → HomC′ F (B), F (C)     y y ( ) HomC (A, D) − → HomC′ F (A), F (D) Solution: The left-hand vertical map is given by composition with α, and the right-hand vertical map is given by composition with F (α). So the composition of the top map and the right-hand map sends β to F (β)F (α), whereas the composition of the left-hand map with the bottom map sends β to F (βα). These two images are always equal if and only if the diagram commutes. Thus (1) holds if and only if the diagram commutes. As to (2), the argument is similar. □ Exercise (6.5). — Let C and C′ be categories, F : C → C′ and F ′ : C′ → C an ∼ Hom (A, F ′ A′ ) denote the natural adjoint pair. Let φA,A′ : HomC′ (F A, A′ ) −→ C bijection, and set ηA := φA,F A (1F A ). Do the following: (1) Prove ηA is natural in A; that is, given g : A → B, the induced square ηA ′ A FA  −−→ F    gy yF ′ F g ηB B −−→ F ′ F B is commutative. We call the natural transformation A 7→ ηA the unit of (F, F ′ ). (2) Given f ′ : F A → A′ , prove φA,A′ (f ′ ) = F ′ f ′ ◦ ηA . (3) Prove the natural map ηA : A → F ′ F A is universal from A to F ′ ; that is, given f : A → F ′ A′ , there is a unique map f ′ : F A → A′ with F ′ f ′ ◦ ηA = f . (4) Conversely, instead of assuming (F, F ′ ) is an adjoint pair, assume given a natural transformation η : 1C → F ′ F satisfying (1) and (3). Prove the equation in (2) defines a natural bijection making (F, F ′ ) an adjoint pair, whose unit is η. (5) Identify the units in the two examples in (6.4): the “free module” functor and the “polynomial ring” functor. (Dually, we can define a counit ε : F F ′ → 1C′ , and prove similar statements.) Solution: For (1), form this canonical diagram, with horizontal induced maps: (F g)∗ (F g)∗ (F ′ F g)∗ g∗ HomC′ (F A, F A) −−−−−→ HomC′ (F A, F B) ←−−−− HomC′ (F B, F B)       φB, F B y φA, F B y φA, F A y HomC (A, F ′ F A) −−−−−→ HomC (A, F ′ F B) ←−−−− HomC (B, F ′ F B) 154 Solutions: 6. Direct Limits It commutes since φ is natural. Follow 1F A out of the upper left corner to find F ′ F g ◦ ηA = φA, F B (g) in HomC (A, F ′ F B). Follow 1F B out of the upper right corner to find φA, F B (g) = ηB ◦ g in HomC (A, F ′ F B). Thus (F ′ F g) ◦ ηA = ηB ◦ g. For (2), form this canonical commutative diagram: f′ ∗ HomC′ (F A, F A) −−−− −→ HomC′ (F A, A′ )    φA,A′  φA, F A y y (F ′ f ′ )∗ HomC (A, F ′ F A) −−−−−→ HomC (A, F ′ A′ ) Follow 1F A out of the upper left-hand corner to find φA,A′ (f ′ ) = F ′ f ′ ◦ ηA . For (3), given an f ′ , note that (2) yields φA,A′ (f ′ ) = f ; whence, f ′ = φ−1 A,A′ (f ). −1 ′ ′ ′ Thus f is unique. Further, an f exists: just set f := φA,A′ (f ). For (4), set ψA,A′ (f ′ ) := F ′ f ′ ◦ ηA . As ηA is universal, given f : A → F ′ A′ , there is a unique f ′ : F A → A′ with F ′ f ′ ◦ ηA = f . Thus ψA,A′ is a bijection: ∼ Hom (A, F ′ A′ ). ψA,A′ : HomC′ (F A, A′ ) −→ C Also, ψA,A′ is natural in A, as ηA is natural in A and F ′ is a functor. And, ψA,A′ is natural in A′ , as F ′ is a functor. Clearly, ψA, F A (1F A ) = ηA . Thus (4) holds. For (5), use the notation of (6.4). Clearly, if F is the “free module” functor, then ηΛ : Λ → R⊕Λ carries an element of Λ to the corresponding standard basis vector. Further, if F is the “polynomial ring” functor and if A is the set of variables X1 , . . . , Xn , then ηA (Xi ) is just Xi viewed in R[X1 , . . . , Xn ]. □ Exercise (6.9). — Let α : L → M and β : L → N be two maps. Their pushout is defined as the universal example of an object P equipped with a pair of maps γ : M → P and δ : N → P such that γα = δβ. In terms of the definitions, express the pushout as a direct limit. Show directly that, in ((Sets)), the pushout is the disjoint union M ⊔ N modulo the smallest equivalence relation ∼ with m ∼ n if there is ℓ ∈ L with α(ℓ) = m and β(ℓ) = n. Show directly that, in ((R-mod)), the pushout is the direct sum M ⊕ N modulo the image of L under the map (α, −β). Solution: Let Λ be the category with three objects λ, µ, and ν and two nonidentity maps λ → µ and λ → ν. Define a functor λ 7→ Mλ by Mλ := L, Mµ := M , Mν := N , αµλ := α, and ανλ := β. Set Q := lim Mλ . Then writing −→ β α 1R 1R N   ←−− L  −−→ M    ηµ y ην y ηλ y Q ←−− Q −−→ Q α as L   −→ M   ηµ y βy ην N −→ Q we see that Q is equal to the pushout of α and β; here γ = ηµ and δ = ην . In ((Sets)), take γ and δ to be the inclusions followed by the quotient map. Clearly γα = δβ. Further, given P and maps γ ′ : M → P and δ ′ : N → P , they define a unique map M ⊔ N → P , and it factors through the quotient if and only if γ ′ α = δ ′ β. Thus (M ⊔ N )/ ∼ is the pushout. In ((R-mod)), take γ and δ to be the inclusions followed by the quotient map. Then for all ℓ ∈ L, clearly ιM α(ℓ) − ιN β(ℓ) = (α(ℓ), −β(ℓ)). So ιM α(ℓ) − ιN β(ℓ) is in Im(L); hence, ιM α(ℓ) and ιN β(ℓ) have the same image in the quotient. Thus γα = δβ. Given γ ′ : M → P and δ ′ : N → P , let φ : M ⊕ N → P be the induced map. Clearly φ factors through the quotient if and only if with γ ′ α = δ ′ β. Thus Solutions: 6. Filtered direct limits 155 (M ⊕ N )/ Im(L) is the pushout. □ Exercise (6.16). — Let C be a category, Σ and Λ small categories. (1) Prove CΣ×Λ = (CΛ )Σ with (σ, λ) 7→ Mσ,λ corresponding to σ 7→ (λ 7→ Mσλ ). (2) Assume C has direct limits indexed by Σ and by Λ. Prove that C has direct limits indexed by Σ × Λ and that limλ∈Λ limσ∈Σ = lim(σ,λ)∈Σ×Λ . −→ −→ −→ Solution: In Σ × Λ, a map (σ, λ) → (τ, µ) factors in two ways: (σ, λ) → (τ, λ) → (τ, µ) and (σ, λ) → (σ, µ) → (τ, µ). So, given a functor (σ, λ) 7→ Mσ,λ , there is a commutative diagram like (6.13.1). It shows that the map σ → τ in Σ induces a natural transformation from λ 7→ Mσ,λ to λ 7→ Mτ,λ . Thus the rule σ 7→ (λ 7→ Mσλ ) is a functor from Σ to CΛ . A map from (σ, λ) 7→ Mσ,λ to a second functor (σ, λ) 7→ Nσ,λ is a collection of maps θσ,λ : Mσ,λ → Nσ,λ such that, for every map (σ, λ) → (τ, µ), the square Mσλ − → Mτ µ    θ θσ,λ y y τ,µ Nσλ −→ Nτ µ is commutative. Factoring (σ, λ) → (τ, µ) in two ways as above, we get a commutative cube. It shows that the θσ,λ define a map in (CΛ )Σ . This passage from CΣ×Λ to (CΛ )Σ is reversible. Thus (1) holds. Assume C has direct limits indexed by Σ and Λ. Then CΛ has direct limits indexed by Σ by (6.13). So the functors limλ∈Λ : CΛ → C and limσ∈Σ : (CΛ )Σ → −→ −→ CΛ exist, and they are the left adjoints of the diagonal functors C → CΛ and CΛ → (CΛ )Σ by (6.6). Hence the composition limλ∈Λ limσ∈Σ is the left adjoint of −→ −→ the composition of the two diagonal functors. But the latter is just the diagonal C → CΣ×Λ owing to (1). So this diagonal has a left adjoint, which is necessarily □ lim(σ,λ)∈Σ×Λ owing to the uniqueness of adjoints. Thus (2) holds. −→ Exercise (6.17). — Let λ 7→ Mλ and λ 7→ Nλ be two functors from a small category Λ to ((R-mod)), and {θλ : Mλ → Nλ } a natural transformation. Show dlim Coker(θλ ) = Coker(lim Mλ → lim Nλ ). −→ −→ Show that the analogous statement for kernels can be false by constructing a counterexample using the following commutative diagram with exact rows: µ2 Z −−→  µ 2 y µ2 Z− → Z/⟨2⟩ − →0   µ 2 µ2 y y Z −−→ Z − → Z/⟨2⟩ − →0 Solution: By (6.8), the cokernel is a direct limit, and by (6.14), direct limits commute; thus, the asserted equation holds. To construct the desired counterexample using the given diagram, view its rows as expressing the cokernel Z/⟨2⟩ as a direct limit over the category Λ of (6.8). View the left two columns as expressing a natural transformation {θλ }, and view the third column as expressing the induced map between the two limits. The latter map is 0, so its kernel is Z/⟨2⟩. However, Ker(θλ ) = 0 for λ ∈ Λ; so lim Ker(θλ ) = 0. □ −→ 156 Solutions: 7. Filtered direct limits 7. Filtered direct limits Exercise (7.2). — ∪ Let R be a ring, M a module, Λ a set, Mλ a submodule for each λ ∈ Λ. Assume Mλ = M . Assume, given λ, µ ∈ Λ, there is ν ∈ Λ such that Mλ , Mµ ⊂ Mν . Order Λ by inclusion: λ ≤ µ if Mλ ⊂ Mµ . Prove that M = lim Mλ . −→ Solution: Let us prove that M has the UMP characterizing lim Mλ . Given −→ homomorphisms βλ : Mλ → P with βλ = βν |Mλ when ∪ λ ≤ ν, define β : M → P by β(m) := βλ (m) if m ∈ Mλ . Such a λ exists as Mλ = M . If also m ∈ Mµ and Mλ , Mµ ⊂ Mν , then βλ (m) = βν (m) = βµ (m); so β is well defined. Clearly, β : M → P is the unique set map such that β|Mλ = βλ . Further, given m, n ∈ M and x ∈ R, there is ν such that m, n ∈ Mν . So β(m+n) = βν (m+n) = β(m)+β(n) and β(xm) = βν (xm) = xβ(m). Thus β is R-linear. Thus M = lim Mλ . □ −→ Exercise (7.3). — Show that every module M is the filtered direct limit of its finitely generated submodules. Solution: Every element m ∈ M belongs to the submodule generated by m; hence, M is the union of all its finitely generated submodules. Any two finitely generated submodules are contained in a third, for example, their sum. So the assertion results from (7.2) with Λ the set of all finite subsets of M . □ Exercise (7.4). — Show that every direct sum of modules is the filtered direct limit of its finite direct subsums. Solution: Consider an element of the direct sum. It has only finitely many nonzero components. So it lies in the corresponding finite direct subsum. Thus the union of the subsums is the whole direct sum. Now, given any two finite direct subsums, their sum is a third. Thus the finite subsets of indices form a directed partially ordered set Λ. So the assertion results from (7.2). □ Exercise (7.6). — Keep the setup of (7.5). For each n ∈ Λ, set Nn := Z/⟨n⟩; if n = ms, define αnm : Nm → Nn by αnm (x) := xs (mod n). Show lim Nn = Q/Z. −→ Solution: For each n ∈ Λ, set Qn := Mn /Z ⊂ Q/Z. If n = ms, then clearly Diagram (7.5.1) induces this one: αm Nm −−n→ Nn   γn y≃ γm y≃ m ηn Qm ֒−−→ Qn ∪ where ηnm is the inclusion. Now, Qn = Q/Z and Qn , Qn′ ⊂ Qnn′ . So (7.2) yields Q/Z = lim Mn . Thus lim Nn = Q/Z. □ −→ −→ Exercise (7.9). — Let R be a filtered direct limit of rings Rλ . Show R = 0 if and only if Rλ = 0 for some λ. Show R is a domain if Rλ is a domain for every λ. Solution: If Rλ = 0, then 1 = 0 in Rλ ; so 1 = 0 in R as αλ : Rλ → R carries 1 to 1 and 0 to 0; hence, R = 0 by (1.1). Conversely, assume R = 0. Then 1 = 0 in R. So αλ 1 = 0 for any given λ. Hence, by (7.8)(3) with Z for R, there is αµλ such that αµλ 1 = 0. But αµλ 1 = 1. Thus 1 = 0 in Rµ , and so Rµ = 0 by (1.1). Suppose every Rλ is a domain. Given x, y ∈ R with xy = 0, we can lift x, y Solutions: 7. Tensor Products 157 back to xλ , yλ ∈ Rλ for some λ by (7.8)(1) and (7.1)(1). Then xλ yλ maps to 0 ∈ R. Hence, by (7.8)(3), there is a transition map αµλ with αµλ (xλ yλ ) = 0 in Rµ . However, αµλ (xλ yλ ) = αµλ (xλ )αµλ (yλ ), and Rµ is a domain. Hence either αµλ (xλ ) = 0 or αµλ (yλ ) = 0. Therefore, either x = 0 or y = 0. Thus R is a domain. □ Exercise (7.11). — Let M := lim Mλ be a filtered direct limit of modules, and −→ N ⊂ M a submodule. For each λ, let αλ : Mλ → M be the insertion, and set −1 Nλ := αλ N ⊂ Mλ . Prove that N = lim Nλ . −→ Solution: The given functor λ 7→ Mλ induces a functor λ 7→ Nλ , and the insertions αλ : Mλ → M induce maps βλ : Nλ → N . So there is β : lim Nλ → N −→ with βαλ = βλ . By (7.10), lim Nλ → M is injective; so β is too. Further, for any −→ m ∈ M , there is an mλ ∈ Mλ such that m = αλ mλ , and if m ∈ N , then mλ ∈ Nλ □ since Nλ := αλ−1 N . Thus β is surjective, so an isomorphism. Exercise (7.13). — Let Λ and Λ′ be small categories, C : Λ′ → Λ a functor. Assume Λ′ is filtered. Assume C is cofinal; that is, (1) given λ ∈ Λ, there is a map λ → Cλ′ for some λ′ ∈ Λ′ , and (2) given ψ, φ : λ ⇒ Cλ′ , there is χ : λ′ → λ′1 with (Cχ)ψ = (Cχ)φ. Let λ 7→ Mλ be a functor from Λ to C whose direct limit exists. Show that limλ′ ∈Λ′ MCλ′ = limλ∈Λ Mλ ; −→ −→ more precisely, show that the right side has the UMP characterizing the left. Solution: Let P be an object of C. For λ′ ∈ Λ′ , take maps γλ′ : MCλ′ → P compatible with the transition maps MCλ′ → MCµ′ . Given λ ∈ Λ, choose a map λ → Cλ′ , and define βλ : Mλ → P to be the composition γ ′ λ → P. βλ : Mλ −→ MCλ′ −− Let’s check that βλ is independent of the choice of λ → Cλ′ . Given a second choice λ → Cλ′′ , there are maps λ′′ → µ′ and λ′ → µ′ for some ′ µ ∈ Λ′ since Λ′ is filtered. So there is a map µ′ → µ′1 such that the compositions λ → Cλ′ → Cµ′ → Cµ′1 and λ → Cλ′′ → Cµ′ → Cµ′1 are equal since C is cofinal. Therefore, λ → Cλ′′ gives rise to the same βλ , as desired. Clearly, the βλ are compatible with the transition maps Mκ → Mλ . So the βλ induce a map β : lim Mλ → P with βαλ = βλ for every insertion αλ : Mλ → lim Mλ . −→ −→ In particular, this equation holds when λ = Cλ′ for any λ′ ∈ Λ′ , as required. □ Exercise (7.14). — Show that every R-module M is the filtered direct limit over a directed set of finitely presented modules. α → R⊕Φ2 − → M → 0. For Solution: By (5.19), there is a presentation R⊕Φ1 − i = 1, 2, let Λi be the set of finite subsets Ψi of Φi , and order Λi by inclusion. Clearly, an inclusion Ψi ֒→ Φi yields an injection R⊕Ψi ֒→ R⊕Φi , which is given by extending vectors by 0. Hence (7.2) yields lim R⊕Ψi = R⊕Φi . −→ Let Λ ⊂ Λ1 × Λ2 be the set of pairs λ := (Ψ1 , Ψ2 ) such that α induces a map αλ : R⊕Ψ1 → R⊕Ψ2 . Order Λ by componentwise inclusion. Clearly, Λ is directed. For λ ∈ Λ, set Mλ := Coker(αλ ). Then Mλ is finitely presented. For i = 1, 2, the projection Ci : Λ → Λi is surjective, so cofinal. Hence, (7.13) □ yields limλ∈Λ R⊕Ci λ = limΨ ∈Λ R⊕Ψi . Thus (6.17) yields lim Mλ = M . −→ i i −→ −→ 158 Solutions: 8. Tensor Products 8. Tensor Products Exercise (8.6). — Let R be a domain. Set K := Frac(R). Given a nonzero submodule M ⊂ K, show that M ⊗R K = K. Solution: Define a map β : a×K → K by β(x, y) := xy. It is clearly R-bilinear. Given any R-bilinear map α : a×K → P , fix a nonzero z ∈ a, and define an R-linear map γ : K → P by γ(y) := α(z, y/z). Then α = γβ as α(x, y) = α(xz, y/z) = α(z, xy/z) = γ(xy) = γβ(x, y). Clearly, β is surjective. So γ is unique with this property. Thus the UMP implies that K = a ⊗R K. (Also, as γ is unique, γ is independent of the choice of z.) Alternatively, form the linear map φ : a ⊗ K → K induced by the∑bilinear map β. Since β is surjective, so is φ.∑ Now, given any w ∈ a ⊗ K, say w = ai ⊗ xi /x with all xi and x in R. Set a := ai xi ∈ a. Then w = a ⊗ (1/x). Hence, if φ(w) = 0, then a/x = 0; so a = 0 and so w = 0. Thus φ is injective, so bijective. □ Exercise (8.8). — Let R be a ring, R′ an R-algebra, M, N two R′ -modules. Show there is a canonical R-linear map τ : M ⊗R N → M ⊗R′ N . Let K ⊂ M ⊗R N denote the R-submodule generated by all the differences (x′ m) ⊗ n − m ⊗ (x′ n) for x′ ∈ R′ and m ∈ M and n ∈ N . Show K = Ker(τ ). Show τ is surjective, and is an isomorphism if R′ is a quotient of R. Solution: The canonical map β ′ : M × N → M ⊗R′ N is R′ -bilinear, so Rbilinear. Hence, by (8.3), it factors: β ′ = τ β where β : M × N → M ⊗R N is the canonical map and τ is the desired map. Set Q := (M ⊗R N )/K. Then τ factors through a map τ ′ : Q → M ⊗R′ N since each generator (x′ m) ⊗ n − m ⊗ (x′ n) of K maps to 0 in M ⊗R′ N . By (8.7), there is an R′ -structure on M ⊗R N with y ′ (m ⊗ n) = m ⊗ (y ′ n), and so by (8.5)(1), another one with y ′ (m ⊗ n) = (y ′ m) ⊗ n. Clearly, K is a submodule for each structure, so Q is too. But on Q the two structures coincide. Further, the canonical map M × N → Q is R′ -bilinear. Hence the latter factors through ∼ M ⊗ ′ N . Hence Ker(τ ) is M ⊗R′ N , furnishing an inverse to τ ′ . So τ ′ : Q −→ R equal to K, and τ is surjective. Finally, suppose R′ is a quotient of R. Then every x′ ∈ R′ is the residue of some x ∈ R. So each (x′ m) ⊗ n − m ⊗ (x′ n) is equal to 0 in M ⊗R N as x′ m = xm and x′ n = xn. Hence Ker(τ ) vanishes. Thus τ is an isomorphism. □ Exercise (8.13). — Let R be a ring, a and b ideals, and M a module. (1) Use (8.11) to show that (R/a) ⊗ M = M/aM . (2) Use (1) to show that (R/a) ⊗ (R/b) = R/(a + b). Solution: To prove (1), view R/a as the cokernel of the inclusion a → R. Then (8.11) implies that (R/a)⊗M is the cokernel of a⊗M → R⊗M . Now, R⊗M = M and x ⊗ m = xm by (8.5)(2). Correspondingly, a ⊗ M → M has aM as image. The assertion follows. (Caution: a ⊗ M → M needn’t be injective; if it’s not, then a ⊗ M ̸= aM . For example, take R := Z, take a := ⟨2⟩, and take M := Z/⟨2⟩; then a ⊗ M → M is just multiplication by 2 on Z/⟨2⟩, and so aM = 0.) To prove (2), apply (1) with M := R/b. Note a(R/b) = (a + b)/b. Hence / R/a ⊗ R/b = (R/b) ((a + b)/b). Solutions: 9. Flatness The latter is equal to R/(a + b) by (4.8). 159 □ Exercise (8.14). — Let k be a field, M and N nonzero vector spaces. Prove that M ⊗ N ̸= 0. Solution: Since k is a field, M and N are free; say M = k ⊕Φ and N = k ⊕Ψ . Then (8.11) yields M ⊗ N = k ⊕(Φ×Ψ) as k ⊗ k = k by (8.5)(2). Thus M ⊗ N ̸= 0. □ Exercise (8.16). — Let F : ((R-mod)) → ((R-mod)) be a linear functor. Show that F always preserves finite direct sums. Show that θ(M ) : M ⊗ F (R) → F (M ) is surjective if F preserves surjections and M is finitely generated, and that θ(M ) is an isomorphism if F preserves cokernels and M is finitely presented. Solution: The first assertion follows immediately from the characterization of finite direct sum in terms of maps (4.15), since F preserves the stated relations. The second assertion follows from the first via the second part of the proof of Watt’s Theorem (8.15), but with Σ and Λ finite. □ √ 3 Exercise (8.21). — Let X be a variable, ω a complex cubic root of 1, and 2 √ the real cube root of 2. Set k := Q(ω) and K := k[ 3 2]. Show K = k[X]/⟨X 3 − 2⟩ and then K ⊗k K = K × K × K. Solution: Note ω is a root of X 2 + X + 1,√which is √ irreducible √ over Q; hence, [k : Q] = 2. But the three roots of X 3 − 2 are 3 2 and ω 3 2 and ω 2 3 2. Therefore, ∼ K. X 3 −2 has no root in k. So X 3 −2 is irreducible over k. Thus k[X]/⟨X 3 −2⟩ −→ Note K[X] = K ⊗k k[X] as k-algebras by (8.20). So (8.5)(2) and (8.10) and (8.13)(1) yield / / k[X] ⟨X 3 − 2⟩ ⊗k K = k[X] ⟨X 3 − 2⟩ ⊗k[X] (k[X] ⊗k K) / / = k[X] ⟨X 3 − 2⟩ ⊗k[X] K[X] = K[X] ⟨X 3 − 2⟩. However, X 3 − 2 factors in K as follows: √ √ √ ( )( )( ) 3 3 3 X 3 − 2 = X − 2 X − ω 2 X − ω2 2 . So the Chinese Remainder Theorem, (1.13), yields / K[X] ⟨X 3 − 2⟩ = K × K × K, √ ∼ K for any i by (1.7). because K[X]/⟨X − ω i 3 2⟩ −→ □ 9. Flatness Exercise (9.7). — Let R be a ring, R′ a flat algebra, and P a flat R′ -module. Show that P is a flat R-module. Solution: Cancellation (8.10) yields • ⊗R P = (• ⊗R R′ ) ⊗R′ P . But • ⊗R R′ and • ⊗R′ P are exact. Hence, •R P is too. Thus P is R-flat. □ Exercise (9.8). — Let R be a ring, M a flat module, and R′ an algebra. Show that M ⊗R R′ is a flat R′ -module. Solution: Cancellation (8.10) yields (M ⊗R R′ ) ⊗R′ • = M ⊗R •. Therefore, (M ⊗R R′ ) ⊗R′ • is exact. Thus M ⊗R R′ is R′ -flat. □ 160 Solutions: 9. Flatness Exercise (9.9). — Let R be a ring, a an ideal. Assume that R/a is R-flat. Show that a = a2 . Solution: Since R/a is flat, tensoring it with the inclusion a ֒→ R yields an injection a ⊗R (R/a) ֒→ R ⊗R (R/a). But the image vanishes: a ⊗ r = 1 ⊗ ar = 0. Further, a ⊗R (R/a) = a/a2 by (8.13). Hence a/a2 = 0. Thus a = a2 . □ Exercise (9.10). — Let R be a ring, R′ a flat algebra with structure map φ. Then R′ is said to be faithfully flat if for every R-module M , the map M → M ⊗ R′ given by x 7→ x ⊗ 1 is injective. Show that the following conditions are equivalent: (1) (2) (3) (4) (5) R′ is faithfully flat. Every ideal a of R is the contraction of its extension, or a = φ−1 (aR′ ). Every prime p of R is the contraction of some prime q of R′ , or p = φ−1 q . Every maximal ideal m of R extends to a proper ideal, or mR′ ̸= R′ . Every nonzero R-module M remains nonzero when tensored with R′ , or M ⊗R R′ ̸= 0. Solution: Assume (1). Then R/a → (R/a) ⊗ R′ is injective. Hence (8.13)(1) implies R/a → R′ /aR′ is injective. Thus (2) holds. Assume (2). Let p be a prime ideal of R. Then (2) yields p = φ−1 (pR′ ). Thus (3.10) yields (3). Assume (3). Let m be a maximal ideal of R. By (3), there is a prime ideal n of R′ with φ−1 (n) = m. So mR′ ⊂ n. Thus (4) holds. Assume (4). Take a nonzero m ∈ M ; set M ′ := Rm. As R′ is flat, the inclusion M ′ ֒→ M yields an injection M ′ ⊗R′ ֒→ M ⊗R′ . So it suffices to show M ′ ⊗R R′ ̸= 0. Note M ′ = R/a for some a by (4.7). So M ′ ⊗R R′ = R′ /aR′ by (8.13)(1). Take a maximal ideal m ⊃ a. Then aR′ ⊂ mR′ . But mR′ ⫋ R′ by (4). Hence M ′ ⊗R R′ ̸= 0. Thus (5) holds. Assume (5). Set K := Ker(M → M ⊗R R′ ). Since R′ is flat, the induced sequence α → M ⊗R R ′ ⊗R R ′ 0 → K ⊗R R ′ → M ⊗R R ′ − is exact. But α has a retraction, namely, m ⊗ x ⊗ y 7→ m ⊗ xy; hence, α is injective. Thus K ⊗R R′ = 0. Hence (5) yields K = 0. Thus (1) holds. □ Exercise (9.11). — Let A and B be local rings, m and n their maximal ideals. Let φ : A → B be a local homomorphism; that is, φ(m) ⊂ n. Assume φ is flat. Show that φ is faithfully flat. Solution: The assertion results from (9.10), as (4) holds since φ(m) ⊂ n. □ Exercise (9.15). — Let R be a ring, R′ an algebra, M and N modules. Show that there is a canonical map σ : HomR (M, N ) ⊗R R′ → HomR′ (M ⊗R R′ , N ⊗R R′ ). Assume R′ is flat over R. Show that if M is finitely generated, then σ is injective, and that if M is finitely presented, then σ is an isomorphism. Solution: Simply put R′ := R and P := R′ in (9.14), put P := N ⊗R R′ in the second equation in (8.10), and combine the two results. □ Solutions: 10. Cayley–Hamilton Theorem 161 Exercise (9.19) (Equational Criterion for Flatness). — Prove ∑ that Condition (9.18)(4) can be reformulated as follows: For every relation i xi yi = 0 with xi ∈ R and yi ∈ M , there are xij ∈ R and yj′ ∈ M such that ∑ ∑ ′ (9.19.1) i xij xi = 0 for all j. j xij yj = yi for all i and Solution: Assume e1 , . . . , em be the standard basis of Rm . ∑m (9.18)(4) holds. Let m Given∑a relation 1 xi yi = 0, define α : R → M by α(ei ) := yi for each i. Set β φ →M → Rn − k := xi ei . Then α(k) = 0. So (9.18)(4) yields a factorization α : Rm − ′ ′ n ′ ′ with φ(k) = 0. Let e1 , . . . , en be the standard basis of R , and set∑ yj := β(ej ) for each ∑ j. Let (xij ) be the n × m matrix of φ; that is, φ(e ) = xji e′j . Then i ∑ yi = xji yj′ . Now, φ(k) = 0; hence, i,j xji xi e′j = 0. Thus (9.19.1) holds. ∑ Conversely, given α : Rm → M and k ∈ Ker(α), write k = xi e i . ∑ Assume (9.19.1). Let∑ φ : Rm → Rn be the map with matrix (xij ); that is, φ(ei ) = xji e′j . ′ n ′ ′ Then φ(k) = xi xji ej = 0. Define β : R → M by β(ej ) := yj . Then βφ(ei ) = yi ; hence, βφ = α. Thus (9.18)(4) holds. □ Exercise (9.22). — Let R be a domain, M a module. Prove that, if M is flat, then M is torsion free; that is, µx : M → M is injective for all nonzero x ∈ R. Prove that, conversely, if R is a PID and M is torsion free, then M is flat. Solution: Since R is a domain, µx : R → R is injective. So if M is flat, then µx ⊗ M : R ⊗ M → R ⊗ M is injective too. But R ⊗ M = M by (8.5). Conversely, assume R is a PID and M is torsion free. Let a be a nonzero ideal, say a = ⟨x⟩. Define α : R → a by α(y) := xy. Then α is injective as R is a domain and x ̸= 0. Further, α is surjective as a = ⟨x⟩. So α is bijective. Consider the composition α⊗M β : M = R ⊗ M −−−→ a ⊗ M → M. Clearly, β = µx . So β is injective since M is torsion free. Hence a ⊗ M → M is injective too. So M is flat by the Ideal Criterion (9.20). □ 10. Cayley–Hamilton Theorem Exercise (10.6). — Let R be a ring, a an ideal. Assume a is finitely generated and idempotent (or a = a2 ). Prove there is a unique idempotent e with ⟨e⟩ = a. Solution: By (10.3) with a for M , there is e ∈ a such that (1 − e)a = 0. So for all x ∈ a, we have (1 − e)x = 0, or x = ex. Thus a = ⟨e⟩ and e = e2 . Finally, e is unique by (1.16)(2). □ Exercise (10.8). — Prove the following conditions on a ring R are equivalent: (1) (2) (3) (4) R is absolutely flat; that is, every module is flat. Every finitely generated ideal is a direct summand of R. Every finitely generated ideal is idempotent. Every principal ideal is idempotent. 162 Solutions: 10. Cayley–Hamilton Theorem Solution: Assume (1). Let a be a finitely generated ideal. Then R/a is flat by hypotheses. So a is a direct summand of R by (10.7). Thus (2) holds. Conditions (2) and (3) are equivalent by (10.7). Trivially, if (3) holds, then (4) does. Conversely, assume (4). Given a finitely generated ideal a, say a = ⟨x1 , . . . , xn ⟩. Then each ⟨xi ⟩ is idempotent by hypothesis. So ⟨xi ⟩ = ⟨fi ⟩ for some idempotent fi by (1.16)(2). Then a = ⟨f1 , . . . , fn ⟩. Hence a is idepotent by (1.16)(5), (1). Thus (3) holds. Assume (2). Let M be a module, and a a finitely generated ideal. Then a is a ∼ aM direct summand of R by hypothesis. So R/a is flat by (9.5). Hence a ⊗ M −→ by (9.12)(1). So M is flat by (9.20). Thus (1) holds. □ Exercise (10.9). — Let R be a ring. (1) (2) (3) (4) Assume Assume Assume Assume R R R R is is is is Boolean. Prove R is absolutely flat. absolutely flat. Prove any quotient ring R′ is absolutely flat. absolutely flat. Prove every nonunit x is a zerodivisor. absolutely flat and local. Prove R is a field. Solution: In (1), as R is Boolean, every element is idempotent. Hence every principal ideal is idempotent by (1.15)(1). Thus (10.8) yields (1). For (2), let b ⊂ R′ be principal, say b = ⟨x⟩. Let x ∈ R lift x. Then ⟨x⟩ is idempotent by (10.8). Hence b is also idempotent. Thus (10.8) yields (2). For (3) and (4), take a nonunit x. Then ⟨x⟩ is idempotent by (10.8). So x = ax2 for some a. Then x(ax − 1) = 0. But x is a nonunit. So ax − 1 ̸= 0. Thus (3) holds. Suppose R is local, say with maximal ideal m. Since x is a nonunit, x ∈ m. So ax ∈ m. So ax − 1 ∈ / m. So ax − 1 is a unit. But x(ax − 1) = 0. So x = 0. Thus 0 is the only nonunit. Thus (4) holds. □ Exercise (10.12). — Let R be a ring, m ⊂ rad(R) an ideal. Let α, β : M → N be two maps of finitely generated modules. Assume α is surjective and β(M ) ⊂ mN . Set γ := α + β. Show that γ is an isomorphism. Solution: As α is surjective, given n ∈ N , there is m ∈ M with α(m) = n. So n = α(m) + β(m) − β(m) ∈ γ(M ) + mN. Hence γ(M ) = N by (10.11). So γ is an isomorphism by (10.4). □ Exercise (10.13). — Let A be a local ring, m the maximal ideal, M a finitely generated A-module, and m1 , . . . , mn ∈ M . Set k := A/m and M ′ := M/mM , and write m′i for the image of mi in M ′ . Prove that m′1 , . . . , m′n ∈ M ′ form a basis of the k-vector space M ′ if and only if m1 , . . . , mn form a minimal generating set of M (that is, no proper subset generates M ), and prove that every minimal generating set of M has the same number of elements. Solution: By (10.11), reduction mod m gives a bijective correspondence between generating sets of M as an A-module, and generating sets of M ′ as an A-module, or equivalently by (4.5), as an k-vector space. This correspondence preserves inclusion. Hence, a minimal generating set of M corresponds to a minimal generating set of M ′ , that is, to a basis. But any two bases have the same number of elements. □ Solutions: 10. Cayley–Hamilton Theorem 163 Exercise (10.14). — Let A be a local ring, k its residue field, M and N finitely generated modules. (1) Show that M = 0 if and only if M ⊗A k = 0. (2) Show that M ⊗A N ̸= 0 if M ̸= 0 and N ̸= 0. Solution: Let m be the maximal ideal. Then M ⊗ k = M/mM by (8.13)(1). So (1) is nothing but a form of Nakayama’s lemma (10.10). In (2), M ⊗ k ̸= 0 and N ⊗ k ̸= 0 by (1). So (M ⊗ k) ⊗ (N ⊗ k) ̸= 0 by (8.14) and (8.8). But (M ⊗ k) ⊗ (N ⊗ k) = (M ⊗ N ) ⊗ (k ⊗ k) by the associative and commutative laws. Finally, k ⊗ k = k by (8.13)(1). □ Exercise (10.17). — Let G be a finite group acting on a domain R, and R′ the ring of invariants. Show every x ∈ R is integral over R′ , in fact, over the subring R′′ generated by the elementary symmetric functions in the conjugates gx for g ∈ G. ∏ Solution: Given an x ∈ R, form F (X) := g∈G (X − gx). Then the coefficients of F (X) are the elementary symmetric functions in the conjugates gx for g ∈ G; hence, they are invariant under the action of G. So F (x) = 0 is a relation of integral dependence for x over R′ , in fact, over its subring R′′ . □ Exercise (10.19). — Let k be a field, P := k[X] the polynomial ring in one variable, f ∈ P . Set R := k[X 2 ] ⊂ P . Using the free basis 1, X of P over R, find an explicit equation of integral dependence of degree 2 on R for f . Solution: Write f = fe + fo , where fe and fo are the polynomials formed by the terms of f of even and odd degrees. Say fo = gX. Then the matrix of µf is ( f gX 2 ) e . Its characteristic polynomial is T 2 − 2fe T + fe2 − fo2 . So the Cayley– g fe Hamilton Theorem (10.1) yields f 2 − 2fe f + fe2 − fo2 = 0. □ Exercise (10.24).∏— Let R1 , . . . , Rn be R-algebras that are integral over R. Show that their product Ri is a integral over R. ∏n Solution: Let y = (y1 , . . . , yn ) ∈ ∏ i=1 Ri . Since Ri /R is integral, R[yi ] is a n module-finite R-subalgebra of R . Hence module-finite R-subalgebra i i=1 R[yi ] is a∏ ∏n n of i=1 Ri by (4.14) ∏ and induction on n. Now, y ∈ i=1 R[yi ]. Therefore, y is n □ integral over R. Thus i=1 Ri is integral over R. Exercise (10.26). i ≤∏r, let Ri be a ring, Ri′ an extension of Ri , and ∏— For 1 ≤ ′ ′ xi ∈ Ri . Set R := Ri , set R := Ri′ , and set x := (x1 , . . . , xr ). Prove (1) x is integral over R if and only if xi is integral over Ri for each i; (2) R is integrally closed in R′ if and only if each Ri is integrally closed in Ri′ . Solution: Assume x is integral over R. Say xn + a1 xn−1 + · · · + an = 0 with aj ∈ R. Say aj =: (a1j , . . . , arj ). Fix i. Then xni + ai1 xn−1 + · · · + ain = 0. So xi is integral over Ri . Conversely, assume each xi is integral over Ri . Say xni i +ai1 xini −1 +· · ·+aini = 0. Set n := max ni , set aij := 0 for j > ni , and set aj := (a1j , . . . , arj ) ∈ R for each j. Then xn + a1 xn−1 + · · · + an = 0. Thus x is integral over R. Thus (1) holds. Assertion (2) is an immediate consequence of (1). □ 164 Solutions: 11. Localization of Rings Exercise (10.30). — Let k be a field, X and Y variables. Set R := k[X, Y ]/⟨Y 2 − X 2 − X 3 ⟩, and let x, y ∈ R be the residues of X, Y . Prove that R is a domain, but not a field. Set t := y/x ∈ Frac(R). Prove that k[t] is the integral closure of R in Frac(R). Solution: As k[X, Y ] is a UFD and Y 2 −X 2 −X 3 is irreducible, ⟨Y 2 −X 2 −X 3 ⟩ is prime by (2.6); however, it is not maximal by (2.27). Hence R is a domain by (2.9), but not a field by (2.17). Note y 2 − x2 − x3 = 0. Hence x = t2 − 1 and y = t3 − t. So k[t] ⊃ k[x, y] = R. Further, t is integral over R; so k[t] is integral over R by (2)⇒(1) of (10.23). Finally, k[t] has Frac(R) as fraction field. Further, Frac(R) ̸= R, so x and y cannot be algebraic over k; hence, t must be transcendental. So k[t] is normal by (10.29)(1). Thus k[t] is the integral closure of R in Frac(R). □ 11. Localization of Rings Exercise (11.2). — Let R be a ring, S a multiplicative subset. Prove S −1 R = 0 if and only if S contains a nilpotent element. Solution: By (1.1), S −1 R = 0 if and only if 1/1 = 0/1. But by construction, 1/1 = 0/1 if and only if 0 ∈ S. Finally, since S is multiplicative, 0 ∈ S if and only if S contains a nilpotent element. □ Exercise (11.3). — Let R be a ring, S a multiplicative subset, S its saturation. Set T := (S −1 R)× . Show T = { x/s | x ∈ S and s ∈ S }. Show φ−1 S T = S. Solution: First, given x ∈ S and s ∈ S, take y ∈ R such that xy ∈ S. Then x/s · sy/xy = 1 in S −1 R. Thus x/s ∈ T . Conversely, say x/s · y/t = 1 in S −1 R with x, y ∈ R and s, t ∈ S. Then there’s u ∈ S with xyu = stu in R. But stu ∈ S. Thus x ∈ S. Thus the first assertion holds. Set U := φ−1 S T . Then U is saturated multiplicative by (3.12). Further, U ⊃ S by (11.1). Thus (1)(c) of (3.14) yields U ⊃ S. Conversely, take x ∈ U . Then x/1 ∈ T . So the first assertion yields x/1 = y/s with y ∈ S and s ∈ S. So there’s t ∈ S with xst = yt in R. But S ⊃ S by (1)(a) of (3.14), and S is multiplicative by (1)(b); so yt ∈ S. But S is saturated by (1)(b). Thus x ∈ S. Thus U = S. □ Exercise (11.5). — Find all intermediate rings Z ⊂ R ⊂ Q, and describe each R as a localization of Z. As a starter, prove Z[2/3] = S −1 Z where S = {3i | i ≥ 0}. Solution: Clearly Z[2/3] ⊂ Z[1/3] as 2/3 = 2·(1/3). But the opposite inclusion holds as 1/3 = 1 − (2/3). Obviously, S −1 Z = Z[1/3]. Let P ⊂ Z be the set of all prime numbers that appear as factors of the denominators of elements of R in lowest terms; recall that x = r/s ∈ Q is in lowest terms if r and s have no common prime divisor. Let S be the multiplicative subset generated by P , that is, the smallest multiplicative subset containing P . Clearly, S is equal to the set of all products of elements of P . First note that, if p ∈ P , then 1/p ∈ R. Indeed, take an element x = r/ps ∈ R in lowest terms. Then sx = r/p ∈ R. Also the Euclidean algorithm yields m, n ∈ Z such that mp + nr = 1. Then 1/p = m + nsx ∈ R, as desired. Hence S −1 Z ⊂ R. But the opposite inclusion holds because, by the very definition of S, every element Solutions: 11. Localization of Rings 165 of R is of the form r/s for some s ∈ S. Thus S −1 Z = R. □ Exercise (11.8). — Let R′ and R′′ be rings. Consider R := R′ × R′′ and set S := { (1, 1), (1, 0) }. Prove R′ = S −1 R. Solution: Let’s show that the projection map π : R′ × R′′ → R′ has the UMP of (11.6). First, note that πS = {1} ⊂ R′× . Let ψ : R′ × R′′ → B be a ring map such that ψ(1, 0) ∈ B × . Then in B, ( ) ψ(1, 0) · ψ(0, x) = ψ (1, 0) · (0, x) = ψ(0, 0) = 0 in B. Hence ψ(0, x) = 0 for all x ∈ R′′ . So ψ factors uniquely through π by (1.5). □ Exercise (11.9). — Take R and S as in (11.8). On R × S, impose this relation: (x, s) ∼ (y, t) if xt = ys. Prove that it is not an equivalence relation. Solution: Observe that, for any z ∈ R′′ , we have ( ) ( ) (1, z), (1, 1) ∼ (1, 0), (1, 0) . However, if z ̸= 0, then ( ) ( ) (1, z), (1, 1) ∼ ̸ (1, 0), (1, 1) . Thus although ∼ is reflexive and symmetric, it is not transitive if R′′ ̸= 0. □ Exercise (11.15). — Let R be a ring, S a multiplicative subset, a and b ideals. Show (1) if a ⊂ b, then aS ⊂ bS ; (2) (aS )S = aS ; and (3) (aS bS )S = (ab)S . Solution: For (1), take x ∈ aS . Then there is s ∈ S with sx ∈ a. If a ⊂ b, then sx ∈ b, and so x ∈ bS . Thus (1) holds. To show (2), proceed by double inclusion. First, note aS ⊃ a by (11.14)(2). So (aS )S ⊃ aS again by (11.14)(2). Conversely, given x ∈ (aS )S , there is s ∈ S with sx ∈ aS . So there is t ∈ S with tsx ∈ a. But ts ∈ S. So x ∈ aS . Thus (2) holds. To show (3), proceed by double inclusion. First, note a ⊂ aS and b ⊂ bS by S S S S (11.14)(2). So ab ⊂ aS bS . Thus (11.14)(2) b ) . ∑ yields (ab) ⊂ (a S S S Conversely, given x ∈ a b , say x := yi zi with yi ∈ a ∏ and zi ∈ bS . Then there are si , ti ∈ S such that si yi ∈ a and ti zi ∈ b. Set u := si ti . Then u ∈ S and ux ∈ ab. So x ∈ (ab)S . Thus aS bS ⊂ (ab)S . So (aS bS )S ⊂ ((ab)S )S by (1). But ((ab)S )S = (ab)S by (2). Thus (3) holds. □ Exercise (11.16). — Let R be a ring, S a multiplicative subset. Prove that nil(R)(S −1 R) = nil(S −1 R). Solution: Proceed by double inclusion. Given an element of nil(R)(S −1 R), put it in the form x/s with x ∈ nil(R) and s ∈ S using (11.12)(1). Then xn = 0 for some n ≥ 1. So (x/s)n = 0. So x/s ∈ nil(S −1 R). Thus nil(R)(S −1 R) ⊂ nil(S −1 R). Conversely, take x/s ∈ nil(S −1 R). Then (x/s)m = 0 with m ≥ 1. So there’s t ∈ S with txm = 0 by (11.14)(1). Then (tx)m = 0. So tx ∈ nil(R). But tx/ts = x/s. So x/s ∈ nil(R)(S −1 R) by (11.12)(1). Thus nil(R)(S −1 R) ⊃ nil(S −1 R). □ Exercise (11.23). — Let R′ /R be an integral extension of rings, and S a multiplicative subset of R. Show that S −1 R′ is integral over S −1 R. 166 Solutions: 11. Localization of Rings Solution: Given x/s ∈ S −1 R′ , let xn + an−1 xn−1 + · · · + a0 = 0 be an equation of integral dependence of x on R. Then (x/s)n + (an−1 /1)(1/s)(x/s)n−1 + · · · + a0 (1/s)n = 0 is an equation of integral dependence of x/s on S −1 R, as required. □ Exercise (11.24). — Let R be a domain, K its fraction field, L a finite extension field, and R the integral closure of R in L. Show that L is the fraction field of R. Show that, in fact, every element of L can be expressed as a fraction b/a where b is in R and a is in R. Solution: Let x ∈ L. Then x is algebraic (integral) over K, say xn + y1 xn−1 + · · · + yn = 0 with yi ∈ K. Write yi = ai /a with a1 , . . . , an , a ∈ R. Then (ax)n + (aa1 )(ax)n−1 + · · · + an a0 = 0. Set b := ax. Then b ∈ R and x = b/a. □ Exercise (11.25). — Let R ⊂ R′ be domains, K and L their fraction fields. Assume that R′ is a finitely generated R-algebra, and that L is a finite dimensional K-vector space. Find an f ∈ R such that Rf′ is module finite over Rf . Solution: Let x1 , . . . , xn generate R′ over R.∏Using (11.24), write xi = bi /ai with bi integral over R and ai in R. Set f := ai . The xi generate Rf′ as an ′ Rf -algebra; so the bi do too. Thus Rf is s module finite over Rf by (10.23). □ Exercise (11.28). — Let R be a ring, S and T multiplicative subsets. (1) Set T ′ := φS (T ) and assume S ⊂ T . Prove T −1 R = T ′−1 (S −1 R) = T −1 (S −1 R). (2) Set U := {st ∈ R | s ∈ S and t ∈ T }. Prove T −1 (S −1 R) = S −1 (T −1 R) = U −1 R. (3) Let S ′ := {t′ ∈ R | t′ t ∈ S for some t ∈ R}. Prove S ′−1 R = S −1 R. Solution: A proof similar to that of (11.26) shows T −1 R = T ′−1 (S −1 R). By (11.22), T ′−1 (S −1 R) = T −1 (S −1 R). Thus (1) holds. As 1 ∈ T , obviously S ⊂ U . So (1) yields U −1 R = U −1 (S −1 R). Now, clearly −1 U (S −1 R) = T −1 (S −1 R). Similarly, U −1 R = S −1 (T −1 R). Thus (2) holds. Finally, in any ring, a product is a unit if and only if each factor is a unit. So a × × homomorphism φ : R → R′ carries S ′ into R′ if and only if φ carries S into R′ . ′−1 −1 Thus S R and S R are universal examples of R-algebras that satisfy equivalent conditions. Thus (3) holds. □ Exercise (11.31) (Localization and normalization commute). — Given a domain R and a multiplicative subset S with 0 ∈ / S. Show that the localization of the normalization S −1 R is equal to the normalization of the localization S −1 R. Solutions: 12. Localization of Modules 167 Solution: Since 0 ∈ / S, clearly Frac(S −1 R) = Frac(R) owing to (11.4). Now, S R is integral over S −1 R by (11.23). Thus S −1 R ⊂ S −1 R. Conversely, given x ∈ S −1 R, consider an equation of integral dependence: −1 xn + a1 xn−1 + · · · + an = 0. ∏ Say ai = bi /si with bi ∈ R and si ∈ S; set s := si . Multiplying by sn yields (sx)n + sa1 (sx)n−1 + · · · + sn an = 0. Hence sx ∈ R. So x ∈ S −1 R. Thus S −1 R ⊃ S −1 R, as desired. □ 12. Localization of Modules Exercise (12.4). — Let R be a ring, S a multiplicative subset, and M a module. Show that M = S −1 M if and only if M is an S −1 R-module. Solution: If M = S −1 M , then obviously M is an S −1 R-module. Conversely, if M is an S −1 R-module, then M equipped with the identity map has the UMP that characterizes S −1 M ; whence, M = S −1 M . □ Exercise (12.5). — Let R be a ring, S ⊂ T multiplicative subsets, M a module. Set T1 := φS (T ) ⊂ S −1 R. Show T −1 M = T −1 (S −1 M ) = T1−1 (S −1 M ). Solution: Let’s check that both T −1 (S −1 M ) and T1−1 (S −1 M ) have the UMP characterizing T −1 M . Let ψ : M → N be an R-linear map into an T −1 R-module. Then the multiplication map µs : N → N is bijective for all s ∈ T by (12.1), so for all s ∈ S since S ⊂ T . Hence ψ factors via a unique S −1 R-linear map ρ : S −1 M → N by (12.3) and by (12.1) again. Similarly, ρ factors through a unique T −1 R-linear map ρ′ : T −1 (S −1 M ) → N . Hence ψ = ρ′ φT φS , and ρ′ is clearly unique, as required. Also, ρ factors through a unique T1−1 (S −1 R)-linear map ρ′1 : T1−1 (S −1 M ) → N . Hence ψ = ρ′1 φT1 φS , and ρ′1 is clearly unique, as required. □ Exercise (12.6). — Let R be a ring, S a multiplicative subset. Show that S becomes a filtered category when equipped as follows: given s, t ∈ S, set Hom(s, t) := {x ∈ R | xs = t}. Given a module M , define a functor S → ((R-mod)) as follows: for s ∈ S, set Ms := M ; to each x ∈ Hom(s, t), associate µx : Ms → Mt . Define βs : Ms → S −1 M ∼ S −1 M . by βs (m) := m/s. Show the βs induce an isomorphism lim Ms −→ −→ Solution: Clearly, S is a category. Now, given s, t ∈ S, set u := st. Then u ∈ S; also t ∈ Hom(s, u) and s ∈ Hom(t, u). Given x, y ∈ Hom(s, t), we have xs = t and ys = t. So s ∈ Hom(t, u) and xs = ys in Hom(s, u). Thus S is filtered. Further, given x ∈ Hom(s, t), we have βt µx = βs since m/s = xm/t as xs = t. So the βs induce a homomorphism β : lim Ms → S −1 M . Now, every element of −→ S −1 M is of the form m/s, and m/s =: βs (m); hence, β is surjective. Each m ∈ lim Ms lifts to an m′ ∈ Ms for some s ∈ S by (7.8)(1). Assume −→ βm = 0. Then βs m′ = 0 as the βs induce β. But βs m′ = m′ /s. So there is t ∈ S with tm′ = 0. So µt m′ = 0 in Mst , and µt m′ 7→ m. So m = 0. Thus β is injective, so an isomorphism. □ 168 Solutions: 13. Support Exercise (12.7). — Let R be a ring, S a multiplicative subset, M a module. Prove S −1 M = 0 if Ann(M ) ∩ S ̸= ∅. Prove the converse if M is finitely generated. Solution: Say f ∈ Ann(M )∩S. Let m/t ∈ S −1 M . Then f /1·m/t = f m/t = 0. Hence m/t = 0. Thus S −1 M = 0. Conversely, assume S −1 M = 0, and ∏ say m1 , . . . mn generate M . Then for each i, there is fi ∈ S with fi mi = 0. Then fi ∈ Ann(M ) ∩ S, as desired. □ Exercise (12.11). — Let R be a ring, S a multiplicative subset, P a projective module. Then S −1 P is a projective S −1 R-module. Solution: By (5.22), there is a module K such that F := K ⊕ P is free. So (12.9) yields that S −1 F = S −1 P ⊕ S −1 L and that S −1 F is free over S −1 R. Hence S −1 P is a projective S −1 R-module again by (5.22). □ Exercise (12.13). — Let R be a ring, S a multiplicative subset, M and N modules. Show S −1 (M ⊗R N ) = S −1 M ⊗R N = S −1 M ⊗S −1 R S −1 N = S −1 M ⊗R S −1 N. Solution: By (12.12), S −1 (M ⊗R N ) = S −1 R ⊗R (M ⊗R N ). The latter is equal to (S −1 R ⊗R M ) ⊗R N by associativity (8.9). Again by (12.12), the latter is equal to S −1 M ⊗R N . Thus the first equality holds. By cancellation (8.10), S −1 M ⊗R N = S −1 M ⊗S −1 R (S −1 R ⊗R N ), and the latter is equal to S −1 M ⊗S −1 R S −1 N by (12.12). Thus the second equality holds. Finally by (8.8), the kernel of the map S −1 M ⊗R S −1 N → S −1 M ⊗S −1 R S −1 N is generated by elements (xm/s) ⊗ (n/1) − (m/1) ⊗ (xn/s) with m ∈ M , n ∈ N , x ∈ R, and s ∈ S. Those elements are zero because µs is an isomorphism on the S −1 R-module S −1 M ⊗R S −1 N . Thus the third equality holds. □ ⊕ Exercise (12.24). — Set R := Z and S = Z − ⟨0⟩. Set M := n≥2 Z/⟨n⟩ and N := M . Show that the map σ of (12.21) is not injective. Solution: Given m > 0, let en be the nth standard basis element for some n > m. Then m · en ̸= 0. Hence µR : R → HomR (M, M ) is injective. But S −1 M = 0, as any x ∈ M has only finitely many nonzero components; so kx = 0 for some nonzero integer k. So Hom(S −1 M, S −1 M ) = 0. Thus σ is not injective. □ 13. Support Exercise (13.2). — Let R be a ring, p ∈ Spec(R). Show that p is a closed point — that is, {p} is a closed set — if and only if p is a maximal ideal. Solution: If p is maximal, then V(p) = {p}; so p is closed. Conversely, suppose p is not maximal. Then p ⫋ m for some maximal ideal m. If p ∈ V(a), then m ∈ V(a) too. So {p} ̸= V(a). Thus {p} is not closed. □ Exercise (13.3). — Let R be a ring, R′ a flat algebra with structure map φ. Show that R′ is faithfully flat if and only if Spec(φ) is surjective. Solution: Owing to the definition of Spec(φ) in (13.1), the assertion amounts to the equivalence of (1) and (3) of (9.10). □ Exercise (13.5). — Let R be a ring, X := Spec(R), and U an open subset. Show U is quasi-compact if and only if X − U = V (a) where a is finitely generated. Solutions: 13. Support 169 ∪ Solution: ∪nAssume U is quasi-compact. By (13.1), ∩ U = λ D(fλ ) for some fλ . Hence U = 1 D(fi ) for some fi . Thus X − U = V(fi ) = ∪nV(⟨f1 , . . . , fn ⟩). Conversely, assume X − U = V(⟨f1 , . . . , fn ⟩). Then U = i=1 D(fi ). By (13.4), each D(fi ) is quasi-compact. Thus U is quasi-compact. □ Exercise (13.6). — Let B be a Boolean ring, and set X := Spec(B). Show X is a compact Hausdorff space. (Following Bourbaki, “quasi-compact” is shortened to “compact” when the space is Hausdorff.) Further, show a subset U ⊂ X is both open and closed if and only if U = D(f ) for some f ∈ B. Solution: Let f ∈ B. Then D(f ) ∪ D(1 − f ) = X whether B is Boolean or not; indeed, if p ∈ X − D(f ), then f ∈ p, so 1 − f ∈ / p, so p ∈ D(1 − f ). Further, D(f ) ∩ D(1 − f ) = ∅; indeed, if p ∈ D(f ), then f ∈ / p, but f (1 − f ) = 0 as B is Boolean, so 1 − f ∈ p, so p ∈ / D(1 − f ). Thus X − D(f ) = D(1 − f ). Thus D(f ) is closed as well as open. Let p, q be prime ideals with p ̸= q. Then there is f ∈ p − q. So p ∈ / D(f ), but q ∈ D(f ). By the above, D(f ) is both open and closed. Thus X is Hausdorff. By (13.4), X is quasi-compact, so compact as it is Hausdorff. Finally, let U ⊂ X be open and closed. Then U is quasi-compact, as U is closed and X is quasi-compact. So X − U = V (a) where a is finitely generated by (13.5). Since B is Boolean, a = ⟨f ⟩ for some f ∈ B by (1.16) (5). Thus U = D(f ). □ Exercise (13.7) (Stone’s Theorem). — Show every Boolean ring B is isomorphic to the ring of continuous functions from a compact Hausdorff space X to F2 with the discrete topology. Equivalently, show B is isomorphic to the ring R of open and ∼ R is given by f 7→ D(f ). closed subsets of X; in fact, X := Spec(B), and B −→ Solution: The two statements are equivalent by (1.2). Further, X := Spec(B) is compact Hausdorff, and its open and closed subsets are precisely the D(f ) by (13.6). Thus f 7→ D(f ) is a well defined function, and is surjective. This function preserves multiplication owing to (13.1.1). To show it preserves addition, we must show that, for any f, g ∈ B, D(f + g) = (D(f ) − D(g)) ∪ (D(g) − D(f )). (13.7.1) Fix a prime p. There are four cases. First, if f ∈ / p and g ∈ p, then f + g ∈ / p. Second, if g ∈ / p but f ∈ p, then again f + g ∈ / p. In both cases, p lies in the open sets on both sides of (13.7.1). Third, if f ∈ p and g ∈ p, then f + g ∈ p. The first three cases do not use the hypothesis that B is Boolean. The fourth does. Suppose f ∈ / p and g ∈ / p. Now, B/p = F2 by (2.18). So the residues of f and g are both equal to 1. But 1 + 1 = 0 ∈ F2 . So again f + g ∈ p. Thus in both the third and fourth cases, p lies in neither side of (13.7.1). Thus (13.7.1) holds. Finally, to show that f 7→ D(f ) is injective, suppose that D(f ) is empty. Then f ∈ nil(B). But nil(B) = ⟨0⟩ by (3.23). Thus f = 0. □ Exercise (13.14). — Let R be a ring, M a module, p ∈ Supp(M ). Prove V(p) ⊂ Supp(M ). Solution: Let q ∈ V(p). Then q ⊃ p. So Mp = (Mq )p by (11.28)(1). Now, p ∈ Supp(M ). So Mp ̸= 0. Hence Mq ̸= 0. Thus q ∈ Supp(M ). □ 170 Solutions: 13. Support Exercise (13.15). — Let Z be the integers, Q the rational numbers, and set M := Q/Z. Find Supp(M ), and show that it is not Zariski closed. Solution: Let p ∈ Spec(R). Then Mp = Qp /Zp since localization is exact by (12.16). Now, Qp = Q by (12.4) and (12.1) since Q is a field. If p ̸= ⟨0⟩, then Zp ̸= Qp since pZp ∩ Z = p by (11.17). If p = ⟨0⟩, then Zp = Qp . Thus Supp(M ) consists of all the nonzero primes of Z. Finally, suppose Supp(M ) = V(a). Then a lies in every nonzero prime; so a = ⟨0⟩. But ⟨0⟩ is prime. Hence ⟨0⟩ ∈ V(a) = Supp(M ), contradicting the above. Thus Supp(M ) is not closed. □ Exercise (13.17). — Let R be a ring, P a module, and M, N submodules. Show M = N if Mm = Nm for every maximal ideal m. First assume M ⊂ N . Solution: If M ⊂ N , then (12.16) yields (N/M )m = Nm /Mm = 0 for each m; so N/M = 0 by (13.16). The general case follows by replacing N by M + N owing to (12.15)(4), (5). □ Exercise (13.18). — Prove these three conditions on a ring R are equivalent: (1) R is reduced. (2) S −1 R is reduced for all multiplicatively closed sets S. (3) Rm is reduced for all maximal ideals m. Solution: Assume (1) holds. Then nil(R) = 0. But nil(R)(S −1 R) = nil(S −1 R) by (11.16). Thus (2) holds. Trivially (2) implies (3). Assume (3) holds. Then nil(Rm ) = 0. Hence nil(R)m = 0 by (11.16) and (12.2). So nil(R) = 0 by (13.16). Thus (1) holds. □ Exercise (13.19). — Let R be a ring, Σ the set of minimal primes. Prove this: (1) If Rp is a domain for any prime p, then the p ∈ Σ are pairwise comaximal. ∏n (2) Rp is a domain for any prime p and Σ is finite if and only if R = i=1 Ri where Ri is a domain. If so, then Ri = R/pi with {p1 , . . . , pn } = Σ. Solution: Consider (1). Suppose p, q ∈ Σ are not comaximal. Then p + q lies in some maximal ideal m. Hence Rm contains two minimal primes, pRm and qRm , by (11.18). However, Rm is a domain by hypothesis, and so ⟨0⟩ is its only minimal prime. Hence pRm = qRm . So p = q. Thus (1) holds. Consider (2). Assume Rp is a domain for any p. Then R∏is reduced by (13.18). Assume, also, Σ is finite. Form the canonical map φ : R → p∈Σ R/p; it is injective by (3.28), and surjective by (1) and the Chinese Remainder Theorem (1.13). Thus R is a finite product of domains. ∏n Conversely,∏assume R = i=1 Ri where Ri is a domain. Let p be a prime of R. Then Rp = (Ri )p by (12.10). Each (Ri )p is a domain by (11.4). But Rp is local. So Rp = (Ri )p for some i by (2.5). ∏ Thus Rp is a domain. Further, owing to (2.12), each pi ∈ Σ has the form pi = aj where, after renumbering, ai = ⟨0⟩ and ∼ R . Thus (2) holds. aj = Rj for j ̸= i. Thus the ith projection gives R/pi −→ □ i Exercise (13.21). — Let R be a ring, M a module. Prove elements mλ ∈ M generate M if and only if, at every maximal ideal m, their images mλ generate Mm . Solutions: 14. Krull–Cohen–Seidenberg Theory 171 Solution: The mλ define a map α : R⊕{λ} → M . By (13.20), it is surjective ) ( ) ( ⊕{λ} if and only if αm : R⊕{λ} m → Mm is surjective for all m. But R⊕{λ} m = Rm by (12.10). Hence (4.10)(1) yields the assertion. □ Exercise (13.24). — Let R be a ring, R′ a flat algebra, p′ a prine in R′ , and p its contraction in R. Prove that Rp′ ′ is a faithfully flat Rp -algebra. Solution: First, Rp′ is flat over Rp by (13.23). Next, Rp′ ′ is flat over Rp′ by (12.17) and (11.28) as R − p ⊂ R′ − p′ . Hence Rp′ ′ is flat over Rp by (9.7). But a flat local homomorphism is faithfully flat by (9.10)(4). □ Exercise (13.28). — Given n, prove an R-module P is locally free of rank n if n holds at each maximal ideal m. and only if P is finitely generated and Pm ≃ Rm Solution: If P is locally free of rank n, then P is finitely generated by (13.27). Also, for any p ∈ Spec(R), there’s f ∈ R − p with Pf ≃ Rfn ; so Pp ≃ Rpn by (12.5). As to the converse, given any prime p, take a maximal ideal m containing it. n . Take a free basis p1 /f1k1 , . . . , pn /fnkn of Pm over Rm . The pi Assume Pm ≃ Rm n n define a map α : R → P , and αm : Rm → Pm is bijective, so surjective. Assume P is finitely generated. Then (12.20)(1) provides f ∈ R − m such that αf : Rfn → Pf is surjective. Hence αq : Rqn → Pq is surjective for every q ∈ D(f ) by (12.5) and (12.16). Assume Pq ≃ Rqn if also q is maximal. So αq is bijective by (10.4). Clearly, αq = (αf )(qRf ) . Hence αf : Rfn → Pf is bijective owing to (13.20) with Rf for R, as desired. □ Exercise (13.29). — Let A be a semilocal ring, P a locally free module of rank n. Show that P is free of rank n. Solution: As P is locally free, P is finitely presented by (13.27), and Pm ≃ Anm at the maximal m by (13.28). But A is semilocal. So P ∼ □ = An by (13.22). 14. Krull–Cohen–Seidenberg Theory Exercise (14.4). — Let R ⊂ R′ be an integral extension of rings, and p a prime of R. Suppose R′ has just one prime p′ over p. Show (a) that p′ Rp′ is the only maximal ideal of Rp′ , (b) that Rp′ ′ = Rp′ , and (c) that Rp′ ′ is integral over Rp . Solution: Since R′ is integral over R, the localization Rp′ is integral over Rp by (11.23). Moreover, Rp is a local ring with unique maximal ideal pRp by (11.20). Hence, every maximal ideal of Rp′ lies over pRp by (14.3)(1). But every maximal ideal of Rp′ is the extension of some prime q′ ⊂ R′ by (11.18)(2), and therefore q′ lies over p in R. So, by hypothesis, q′ = p′ . Thus p′ Rp′ is the only maximal ideal of Rp′ ; that is, (a) holds. So Rp′ − p′ Rp′ consists of units. Hence (11.28) and (11.7) □ yield (b). But Rp′ is integral over Rp ; so (c) holds too. Exercise (14.5). — Let R ⊂ R′ be an integral extension of domains, and p a prime of R. Suppose R′ has at least two distinct primes p′ and q′ lying over p. Show that Rp′ ′ is not integral over Rp . Show that, in fact, if y lies in q′ , but not in p′ , then 1/y ∈ Rp′ ′ is not integral over Rp . 172 Solutions: 14. Krull–Cohen–Seidenberg Theory Solution: Suppose 1/y is integral over Rp . Say (1/y)n + a1 (1/y)n−1 + · · · + an = 0 with n ≥ 1 and ai ∈ Rp . Multiplying by y n−1 , we obtain 1/y = −(a1 + · · · + an y n−1 ) ∈ Rp′ . However, y ∈ q′ , so y ∈ q′ Rp′ . Hence 1 ∈ q′ Rp′ . So q′ ∩ (R − p) ̸= ∅ by (11.17)(3). But q′ ∩ R = p, a contradiction. So 1/y is not integral over Rp . □ Exercise (14.6). — Let k be a field, and X an indeterminate. Set R′ := k[X], and Y := X 2 , and R := k[Y ]. Set p := (Y − 1)R and p′ := (X − 1)R′ . Is Rp′ ′ integral over Rp ? Explain. Solution: Note that R′ is a domain, and that the extension R ⊂ R′ is integral as R′ is generated by 1 and X as an R-module. Suppose the characteristic is not 2. Set q′ := (X + 1)R′ . Then both p′ and q′ contain Y − 1, so lie over the maximal ideal p of R. Further X + 1 lies in q′ , but not in p′ . Hence Rp′ ′ is not integral over Rp by (14.5). Suppose the characteristic is 2. Then (X − 1)2 = Y − 1. Let q′ ⊂ R′ be a prime over p. Then (X − 1)2 ∈ q′ . So p′ ⊂ q′ . But p′ is maximal. So q′ = p′ . Thus R′ has just one prime p′ over p. Hence Rp′ ′ is integral over Rp by (14.4). □ Exercise (14.12). ∪ — Let R be a reduced ring, Σ the set of minimal primes. Prove that z.div(R) = p∈Σ p and that Rp = Frac(R/p) for any p ∈ Σ. ∪ Solution: If p ∈ Σ, then p ⊂ z.div(R) by (14.10). Thus z.div(R) ⊃ p∈Σ p. ∪ Conversely, say xy = 0. If x ∈ / p for some p ∈ Σ, then y ∈ p. So if x ∈ / p∈Σ p, ∩ ∩ then y ∈ p∈Σ p. But p∈Σ p = ⟨0⟩ by the Scheinnullstellensatz (3.22) and (3.11). ∪ ∪ So y = 0. Hence if x ∈ / p∈Σ p, then x ∈ / z.div(R). Thus z.div(R) ⊂ p∈Σ p. Thus ∪ z.div(R) = p∈Σ p. Fix p ∈ Σ. Then Rp is reduced by (13.18). Further, Rp has only one prime, namely pRp , by (11.18)(2). Hence Rp is a field, and pRp = ⟨0⟩. But by (12.19), Rp /pRp = Frac(R/p). Thus Rp = Frac(R/p). □ Exercise (14.13). — Let R be a ring, Σ the set of minimal primes, and K the total quotient ring. Assume Σ is finite. Prove these three conditions are equivalent: (1) R is reduced. ∪ (2) z.div(R) = p∈Σ p, and Rp = Frac(R/p) for each p ∈ Σ. ∏ (3) K/pK = Frac(R/p) for each p ∈ Σ, and K = p∈Σ K/pK. Solution: Assume (1) holds. Then (14.12) yields (2). Assume ∪ (2) holds. Set S := R − z.div(R). Let q be a prime of R with q ∩ S = ∅. Then q ⊂ p∈Σ p. But Σ is finite. So q ⊂ p for some p ∈ Σ by Prime Avoidance (3.15). Hence q = p since p is minimal. But K = S −1 R. Therefore, by (11.18)(2), for p ∈ Σ, the extensions pK are the only primes of K, and they all are both maximal and minimal. Fix p ∈ Σ. Then K/pK = S −1 (R/p) by (12.18). So S −1 (R/p) is a field. But clearly S −1 (R/p) ⊂ Frac(R/p). Therefore, K/pK = Frac(R/p) by (2.3). Further, −1 S ⊂ R−p. Hence (11.18)(2) yields p = φ−1 S (pK). Therefore, φS (K −pK) = R−p. So KpK = Rp by (11.26). But Rp = Frac(R/p) by hypothesis. Thus K has only finitely many primes, the pK; each pK is minimal, and each KpK is a domain. Solutions: 14. Krull–Cohen–Seidenberg Theory 173 ∏ Therefore, (13.19)(2) yields K = p∈Σ K/pK. Thus (3) holds. Assume (3) holds. Then K is a finite product of fields, and fields are reduced. But clearly, a product of reduced ring is reduced. Further, R ⊂ K, and trivially, a subring of a reduced ring is reduced. Thus (1) holds. □ Exercise (14.14). — Let A be a reduced local ring with residue field k and a finite set Σ of minimal primes. For each p ∈ Σ, set K(p) := Frac(A/p). Let P be a finitely generated module. Show that P is free of rank r if and only if dimk (P ⊗A k) = r and dimK(p) (P ⊗A K(p)) = r for each p ∈ Σ. Solution: If P is free of rank r, then dim(P ⊗ k) = r and dim(P ⊗ K(p)) = r owing to (8.11). Conversely, suppose dim(P ⊗ k) = r. As P is finitely generated, (10.13) implies P is generated by r elements. So (5.19) yields an exact sequence α 0→M − → Ar → P → 0. Momentarily, fix a p ∈ Σ. Since A is reduced, K(p) = Rp by (14.12). So K(p) is flat by (12.17). So the induced sequence is exact: 0 → M ⊗ K(p) → K(p)r → P ⊗ K(p) → 0. Suppose dim(P ⊗ K(p)) = r too. It then follows that M ⊗A K(p) = 0. Let K be the total quotient ring of A, and form this commutative square: α M  −−−→  φM y Ar φ r y A M ⊗K − → Kr Here α is injective. And φAr is injective as φA : A → K∏is. Hence, φM is injective. By hypothesis, A is reduced and Σ is finite; so K = p∈Σ K(p) by (14.13). So ∏ M ⊗ K = (M ⊗ K(p)). But M ⊗A K(p) = 0 for each p ∈ Σ. So M ⊗ K = 0. But ∼ P , as desired. □ φM : M → M ⊗ K is injective. So M = 0. Thus Ar −→ Exercise (14.15). — Let A be a reduced local ring with residue field k and a finite set of minimal primes. Let P be a finitely generated module, B an A-algebra with Spec(B) → Spec(A) surjective. Show that P is a free A-module of rank r if and only if P ⊗ B is a free B-module of rank r. Solution: If P is a free A-module of rank r, then P ⊗ B is a free B-module of rank r owing to (8.11). Conversely, let p ⊂ A be a prime. Since Spec(B) → Spec(A) is surjective, there is a prime q ⊂ B whose trace is p. Set K := Frac(A/p) and L := Frac(B/q). Then the structure map A → B induces a map K → L. Moreover, (P ⊗A B) ⊗B L = (P ⊗A K) ⊗K L. (14.15.1) Suppose P ⊗ B is a free B-module of rank r. Then dimL ((P ⊗A B) ⊗B L) = r owing to (8.11). Hence (14.15.1) implies dimK (P ⊗A K) = r. But p is arbitrary. Thus P is a free A-module of rank r by (14.14), as desired. □ Exercise (14.17). — Let R be a ring, p1 . . . , pr all its minimal primes, and K the total quotient ring. Prove that these three conditions are equivalent: (1) R is normal. (2) R is reduced and integrally closed in K. 174 Solutions: 15. Noether Normalization (3) R is a finite product of normal domains Ri . If so, then the Ri are equal to the R/pj up to order. Solution: Assume (1). Then R is reduced by (13.18). Let x ∈ K be integral over R, and m any maximal ideal. Then x/1 is integral over Rm . So x/1 ∈ Rm by hypothesis. Hence (R[x]/R)m = 0. Therefore, R[x]/R = 0 by (13.16). So x ∈ R Thus (2) holds. ∏ Assume (2). Set Ri := R/pi and Ki := Frac(R∏ K = Ki by (14.13). i ). Then∏ Let Ri′ be the normalization of Ri . Then R ⊂ Ri ⊂ Ri′ . Further, the∏first extension is integral by (10.24), and the second, by (10.26); whence, R ⊂ Ri′ is integral by the tower ∏ property∏(10.22). However, R is integrally closed in K by hypothesis. Hence R = Ri = Ri′ . Thus (3) and the ∏ last assertion hold. Assume (3). Let p be any prime of R. Then Rp = (Ri )p by (12.10), and each (Ri )p is normal by (11.31). But Rp is local. So Rp = (Ri )p for some i by (3.5). Hence Rp is a normal domain. Thus (1) holds. □ 15. Noether Normalization Exercise (15.2). — Let k := Fq be the finite field with q elements, and k[X, Y ] / the polynomial ring. Set f := X q Y − XY q and R := k[X, Y ] ⟨f ⟩. Let x, y ∈ R be the residues of X, Y . For every a ∈ k, show that R is not module finite over P := k[y −ax]. (Thus, in (15.1), no k-linear combination works.) First, take a = 0. Solution: Take a = 0. Then P = k[y]. Any algebraic relation over P satisfied by x is given by a polynomial in k[X, Y ], which is a multiple of f . However, no multiple of f is monic in X. So x is not integral over P . By (10.18), R is not module finite over P . Consider an arbitrary a. Since aq = a, after the change of variable Y ′ := Y − aX, our f still has the same form. Thus, we have reduced to the previous case. □ Exercise (15.3). — Let k be a field, and X, Y, Z variables. Set / R := k[X, Y, Z] ⟨X 2 − Y 3 − 1, XZ − 1⟩, and let x, y, z ∈ R be the residues of X, Y, Z. Fix a, b ∈ k, and set t := x + ay + bz and P := k[t]. Show that x and y are integral over P for any a, b and that z is integral over P if and only if b ̸= 0. Solution: To see x is integral, notice xz = 1, so x2 −tx+b = −axy. Raising both sides of the latter equation to the third power, and using the equation y 3 = x2 − 1, we obtain an equation of integral dependence of degree 6 for x over P . Now, y 3 − x2 − 1 = 0, so y is integral over P [x]. Hence, the Tower Property, (10.22), implies that y too is integral over P . If b ̸= 0, then z = b−1 (t − x − ay) ∈ P [x, y], and so z is integral over P by (10.23). Assume b = 0 and z is integral over P . Now, P ⊂ k[x, y]. So z is integral over k[x, y] as well. But y 3 − x2 + 1 = 0. So y is integral over k[x]. Hence z is too. However, k[x] is a polynomial ring, so integrally closed in its fraction field k(x) by (10.29)(1). Moreover, z = 1/x ∈ k(x). Hence, 1/x ∈ k[x], which is absurd. Thus z is not integral over P if b = 0. □ Solutions: 15. Noether Normalization 175 Exercise (15.7). — Let k be a field, K an algebraically closed extension field. (So K contains a copy of every finite extension field.) Let P := k[X1 , . . . , Xn ] be the polynomial ring, and f, f1 , . . . , fr ∈ P . Assume f vanishes at every zero in K n of f1 , . . . , fr ; in other words, if (a) := (a1 , . . . , an ) ∈ K n and f1 (a) = 0, . . . , fr (a) = 0, then f (a) = 0 too. Prove that there are polynomials g1 , . . . , gr ∈ P and an integer N such that f N = g1 f1 + · · · + gr fr . √ Solution: Set √ a := ⟨f1 , . . . , fr ⟩. We have to show f ∈ a. But, by the Hilbert Nullstellensatz, a is equal to the intersection of all the maximal ideals m containing a. So given an m, we have to show that f ∈ m. Set L := P/m. By the weak Nullstellensatz, L is a finite extension field of k. So we may embed L/k as a subextension of K/k. Let ai ∈ K be the image of the variable Xi ∈ P , and set (a) := (a1 , . . . , an ) ∈ K n . Then f1 (a) = 0, . . . , fr (a) = 0. Hence f (a) = 0 by hypothesis. Therefore, f ∈ m, as desired. □ Exercise (15.10). — Let R be a domain of (finite) dimension r, and p a nonzero prime. Prove that dim(R/p) < r. Solution: Every chain of primes of R/p is of the form p0 /p ⫋ · · · ⫋ ps /p where 0 ⫋ p0 ⫋ · · · ⫋ ps is a chain of primes of R. So s < r. Thus dim(R/p) < r. □ Exercise (15.11). — Let R′ /R be an integral extension of rings. Prove that dim(R) = dim(R′ ). Solution: Let p0 ⫋ · · · ⫋ pr be a chain of primes of R. Set p′−1 := 0. Given ′ pi−1 for 0 ≤ i ≤ r, Going up, (14.3)(4), yields a prime p′i of R′ with p′i−1 ⊂ p′i and p′ i ∩ R = pi . Then p′0 ⫋ · · · ⫋ p′r as p0 ⫋ · · · ⫋ pr . Thus dim(R) ≤ dim(R′ ). Conversely, let p′ 0 ⫋ · · · ⫋ p′ r be a chain of primes of R′ . Set pi := p′ i ∩ R. Then p0 ⫋ · · · ⫋ pr by Incomparability, (14.3)(2). Thus dim(R) ≥ dim(R′ ). □ Exercise (15.16). — Let k be a field, R a finitely generated k-algebra, f ∈ R nonzero. Assume R is a domain. Prove that dim(R) = dim(Rf ). Solution: Note that Rf is a finitely generated R-algebra by (11.11), as Rf is, by (11.11), obtained by adjoining 1/f . So since R is a finitely generated k-algebra, Rf is one too. Moreover, R and Rf have the same fraction field K. Hence both dim(R) and dim(Rf ) are equal to tr. degk (K) by (15.12). □ Exercise (15.17). — Let k be a field, P := k[f ] the polynomial ring in one variable f . Set p := ⟨f ⟩ and R := Pp . Find dim(R) and dim(Rf ). Solution: In P , the chain of primes 0 ⊂ p is of maximal length by (2.6) and (2.23) or (15.12). So ⟨0⟩ and pR are the only primes in R by (11.18). Thus dim(R) = 1. / Set K := Frac(P ). Then/ Rf = K since, if a (bf n ) ∈ K with a, b ∈ P and f ∤ b, then a/b ∈ R and so (a/b) f n ∈ Rf . Thus dim(Rf ) = 0. □ Exercise (15.18). — Let R be a ring, R[X] the polynomial ring. Prove 1 + dim(R) ≤ dim(R[X]) ≤ 1 + 2 dim(R). 176 Solutions: 15. Noether Normalization Solution: Let p0 ⫋ · · · ⫋ pn be a chain of primes in R. Then p0 R[X] ⫋ · · · ⫋ pn R[X] ⫋ pn R[X] + ⟨X⟩ is a chain of primes in R[X] by (2.10). Thus 1 + dim(R) ≤ dim(R[X]). Let p be a prime of R, and q0 ⫋ · · · ⫋ qr be a chain of primes of R[X] with qi ∩R = p for each i. Then (1.8) yields a chain of primes of length r in R[X]/pR[X]. Further, as qi ∩ R = p for each i, the latter chain gives rise to a chain of primes of length r in k(p)[X] where k(p) = (R/p)p by (11.29) and (11.18). But k(p)[X] is a PID. Hence r ≤ 1. Take any chain P0 ⫋ · · · ⫋ Pm of primes in R[X]. Then it contracts to a chain p0 ⫋ · · · ⫋ pn in R. But at most two Pj can contract to a given pi by the above discussion. Thus m ≤ 2n + 1. □ Exercise (15.22). — Let X be a topological space. We say a subset Y is locally closed if Y is the intersection of an open set and a closed set; equivalently, Y is open in its closure Y ; equivalently, Y is closed in an open set containing it. We say a subset X0 of X is very dense if X0 meets every nonempty locally closed subset Y . We say X is Jacobson if its set of closed points is very dense. Show that the following conditions on a subset X0 of X are equivalent: (1) X0 is very dense. (2) Every closed set F of X satisfies F ∩ X0 = F . (3) The map U 7→ U ∩ X0 from the open sets of X to those of X0 is bijective. Solution: Assume (1). Given a closed set F , take any x ∈ F , and let U be an open neighborhood of x in X. Then F ∩ U is locally closed, so meets X0 . Hence x ∈ F ∩ X0 . Thus F ⊂ F ∩ X0 . The opposite inclusion is trivial. Thus (2) holds. Assume (2). In (3), the map is trivially surjective. To check it’s injective, suppose U ∩X0 = V ∩X0 . Then (X −U )∩X0 = (X −V )∩X0 . So (2) yields X −U = X −V . So U = V . Thus (3) holds. Assume (3). Then the map F 7→ F ∩ X0 of closed sets is bijective too; whence, so is the map Y 7→ Y ∩ X0 of locally closed sets. In particular, if a locally closed set Y is nonempty, then so is Y ∩ X0 . Thus (1) holds. □ Exercise (15.23). — Let R be a ring, X := Spec(R), and X0 the set of closed points of X. Show that the following conditions are equivalent: (1) R is a Jacobson ring. (2) X is a Jacobson space. (3) If y ∈ X is a point such that {y} is locally closed, then y ∈ X0 . Solution: Assume (1). Let F ⊂ X be closed. Trivially, F ⊃ F ∩ X0 . To prove F ⊂ F ∩ X0 , say F = V(a) and F ∩ X0 = V(b). Then F ∩ X0 is the set of maximal √ ideals m containing a by (13.2), and every such m contains b. So (1) √ implies b ⊂ a. But V( a) = F . Thus F ⊂ F ∩ X0 . Thus (15.22) yields (2). ∩ Assume (2). Let y ∈ X a point ( be ∩ ) such that {y} is locally closed. Then {y} X0 is nonempty by (2). So {y} X0 ∋ y. Thus (3) holds. Assume (3). Let p be a prime ideal of R such that pRf is maximal for some f∈ / p. Then {p} is closed in D(f ) by (13.1). So {p} is locally closed in X. Hence {p} is closed in X by (3). Thus p is maximal. Thus (15.21) yields (1). □ Solutions: 16. Chain Conditions 177 Exercise (15.25). — Let P := Z[X1 , . . . , Xn ] be the polynomial ring. Assume f ∈ P vanishes at every zero in K n of f1 , . . . , fr ∈ P for every finite field K; that is, if (a) := (a1 , . . . , an ) ∈ K n and f1 (a) = 0, . . . , fr (a) = 0 in K, then f (a) = 0 too. Prove there are g1 , . . . , gr ∈ P and N ≥ 1 such that f N = g1 f1 + · · · + gr fr . √ Solution: Set a := ⟨f1 , . . . , fr ⟩. Suppose f ∈ / a. Then f lies outside some maximal ideal m containing a by (15.24)(2) and (15.19). Set K := P/m. Then K is a finite extension of Fp for some prime p by (15.24)(1). So K is finite. Let ai be the residue of Xi , set (a) := (a1 , . . . , an ) ∈ K n . Then f1 (a) = 0,√ . . . , fr (a) = 0. □ So f (a) = 0 by hypothesis. Thus f ∈ m, a contradiction. Thus f ∈ a. Exercise (15.26). — Let R be a ring, R′ an algebra. Prove that if R′ is integral over R and R is Jacobson, then R′ is Jacobson. √ Solution: Given an ideal a′ ⊂ R′ and an f outside a, set R′′ := R[f ]. Then R′′ is Jacobson by (15.24). So R′′ has a maximal ideal m′′ that avoids f and contains a′ ∩ R′′ . But R′ is integral over R′′ . So R′ contains a prime m′ that contains a′ and that contracts to m′′ by Going Up (14.3)(4). Then m′ avoids f as m′′ does, and m′ is maximal by Maximality, (14.3)(1). Thus R′ is Jacobson. □ Exercise (15.27). — Let R be a Jacobson ring, S a multiplicative subset, f ∈ R. True or false: prove or give a counterexample to each of the following statements: (1) The localized ring Rf is Jacobson. (2) The localized ring S −1 R is Jacobson. (3) The filtered direct limit lim Rλ of Jacobson rings Rλ is Jacobson. −→ Solution: (1) True: Rf = R[1/f ]; so Rf is Jacobson by (15.24). (2) False: by (15.20), Z is Jacobson, but Z⟨p⟩ isn’t for any prime number p. (3) False: Z⟨p⟩ isn’t Jacobson by (2), but Z⟨p⟩ = lim Z by (12.6). □ −→ Exercise (15.28). — Let R be a reduced Jacobson ring with a finite set Σ of minimal primes, and P a finitely generated module. Show that P is locally free of rank r if and only if dimR/m (P/mP ) = r for any maximal ideal m. Solution: Suppose P is locally free of rank r. Then given any maximal ideal m, there is an f ∈ R − m such that Pf is a free Rf -module of rank r by (13.25). But Pm is a localization of Pf by (12.5). So Pm is a free Rm -module of rank r by (12.10). But Pm /mPm = (P/mP )m by (12.18). Also Rm /mRm = R/m by (12.19). Thus dimR/m (P/mP ) = r. Consider the converse. Given a p ∈ Σ, set K := Frac(R/p). Then P ⊗R K is a K-vector space, say of dimension n. Since R is reduced, K = Rp by (14.12). So by (12.20), there is an h ∈ R − p with Ph free of rank n. As R is Jacobson, there is a maximal ideal m avoiding h, by (15.19). Hence, as above, dimR/m (P/mP ) = n. But, by hypothesis, dimR/m (P/mP ) = r. Thus n = r. Given a maximal ideal m, set A := Rm . Then A is reduced by (13.18). Each minimal prime of A is of the form pA where p ∈ Σ by (11.18)(2). Further, it’s not hard to see, essentially as above, that Pm ⊗ Frac(A/pA) = P ⊗ Frac(R/p). Hence (14.14) implies Pm is a free A-module of rank r. Finally, (13.28) implies P is locally free of rank r. □ 16. Chain Conditions 178 Solutions: 16. Chain Conditions Exercise (16.2). — Let a be a finitely generated ideal in an arbitrary ring. Show every set that generates a contains a finite subset that generates a. Solution: Say a is generated by x1 , . . . , xr and also by the yλ for λ ∈ Λ. Write ∑ xi = j zj yλij . Then the yλij generate a. □ Exercise (16.8). — Let R be a ring, X a variable, R[X] the polynomial ring. Prove this statement or find a counterexample: if R[X] is Noetherian, then so is R. Solution: It’s true. Since R[X] is Noetherian, so is R[X]/⟨X⟩ by (16.7). But the latter ring is isomorphic to R by (1.7); so R is Noetherian. □ β α Exercise (16.14). — Let 0 → L − →M − → N → 0 be a short exact sequence of R-modules, and M1 , M2 two submodules of M . Prove or give a counterexample to this statement: if β(M1 ) = β(M2 ) and α−1 (M1 ) = α−1 (M2 ), then M1 = M2 . Solution: The statement is false: form the exact sequence β α 0→R− →R⊕R− →R→0 with α(r) := (r, 0) and β(r, s) := s, and take M1 := {(t, 2t) | t ∈ R} and M2 := {(2t, t) | t ∈ R}. (Geometrically, we can view M1 as the line determined by the origin and the point (1, 2), and M2 as the line determined by the origin and the point (2, 1). Then β(M1 ) = β(M2 ) = R, and α−1 (M1 ) = α−1 (M2 ) = 0, but M1 ̸= M2 in R ⊕ R.) □ Exercise (16.17). — Let R be⊕ a ring, a1 , . . . , ar ideals such that each R/ai is a Noetherian ring. Prove (1) that R/ai is a Noetherian R-module, and (2) that, ∩ if ai = 0, then R too is a Noetherian ring. Solution: Any R-submodule of R/ai is an ideal of R/ai . Since R/ai is a Noetherian ring, such an ideal is finitely generated as an (R/a⊕ i )-module, so as an Rmodule as well. Thus R/ai is a Noetherian R-module. So R/ai is a Noetherian R-module by (16.16). Thus (1) holds. ⊕ ∩ To prove (2), note that the kernel of the natural map R → R/ai is ai , which is 0 by hypothesis. So R can be identified with a submodule of the Noetherian ⊕ R-module R/ai . Hence R itself is a Noetherian R-module by (16.15)(2). So R is a Noetherian ring by (16.12). □ Exercise (16.20). — Let G be a finite group acting on a domain R, and R′ the subring of invariants. Let k ⊂ R′ be a field. Using (10.17), prove this celebrated theorem of E. Noether (1926): if R is algebra finite over k, then so is R′ . Solution: By (10.17), R is integral over R′′ . But it’s algebra finite. So it’s module finite by (10.23). Hence (16.19) yields the assertion. □ Exercise (16.24). — Let k be a field, R an algebra. Assume that R is finite dimensional as a k-vector space. Prove that R is Noetherian and Artinian. Solution: View R as a vector space, and ideals as subspaces. Now, by a simple dimension argument, any ascending or descending chain of subspaces of R stabilizes. Thus R is Noetherian by (16.5) and is Artinian by definition. □ Solutions: 17. Associated Primes 179 / Exercise (16.25). — Let p be a prime number, and set M := Z[1/p] Z. Prove that any Z-submodule N ⊂ M is either finite or all of M . Deduce that M is an Artinian Z-module, and that it is not Noetherian. Solution: Given q ∈ N , write q = n/pe where n is relatively prime to p. Then there is an m ∈ Z with nm ≡ 1 (mod pe ). Hence N ∋ m(n/pe ) = 1/pe , and so 1/pr = pe−r (1/pe ) ∈ N for any 0 ≤ r ≤ e. Therefore, either N = M , or there is a largest integer e ≥ 0 with 1/pe ∈ N . In the second case, N is finite. Let M ⊋ N1 ⊃ N2 ⊃ · · · be a descending chain. By what we just proved, each Ni is finite, say with ni elements. Then the sequence n1 ≥ n2 ≥ · · · stabilizes; say ni = ni+1 = · · · . But Ni ⊃ Ni+1 ⊃ · · · , so Ni = Ni+1 = · · · . Thus M is Artinian. Finally, suppose m1 , . . . , mr generate M , say mi = ni /pei . Set e := max ei . Then 1/pe generates M , a contradiction since 1/pe+1 ∈ M . Thus M is not finitely generated, and so not Noetherian. □ Exercise (16.26). — Let R be an Artinian ring. Prove that R is a field if it is a domain. Deduce that in general every prime ideal p of R is maximal. Solution: Take any nonzero element x ∈ R, and consider the chain of ideals ⟨x⟩ ⊃ ⟨x2 ⟩ ⊃ · · · . Since R is Artinian, the chain stabilizes; so ⟨xe ⟩ = ⟨xe+1 ⟩ for some e. Hence xe = axe+1 for some a ∈ R. If R is a domain, then we can cancel to get 1 = ax; thus R is then a field. In general, R/p is Artinian by (16.23)(2). Now, R/p is also a domain by (2.9). Hence, by what we just proved, R/p is a field. Thus p is maximal by (2.17). □ 17. Associated Primes Exercise (17.6). — Given modules M1 , . . . , Mr , set M := M1 ⊕ · · · ⊕ Mr . Prove Ass(M ) = Ass(M1 ) ∪ · · · ∪ Ass(Mr ). Solution: Set N := M2 ⊕ · · · ⊕ Mr . Then N, M1 ⊂ M . Also, M/N = M1 . So (17.5) yields Ass(N ), Ass(M1 ) ⊂ Ass(M ) ⊂ Ass(N ) ∪ Ass(M1 ). So Ass(M ) = Ass(N ) ∪ Ass(M1 ). The assertion follows by induction on r. □ Exercise (17.7). — Take R := Z and M := Z/⟨2⟩ ⊕ Z. Find Ass(M ) and find two submodules L, N ⊂ M with L + N = M but Ass(L) ∪ Ass(N ) ⫋ Ass(M ). Solution: First, we have Ass(M ) = {⟨0⟩, ⟨2⟩} by (17.6) and (17.4)(2). Next, take L := R · (1, 1) and N := R · (0, 1). Then the canonical maps Z → L and Z → N are isomorphisms. Hence both Ass(L) and Ass(N ) are {⟨0⟩} by (17.4)(2). Finally, L + N = M because (a, b) = a · (1, 1) + (b − a) · (0, 1). □ Exercise (17.10). — Let R be a ring, and suppose Rp is a domain for every prime p. Prove every associated prime of R is minimal. 180 Solutions: 18. Primary Decomposition Solution: Let p ∈ Ass(R). Then pRp ∈ Ass(Rp ) by (17.9). By hypothesis, Rp is a domain. So pRp = ⟨0⟩ by (17.4). Hence p is a minimal prime of R by (11.18)(2). Alternatively, say p = Ann(x) with x ∈ R. Then x/1 ̸= 0 in Rp ; otherwise, there would be some s ∈ R − p such that sx = 0, contradicting p = Ann(x). However, for any y ∈ p, we have xy/1 = 0 in Rp . Since Rp is a domain and since x/1 ̸= 0, we must have y/1 = 0 in Rp . So there exists some t ∈ R − p such that ty = 0. Now, p ⊃ q for some minimal prime q by (3.11). Suppose p ̸= q. Then there is some y ∈ p − q. So there exists some t ∈ R − p such that ty = 0 ∈ q, contradicting the primeness of q. Thus p = q; that is, p is minimal. □ Exercise (17.15). — Let R be a Noetherian ring, M a module, N a submodule, x ∈ R. Show that, if x ∈ / p for any p ∈ Ass(M/N ), then xM ∩ N = xN . ∩ Solution: Trivially, xN ⊂ xM N . Conversely, take m ∈ M with xm ∈ N . Let m′ be the residue of m in M/N . Then xm′ = / z.div(M/N ). ∩ 0. By (17.14), x ∈ So m′ = 0. So m ∈ N . So xm ∈ xN . Thus xM N ⊂ xN , as desired. □ Exercise (17.21). — Let R be a Noetherian ring, a an ideal. Prove the primes minimal containing a are associated to a. Prove such primes are finite in number. Solution: Since a = Ann(R/a), the primes in question are the primes minimal in Supp(R/a) by (13.9)(3). So they are associated to a by (17.17), and they are finite in number by (17.20). □ Exercise (17.22). — Take R := Z and M := Z in (17.19). Determine when a chain 0 ⊂ M1 ⫋ M is acceptable, and show that then p2 ∈ / Ass(M ). Solution: If the chain is acceptable, then M1 ̸= 0 as M1 /0 ≃ R/p1 , and M1 is a prime ideal as M1 = Ann(M/M1 ) = p2 . Conversely, the chain is acceptable if M1 is a nonzero prime ideal p, as then M1 /0 ≃ R/0 and M/M1 ≃ R/p. Finally, Ass(M ) = 0 by (17.4). Further, as just observed, given any acceptable chain, p2 = M1 ̸= 0. So p2 ∈ / Ass(M ). □ Exercise (17.23). — Take R := Z and M := Z/⟨12⟩ in (17.19). Find all three acceptable chains, and show that, in each case, {pi } = Ass(M ). Solution: An acceptable chain in M corresponds to chain ⟨12⟩ ⊂ ⟨a1 ⟩ ⊂ ⟨a2 ⟩ ⊂ · · · ⊂ ⟨an ⟩ = Z. Here ⟨a1 ⟩/⟨12⟩ ≃ Z/⟨p1 ⟩ with p1 prime. So a1 p1 = 12. Hence the possibilities are p1 = 2, a1 = 6 and p1 = 3, a1 = 4. Further, ⟨a2 ⟩/⟨a1 ⟩ ≃ Z/⟨p2 ⟩ with p2 prime. So a2 p2 = a1 . Hence, if a1 = 6, then the possibilities are p2 = 2, a2 = 3 and p2 = 3, a2 = 2; if a1 = 4, then the only possibility is p2 = 2 and a2 = 2. In each case, a2 is prime; hence, n = 3, and these three chains are the only possibilities. Conversely, each of these three possibilities, clearly, does arise. In each case, {pi } = {⟨2⟩, ⟨3⟩}. Hence (17.19.1) yields Ass(M ) ⊂ {⟨2⟩, ⟨3⟩}. For any M , if 0 ⊂ M1 ⊂ · · · ⊂ M is an acceptable chain, then (17.5) and (17.4)(2) yield Ass(M ) ⊃ Ass(M1 ) = {p1 }. Here, there’s one chain with p1 = ⟨2⟩ and another with p1 = ⟨3⟩; hence, Ass(M ) ⊃ {⟨2⟩, ⟨3⟩}. Thus Ass(M ) = {⟨2⟩, ⟨3⟩}. □ 18. Primary Decomposition Solutions: 18. Primary Decomposition 181 Exercise (18.6). — Let R be a ring, and p = ⟨p⟩ a principal prime generated by a nonzerodivisor p. Show every positive power pn is p-primary, and conversely, if R is Noetherian, then every p-primary ideal q is equal to some power pn . Solution: Let’s proceed by induction. Form the exact sequence 0 → pn /pn+1 → R/pn+1 → R/pn → 0. Consider the map R → pn /pn+1 given by x 7→ xpn . It is surjective, and its kernel ∼ pn /pn+1 . But Ass(R/p) = {p} is p as p is a nonzerodivisor. Hence R/p −→ by (17.4)(2). Hence (17.5) yields Ass(R/pn ) = {p} for every n ≥ 1, as desired. √ Conversely, p = q by (18.5). So pn ∈ q for some n; take n minimal. Then pn ⊂ q. Suppose there is an x ∈ q − pn . Say x = ypm for some y and m ≥ 0. Then m < n as x ∈ / pn . Take m maximal. Now, pm ∈ / q as n is minimal. So (18.5) yields y ∈ q ⊂ p. Hence y = zp for some z. Then x = zpm+1 , contradicting the maximality of m. Thus q = pn . □ Exercise (18.7). — Let k be a field, and k[X, √ Y ] the polynomial ring. Let a be the ideal ⟨X 2 , XY ⟩. Show a is not primary, but a is prime. Show a satisfies this condition: ab ∈ a implies a2 ∈ a or b2 ∈ a. √ Solution: First, ⟨X⟩ is prime by (2.11). But ⟨X 2 ⟩ ⊂ a ⊂ ⟨X⟩. √ So a = ⟨X⟩ by (3.26). On the other hand, XY ∈ a, but X ∈ / a and Y ∈ / a; thus a is not primary by (18.5). If ab ∈ a, then X | a or X | b, so a2 ∈ a or b2 ∈ a. □ Exercise (18.8). — Let φ : R → R′ be a homomorphism of Noetherian rings, and q ⊂ R′ a p-primary ideal. Show that φ−1 q ⊂ R is φ−1 p-primary. Show that the converse holds if φ is surjective. Solution: Let xy ∈ φ−1 q, but x ∈ / φ−1 q. Then φ(x)φ(y) ∈ q, but φ(x) ∈ / q. n So φ(y) ∈ q for some n ≥ 1 by (18.5). Hence, y n ∈ φ−1 q. So φ−1 q is primary √ by (18.5). Its radical is φ−1 p as p = q, and taking the radical commutes with taking the inverse image by (3.27). The converse can be proved similarly. □ Exercise (18.16). — Let k be a field, R := k[X, Y, Z] be the polynomial ring. Set a := ⟨XY, X − Y Z⟩, set q1 := ⟨X, Z⟩ and set q2 := ⟨Y 2 , X − Y Z⟩. Show that a = q1 ∩q2 holds and that this expression is an irredundant primary decomposition. Solution: First, XY = Y (X −Y Z)+Y 2 Z ∈ q2 . Hence a ⊂ q1 ∩q2 . Conversely, take F ∈ q1 ∩ q2 . Then F ∈ q2 , so F = GY 2 + H(X − Y Z) with G, H ∈ R. But F ∈ q1 , so G ∈ q1 ; say G = AX + BZ with A, B ∈ R. Then F = (AY + B)XY + (H − BY )(X − ZY ) ∈ a. Thus a ⊃ q1 ∩ q2 . Thus a = q1 ∩ q2 holds. Finally, q1 is prime by (2.11). Now, using (18.8), let’s show q2 is ⟨X, Y ⟩primary. Form φ : k[X, Y, Z] → k[Y, Z] with φ(X) := Y Z. Clearly, q2 = φ−1 ⟨Y 2 ⟩ and ⟨X, Y ⟩ = φ−1 ⟨Y ⟩; also, ⟨Y 2 ⟩ is ⟨Y ⟩-primary by (18.2). Thus a = q1 ∩ q2 is a primary decomposition. It is irredundant as q1 and ⟨X, Y ⟩ are distinct. □ Exercise (18.17). — Let R := R′ × R′′ be a product of two domains. Find an irredundant primary decomposition of ⟨0⟩. 182 Solutions: 18. Primary Decomposition Solution: Set p′ := ⟨0⟩ × R′′ and p′′ := R′′ × ⟨0⟩. Then p′ and p′′ are prime by (2.12), so primary by (17.4)(2). Clearly ⟨0⟩ = p′ ∩ p′′ . Thus this representation is a primary decomposition; it is irredundant as both p′ and p′′ are needed. □ Exercise (18.21). — Let R be a Noetherian ring, a an ideal, and M a finitely generated module. Consider the following submodule of M : ∪ Γa (M ) := n≥1 {m ∈ M | an m = 0 for some n ≥ 1}. ∩ ∩ (1) For any decomposition 0 = Qi with Qi pi -primary, show Γa (M ) = a̸⊂pi Qi . (2) Show Γa (M ) is the set of all m ∈ M such that m/1 ∈ Mp vanishes for every prime p with a ̸⊂ p. (Thus Γa (M ) is the set of all m whose support lies in V(a).) Solution: For (1), given m ∈ Γa (M ), say an m = 0. Given i with ∩ a ̸⊂ pi , take a ∈ a − pi . Then an m = 0 ∈ Qi . Hence m ∈ Qi by (18.4). Thus m ∈ a̸⊂pi Qi . ∩ Conversely, given m ∈ a̸⊂pi Qi , take any j with a ⊂ pj . Now, pj = nil(M/Qj ) by (18.3). So there is nj with anj m∩∈ Qj . Set n := max{nj }. Then an m ∈ Qi for all i, if a ⊂ pi or not. Hence an m ∈ Qi = 0. Thus m ∈ Γa (M ). For (2), given m ∈ Γa (M ), say an m = 0. Given a prime p with a ̸⊂ p, take a ∈ a − p. Then an m = 0 and an ∈ / p. So m/1 ∈ Mp vanishes. Conversely, given an m ∈ M such that ∩ m/1 ∈ Mp vanishes for every prime p with a ̸⊂ p, consider a decomposition 0 = Qi with Qi pi -primary; one exists by (18.20). By (1), it suffices to show m ∈ Qi if a ̸⊂ pi . But m/1 ∈ Mpi vanishes. So there’s an a ∈ R − pi with am = 0 ∈ Qi . So (18.4) yields m ∈ Qi , as desired. □ Exercise (18.25). — Let R∩be a Noetherian ring, M a finitely generated module, N a submodule. Prove N = p∈Ass(M/N ) φ−1 p (Np ). ∩r Solution: (18.20) yields an irredundant primary decomposition N = 1 Qi . Say Qi is pi -primary. Then {p∩i }r1 = Ass(M/N∩) (by (18.19).) Also, ∩ ∩r (18.23) yields r r ∩ −1 (N ) = Q = □ φpi (Npi ) = pj ⊂pi Qj . Thus 1 φ−1 pi j pi 1 pj ⊂pi 1 Qi = N . Exercise (18.26). — Let R be a Noetherian ring, p a prime. Its nth symbolic power p(n) is defined as the saturation (pn )S where S := R − p. (1) Show p(n) is the p-primary component of pn . (2) Show p(m+n) is the p-primary component of p(n) p(m) . (3) Show p(n) = pn if and only if pn is p-primary. (4) Given a p-primary ideal q, show q ⊃ p(n) for all large n. Solution: Clearly, p is minimal in V(pn ). But V(pn ) = Supp(R/pn ) by (13.9). Hence p is minimal in Ass(R/pn ) by (17.17) and (17.3). Thus (18.24) yields (1). Clearly, (11.15)(3) yields (p(m) p(n) )S = p(m+n) . Thus (18.24) yields (2). If p(n) = pn , then pn is p-primary by (1). Conversely, if pn is p-primary, then pn = p(n) because primary ideals are saturated by (18.22). Thus (3) holds. √ For (4), recall p = q by (18.5). So q ⊃ pn for all large n by (3.26). Hence qS ⊃ p(n) . But qS = q by (18.22) since p ∩ (R − p) = ∅. Thus (4) holds. □ Exercise (18.27). — Let R be a Noetherian ring, ⟨0⟩ = q1 ∩· · ·∩qn an irredundant √ primary decomposition. Set pi := qi for i = 1, . . . , n. (r) (1) Suppose pi is minimal for some i. Show qi = pi for all large r. (r) (2) Suppose pi is not minimal for some i. Show that replacing qi by pi for large r gives infinitely many distinct irredundant primary decompositions of ⟨0⟩. Solutions: 19. Length 183 Solution: Set A := Rpi and m := pi A. Then A is Noetherian√by (16.7). Suppose pi is minimal. Then m is the only prime in A. So m = ⟨0⟩ by the Scheinnullstellensatz (3.22). So mr = 0 for all large r by (3.25). So p(r) = qi by the Second Uniqueness Theorem (18.24) and Lemma (18.22). Thus (1) holds. Suppose pi is not minimal. Then the mr are distinct. Otherwise, mr = mr+1 for some r. So mr = 0 by Nakayama’s Lemma (10.10). But then m is minimal; so pi (r) is too, contrary to hypothesis. So (11.17)(1) implies the pi are distinct. ∩ (r) (r) However, qi ⊃ pi for all large r by (18.26)(4). Hence ⟨0⟩ = pi ∩ j̸=i qj . (r) (r) But pi is pi -primary by (18.26)(1). Thus replacing qi by pi for large r gives infinitely many distinct primary decompositions of ⟨0⟩. These decompositions are irredundant owing to two applications of (18.18). A first yields {pi } = Ass(R) as ⟨0⟩ = q1 ∩ · · · ∩ qn is irredundant. So a second yields the desired irredundancy. □ Exercise (18.29). — Let R be a Noetherian ring, m ⊂ rad(R) an ideal, M a finitely generated module, and M ′ a submodule. Considering M/N , show that ∩ M ′ = n≥0 (mn M + M ′ ). ∩ Solution: Set N := n≥0 mn (M/M ′ ). Then by (18.28), there is x ∈ m such that (1 + x)N = 0. By (3.2), 1 + x is a unit since m ⊂ rad(R). Therefore, n ′ n ′ ′ N = (1 + x−1 )(1 / +′ x)N = ⟨0⟩. However, m (M/M ) = (m M + M )/M . Thus ∩ n ′ (m M + M ) M = 0, as desired. □ 19. Length Exercise (19.2). — Let R be a ring, M a module. Prove these statements: (1) If M is simple, then any nonzero element m ∈ M generates M . (2) M is simple if and only if M ≃ R/m for some maximal ideal m, and if so, then m = Ann(M ). (3) If M has finite length, then M is finitely generated. Solution: Obviously, Rm is a nonzero submodule. So it is equal to M , because M is simple. Thus (1) holds. Assume M is simple. Then M is cyclic by (1). So M ≃ R/m for m := Ann(M ) by (4.7). Since M is simple, m is maximal owing to the bijective correspondence of (1.8). By the same token, if, conversely, M ≃ R/m with m maximal, then M is simple. Thus (2) holds. Assume ℓ(M ) < ∞. Let M = M0 ⊃ M1 ⊃ · · · ⊃ Mm = 0 be a composition series. If m = 0, then M = 0. Assume m ≥ 1. Then M1 has a composition series of length m − 1. So, by induction on m, we may assume M1 is finitely generated. Further, M/M1 is simple, so finitely generated by (1). Hence M is finitely generated by (16.15)(1). Thus (3) holds. □ Exercise (19.4). — Let R be a Noetherian ring, M a finitely generated module. Prove the equivalence of the following three conditions: (1) that M has finite length; (2) that Supp(M ) consists entirely of maximal ideals; (3) that Ass(M ) consists entirely of maximal ideals. 184 Solutions: 19. Length Prove that, if the conditions hold, then Ass(M ) and Supp(M ) are equal and finite. Solution: If (1) holds, then (2) holds owing to (19.3). If (2) holds, then (1) holds owing to (17.19) and (19.2)(2). Finally, (17.16) and (17.20) imply that (2) and (3) are equivalent and that the last assertion holds. □ Exercise (19.5). — Let R be a Noetherian ring, q a p-primary ideal. Consider chains of primary ideals from q to p. Show (1) all such chains have length at most ℓ(A) where A := (R/q)p and (2) all maximal chains have length exactly ℓ(A). Solution: There is a natural bijective correspondence between the p-primary ideals containing q and the (p/q)-primary ideals of R/q, owing to (18.8). In turn, there is one between the latter ideals and the ideals of A primary for its maximal ideal m, owing to (18.8) again and also√ to (18.22) with M := A. √ However, p = q by (18.5). So m = ⟨0⟩. Hence every ideal of A is m-primary by (18.10). Further, m is the only prime of A; so ℓ(A) is finite by (19.4) with M := A. Hence (19.3) with M := A yields (1) and (2). □ Exercise (19.8). — Let k be a field, and R a finitely generated k-algebra. Prove that R is Artinian if and only if R is a finite-dimensional k-vector space. Solution: Since k is Noetherian by (16.1) and since R is a finitely generated k-algebra, R is Noetherian by (16.11). Assume R is Artinian. Then ℓ(R) < ∞ by (19.6). So R has a composition series. The successive quotients are isomorphic to residue class fields by (19.2)(2). These fields are finitely generated k-algebras, since R is so. Hence these fields are finite extension fields of k by the Weak Nullstellensatz. Thus R is a finite-dimensional k-vector space. The converse holds by (16.24). □ Exercise (19.10). — Let k be a field, A a local k-algebra. Assume the map from k to the residue field is bijective. Given an A-module M , prove ℓ(M ) = dimk (M ). Solution: If M = 0, then ℓ(M ) = 0 and dimk (M ) = 0. If M ∼ = k, then ℓ(M ) = 1 and dimk (M ) = 1. Assume 1 ≤ ℓ(M ) < ∞. Then M has a submodule M ′ with M/M ′ ∼ = k. So Additivity of Length, (19.9), yields ℓ(M ′ ) = ℓ(M ) − 1 ′ and dimk (M ) = dimk (M ) − 1. Hence ℓ(M ′ ) = dimk (M ′ ) by induction on ℓ(M ). Thus ℓ(M ) = dimk (M ). If ℓ(M ) = ∞, then for every m ≥ 1, there exists a chain of submodules, Hence dimk (M ) = ∞. M = M0 ⫌ M1 ⫌ · · · ⫌ Mm = 0. □ Exercise (19.12). — Prove these conditions on a Noetherian ring R equivalent: (1) that R is Artinian; (2) that Spec(R) is discrete and finite; (3) that Spec(R) is discrete. Solution: Condition (1) holds, by (19.11), if and only if Spec(R) consists of finitely points and each is a maximal ideal. But a prime p is a maximal ideal if and only if {p} is closed in Spec(R) by (13.2). It follows that (1) and (2) are equivalent. Trivially, (2) implies (3). Conversely, (3) implies (2), since Spec(R) is quasicompact by (13.4). Thus all three conditions are equivalent. □ Exercise (19.13). — Let R be an Artinian ring. Show that rad(R) is nilpotent. Solutions: 20. Hilbert Functions 185 Solution: Set m := rad(R). Then m ⊃ m2 ⊃ · · · is a descending chain. So m = mr+1 for some r. But R is Noetherian by Akizuki’s Theorem (19.11). So m is finitely generated. Thus Nakayama’s Lemma (10.10) yields mr = 0. □ r Exercise (19.16). — Let R be a ring, p a prime ideal, and R′ a module-finite R-algebra. Show that R′ has only finitely many primes p′ over p, as follows: reduce to the case that R is a field by localizing at p and passing to the residue rings. Solution: First note that, if p′ ⊂ R′ is a prime lying over p, then p′ Rp′ ⊂ Rp′ is a prime lying over the maximal ideal pRp . Hence, by (11.18)(2), it suffices to show that Rp′ has only finitely many such primes. Note also that Rp′ is module-finite over Rp . Hence we may replace R and R′ by Rp and Rp′ , and thus assume that p is the unique maximal ideal of R. Similarly, we may replace R and R′ by R/p and R′ /pR′ , and thus assume that R is a field. There are a couple of ways to finish. First, R′ is now Artinian by (19.15) or by (16.24); hence, R′ has only finitely many primes by (19.11). Alternatively, every prime is now minimal by incomparability (14.3)(2). Further, R′ is Noetherian by (16.11); hence, R′ has only finitely many minimal primes by (17.21). □ Exercise (19.18). — Let R be a Noetherian ring, and M a finitely generated module. Prove the following four conditions are equivalent: (1) that M has finite length; ∏ (2) that M is annihilated by some finite product of maximal ideals mi ; (3) that every prime p containing Ann(M ) is maximal; (4) that R/Ann(M ) is Artinian. Solution: Assume (1) holds. Let M = M0 ⊃ · · · ⊃ Mm = 0 be a composition series, and set mi := Ann(Mi−1 /Mi ). Then mi is maximal by (19.2)(2). Further, mi Mi−1 ⊂ Mi . Hence mi · · · m1 M0 ⊂ Mi . Thus (2) holds. ∏ If (2) holds, then (3) does too. Indeed, if p ⊃ Ann(M ) ⊃ mi , then p ⊃ mi for some i by (2.2) as p is prime, and so p = mi as mi is maximal. Assume (3) holds. Then dim(R/Ann(M )) = 0. But, by (16.7), any quotient of R is Noetherian. Hence Akizuki’s Theorem (19.11) yields (4). If (4) holds, then (19.14) yields (1), because M is a finitely generated module over R/ Ann(M ) owing to (4.5). □ 20. Hilbert Functions Exercise (20.5). — Let k be a field, k[X, Y ] the polynomial ring. Show ⟨X, Y 2 ⟩ and ⟨X 2 , Y 2 ⟩ have different Hilbert Series, but the same Hilbert Polynomial. Solution: Set m := ⟨X, Y ⟩ and a := ⟨X, Y 2 ⟩ and b := ⟨X 2 , Y 2 ⟩. They are graded by degree. So ℓ(a1 ) = 1, and ℓ(an ) = ℓ(mn ) for all n ≥ 2. Further, ℓ(b1 ) = 0, ℓ(b2 ) = 2, and ℓ(bn ) = ℓ(mn ) for n ≥ 3. Thus the three ideals have the same Hilbert Polynomial, namely h(n) = n + 1, but different Hilbert Series. □ ⊕ ⊕ Exercise (20.6).⊕ — Let R = Rn be a graded ring, M = Mn a graded Rmodule. Let N = Nn be a homogeneous submodule; that is, Nn = N ∩ Mn . Assume R0 is Artinian, R is a finitely generated R0 -algebra, and M is a finitely generated R-module. Set N ′ := { m ∈ M | there is k0 such that Rk m ∈ N for all k ≥ k0 }. 186 Solutions: 20. Hilbert Functions (1) Prove that N ′ is a homogeneous submodule of M with the same Hilbert Polynomial as N , and that N ′ is the largest such submodule. ∩ ⊕ (2) Let N = ∩ Qi be a decomposition with Qi pi -primary. Set R+ := n>0 Rn . Prove that N ′ = pi ̸⊃R+ Qi . ∑ Solution: Given m = mi ∈ N ′ , say Rk m ⊂ N . Then Rk mi ⊂ N since N is homogeneous. Hence mi ∈ N ′ . Thus N ′ is homogeneous. By (19.11) and (16.11), R is Noetherian. So N ′ is finitely generated by (16.18). Let n1 , . . . , nr be homogeneous generators of N ′ with ni ∈ Nki ; set k ′ := max{ki }. ′ ′ There ∑is k such that Rk ni ∈ N for all i. Given ℓ ≥ k + k , take n ∈ Nℓ , and′ write n= yi ni with yi ∈ Rℓ−ki . Then yi ni ∈ Nℓ for all i. So n ∈ Nℓ . Thus Nℓ = Nℓ for all ℓ ≥ k + k ′ . Thus N and N ′ have the same Hilbert polynomial. Say N ′′ ⊃ N , and both have the same Hilbert Polynomial. Then there is k0 with ℓ(Nk′′ ) = ℓ(Nk ) for all k ≥ k0 . So Nk′′ = Nk for all k ≥ k0 . So, if n ∈ N ′′ , then Rk n ∈ N for all k ≥ k0 . Thus N ′′ ⊂ N ′ . Thus (1) holds. ∩ To prove (2), note 0 = (Qi /N ) in M/N . By (18.21), ∩ (Qi /N ). ΓR+ (M/N ) = pi ̸⊃R+ ′ But clearly ΓR+ (M/N ) = N /N . Thus N ′ = ∩ pi ̸⊃R+ Qi . □ Exercise (20.9). — Let k be a field, P := k[X, Y, Z] the polynomial ring in three variables, f ∈ P a homogeneous polynomial of degree d ≥ 1. Set R := P/⟨f ⟩. Find the coefficients of the Hilbert Polynomial h(R, n) explicitly in terms of d. Solution: Clearly, the following sequence is exact: µf 0 → P (−d) −−→ P → R → 0. Hence, Additivity of Length, (19.9), yields h(R, n) = h(P, n) − h(P (−d), n). But P (−d)n = P (n − d), so h(P (−d), n) = h(P, n − d). Therefore, (20.4) yields ) (2−d+n) ( = dn − (d − 3)d/2. □ − h(R, n) = 2+n 2 2 Exercise (20.10). — Under the conditions of (20.8),( assume) there is ( a homo) geneous nonzerodivisor f ∈ R with Mf = 0. Prove deg h(R, n) > deg h(M, n) ; start with the case M := R/⟨f k ⟩. Solution: Suppose M := R/⟨f k ⟩. Set c := k deg(f ). Form the exact sequence µ 0 → R(−c) − → R → M → 0 where µ is multiplication by f k . Then Additivity of Length (19.9) yields h(M, n) = h(R, n) − h(R, n − c). But e(1) + · · · and h(R, n − c) = (d−1)! (n − c)d−1 + · · · . ( ) ( ) by (20.8). Thus deg h(R, n) > deg h(M, n) . In the general case, there is k with f k M = 0 by (12.7). ⊕ Set M ′ := R/⟨f k ⟩. Then generators mi ∈ Mci for 1 ≤ i ≤ r yield( a surjection ) i M ′ (−c → M). ( i) → ∑ ′ ′ ′ Hence i (ℓ(Mn−c ) ≥ ℓ(Mn() for all n. But deg h(M (−c ), n) = deg h(M i i ) ) ( ) ( ) , n) . ′ ′ Hence deg h(M , n)( ≥ deg)h(M, n) ( . But )deg h(R, n) > deg h(M , n) by the first case. Thus deg h(R, n) > deg h(M, n) . □ h(R, n) = e(1) d−1 (d−1)! n Solutions: 20. Appendix: Homogeneity 187 Exercise (20.15). — Let R be a Noetherian ring, q an ideal, and M a finitely √ generated module. Assume ℓ(M/qM ) < ∞. Set m := q. Show deg pm (M, n) = deg pq (M, n). Solution: There is an m such that m ⊃ q ⊃ mm by (3.25). Hence mn M ⊃ qn M ⊃ mmn M for all n ≥ 0. Dividing into M and extracting lengths yields ℓ(M/mn M ) ≤ ℓ(M/qn M ) ≤ ℓ(M/mmn M ). Therefore, for large n, we get pm (M, n) ≤ pq (M, n) ≤ pm (M, nm). The two extremes are polynomials in n with the same degree, say d, (but not the same leading coefficient). Dividing by nd and letting n → ∞, we conclude that the polynomial pq (M, n) also has degree d. □ Exercise (20.19). — Derive the Krull Intersection Theorem, (18.28), from the Artin–Rees Lemma, (20.18). Solution: In the notation of (18.28), we must prove that N = aN . So apply the Artin–Rees Lemma to N and the a-adic filtration of M ; we get an m such that a(N ∩ am M ) = N ∩ am+1 M . But N ∩ an M = N for all n ≥ 0. Thus N = aN . □ 20. Appendix: Homogeneity ⊕ ⊕ Exercise (20.22). — Let R = Rn be a graded ring, M = n≥n0 Mn a graded ⊕ module, a ⊂ n>0 Rn a homogeneous ideal. Assume M = aM . Show M = 0. ⊕ Solution: Suppose M ̸= 0; say Mn0 ̸= 0. Note M = aM ⊂ n>n0 Mn ; hence Mn0 = 0, a contradiction. Thus M = 0. □ ⊕ ⊕ Exercise (20.23). — Let R = Rn be ⊕ a Noetherian graded ring, M = Mn a finitely generated graded R-module, N = Nn a homogeneous submodule. Set N ′ := { m ∈ M | Rn m ∈ N for all n ≫ 0 }. Show that N ′ is the largest homogeneous submodule of M containing N and having, for all n ≫ 0, its degree-n homogeneous component Nn′ equal to Nn . Solution: Given m, m′ ∈ N ′ , say Rn m, Rn m′ ∈ N for n ≫ 0. Let x ∈ R. Then Rn (m + m′ ), Rn xm ∈ N for n ≫ 0. So N ′ ⊂ M is a submodule. Trivially N ⊂ N ′ . Let mi be a homogeneous component of m. Then Rn mi ∈ N for n ≫ 0 as N is homogeneous. Thus N ′ ⊂ M is a homogeneous submodule containing N . Since R is Noetherian and M is finitely generated, N ′ is finitely generated, say by g, g ′ , . . . , g (r) . Then there is n0 with Rn g, Rn g ′ , . . . , Rn g (r) ∈ N for n ≥ n0 . Replace g, g ′ , . . . , g (r) by their homogeneous components. Say g, g ′ , . . . , g (r) are now of degrees d, d′ , . . . , d(r) with d ≥ d′ ≥ · · · ≥ d(r) . Set n1 := d + n0 . Given m ∈ Nn′ with n ≥ n1 , say m = xg + x′ g ′ + · · · with x ∈ Rn−d and ′ x ∈ Rn−d′ and so on. Then n0 ≤ n − d ≤ n − d′ ≤ · · · . Hence m ∈ Nn . Thus ′ ′ Nn′ ⊂ Nn . But ⊕ N ′′⊃ N . Thus Nn = Nn for n ≥ n′′1 , as desired. ′′ with Nn = Nn for n ≥ n2 . Let m ∈ N ′′ Let N = Nn ⊂ M be ⊕homogeneous ′′ and p ≥ n2 . Then Rp m ∈ n≥n2 Nn ⊂ N . So m ∈ N ′ . Thus N ′′ ⊂ N ′ . □ 188 Solutions: 21. Dimension Exercise (20.25). — Let √ R be a graded ring, a a homogeneous ideal, and M a graded module. Prove that a and Ann(M ) and nil(M ) are homogeneous. ∑r+n Solution: Take x = i≥r xi ∈ R with the xi the homogeneous components. √ First, suppose x ∈ a. Say xk ∈ a. Either xkr vanishes√or it is the initial √ component of xk . But a is homogeneous. So xkr ∈ a. So xr ∈√ a. So x − xr ∈ a √ by (3.24). So all the xi are in a by induction on n. Thus a is homogeneous. ∑ Second, suppose x ∈ Ann(M ). Let m ∈ M . Then 0 = xm = xi m. If m is homogeneous, then xi m = 0 for all i, since M is graded. But M has a set of homogeneous generators. Thus xi ∈ Ann(M ) for√all i, as desired. □ Finally, nil(M ) is homogeneous, as nil(M ) = Ann(M ) by (13.10). Exercise (20.26). — Let R be a Noetherian graded ring, M a finitely generated graded module, Q a submodule. Let Q∗ ⊂ Q be the submodule generated by the homogeneous elements of Q. Assume Q is primary. Then Q∗ is primary too. Solution: Let x ∈ R and m ∈ M be homogeneous with xm ∈ Q∗ . Assume x∈ / nil(M/Q∗ ). Then, given ℓ ≥ 1, there is m′ ∈ M with xℓ m′ ∈ / Q∗ . So m′ has ′′ ℓ ′′ ∗ ℓ ′′ a homogeneous component m with x m ∈ / Q . Then x m ∈ / Q by definition of Q∗ . Thus x ∈ / nil(M/Q). Since Q is primary, m ∈ Q by (18.4). Since m is homogeneous, m ∈ Q∗ . Thus Q∗ is primary by (20.24). □ Exercise (20.30). — Under the conditions of (20.8), assume that R is a domain and that its integral closure R in Frac(R) is a finitely generated R-module. (1) Prove that there is a homogeneous f ∈ R with Rf = Rf . (2) Prove that the Hilbert Polynomials of R and R have the same degree and same leading coefficient. Solution: Let x1 , . . . , xr be homogeneous generators of∏R as an R-module. Write xi = ai /bi with ai , bi ∈ R homogeneous. Set f := bi . Then f xi ∈ R for each i. So Rf = Rf . Thus (1) holds. Consider the short 0(→ R →)R → R/R → 0. Then (R/R)f = 0 ) ( exact sequence by (12.16). So deg h(R/R, n) < deg h(R, n) by (20.10) and (1). But h(R, n) = h(R, n) + h(R/R, n) by (19.9) and (20.8). Thus (2) holds. □ 21. Dimension Exercise (21.8). — Let R be a Noetherian ring, and p be a prime minimal containing x1 , . . . , xr . Given r′ with 1 ≤ r′ ≤ r, set R′ := R/⟨x1 , . . . , xr′ ⟩ and p′ := p/⟨x1 , . . . , xr′ ⟩. Assume ht(p) = r. Prove ht(p′ ) = r − r′ . Solution: Let x′i ∈ R′ be the residue of xi . Then p′ is minimal containing x′r′ +1 , . . . x′r by (1.8) and (2.7). So ht(p′ ) ≤ r − r′ by (21.7). On the other hand, Rp′ ′ = Rp′ by (11.22), and Rp′ = Rp /⟨x1 /1, . . . , xr′ /1⟩ by (12.18) Hence dim(Rp′ ′ ) ≥ dim(Rp ) − r′ by repeated application of (21.5). So ht(p′ ) ≥ r − r′ by (21.6.1), as required. □ Exercise (21.10). — Let R be a Noetherian ring, p a prime ideal with ht(p) ≥ 2. Prove p is the union of infinitely many distinct prime ideals q with ht(q) = 1. Solutions: 21. Dimension 189 Solution: Dividing R by a minimal prime ideal contained in p, we may, plainly, assume R is a domain. Given a nonzero x ∈ p, let qx ⊂ p be a prime ideal minimal containing x. Then ht(qx ) = 1 by the Krull Principal Theorem (21.9). Plainly ∪ qx = p. Finally, if there were only finitely many distinct qx , then by Prime Avoidance (3.15), one qx would be equal to p, a contradiction. □ Exercise (21.11). — Let R be a Noetherian ring with only finitely many prime ideals. Show dim(R) ≤ 1. Solution: By (21.10), there’s no prime p with ht(p) ≥ 2. So dim(R) ≤ 1. □ Exercise (21.12). — Let R be a domain. Prove that, if R is a UFD, then every height-1 prime is principal, and that the converse holds if R is Noetherian. Solution: Let p be a height-1 prime. Then there’s a nonzero x ∈ p. Factor x. One prime factor p must lie in p as p is prime. Clearly, ⟨p⟩ is a prime ideal as p is a prime element. But ⟨p⟩ ⊂ p and ht(p) = 1. Thus, ⟨p⟩ = p. Conversely, assume every height-1 prime is principal and assume R is Noetherian. To prove R is a UFD, it suffices to prove every irreducible element p is prime (see [1, Ch. 11, Sec. 2, pp. 392–396]). Let p be a prime minimal containing p. By Krull’s Principal Ideal Theorem, ht(p) = 1. So p = ⟨x⟩ for some x. Then x is prime by (2.6). And p = xy for some y as p ∈ p . But p is irreducible. So y is a unit. Thus p is prime, as desired. □ Exercise (21.13). — (1) Let A be a Noetherian local ring with a principal prime p of height at least 1. Prove that A is a domain. (2) Let k be a field, P := k[[X]] the formal power series ring in one variable. Set R := P × P . Prove that P is Noetherian and semilocal, and that P contains a principal prime p of height 1, but that P is not a domain. Solution: To prove (1), say p = ⟨x⟩, and let q ⊂ p be a minimal prime. Take y ∈ q. Then y = ax for some a. But x ∈ / q since ht p ≥ 1. Hence a ∈ q. Thus q = qx. But x lies in the maximal ideal of the local ring A, and q is finitely generated since A is Noetherian. Hence Nakayama’s Lemma (10.10) yields q = ⟨0⟩. Thus ⟨0⟩ is prime, and so A is a domain. Alternatively, as a ∈ q, also a = a1 x with a1 ∈ q. Repeating yields an ascending chain of ideals ⟨a⟩ ⊂ ⟨a1 ⟩ ⊂ ⟨a2 ⟩ ⊂ · · · . It must stabilize as A is Noetherian: there’s a k such that ak ∈ ⟨ak−1 ⟩. Then ak = bak−1 = bak x for some b. So ak (1 − bx) = 0. But 1 − bx is a unit by (3.4) as A is local. So ak = 0. Hence y = 0 and so q = ⟨0⟩. Thus A is a domain. As to (2), every nonzero ideal of P is of the form ⟨X n ⟩ by (3.8). Hence P is Noetherian. Thus R is Noetherian by (16.16). The primes of R are of the form q × P or P × q where q is a prime of P by (2.11). Further, m := ⟨X⟩ is the unique maximal ideal by (3.7). Hence R has just two maximal ideals m × P and P × m. Thus R is semilocal. Set p := ⟨(X, 1)⟩. Then p = m × P . So p is a principal prime. Further, p contains just one other prime 0 × P . Thus ht(p) = 1. Finally, R is not a domain as (1, 0) · (0, 1) = 0. □ Exercise (21.14). — Let R be a finitely generated algebra over a field. Assume R is a domain of dimension r. Let x ∈ R be neither 0 nor a unit. Set R′ := R/⟨x⟩. Prove that r − 1 is the length of any chain of primes in R′ of maximal length. 190 Solutions: 22. Completion Solution: A chain of primes in R′ of maximal length lifts to a chain of primes pi in R of maximal length with ⟨x⟩ ⊆ p1 ⫋ · · · ⫋ pd . As x is not a unit, d ≥ 1. As x ̸= 0, also p1 ̸= 0. But R is a domain. So Krull’s Principal Ideal Theorem, (21.8), yields ht p1 = 1. So 0 ⫋ p1 ⫋ · · · ⫋ pr is of maximal length in R. But R is a finitely generated algebra over a field. Hence d = dim R by (15.8). □ Exercise (21.16). — Let R be a Noetherian ring. Show that dim(R[X]) = dim(R) + 1. Solution: Let P be a prime ideal of R[X], and p its contraction in R. Then Rp → R[X]P is a flat local homomorphism by (13.24). Hence (21.15) yields dim(R[X]P ) = dim(Rp ) + dim(R[X]P /pR[X]P ). (21.16.1) But R[X]P /pR[X]P = k(p)[X]P , owing to (1.6) and (11.29). So its dimension is 1, as k(p)[X] is a PID. Hence if P has dim(R[X]P ) = dim(R[X]), then (21.16.1) yields dim(R[X]) ≤ dim(R)+1. Thus the desired equality follows from (15.18). □ Exercise (21.17). — Let A be a Noetherian local ring of dimension r. Let m be the maximal ideal, and k := A/m the residue class field. Prove that r ≤ dimk (m/m2 ), with equality if and only if m is generated by r elements. Solution: By (21.4), dim(A) is the smallest number of elements that generate a parameter ideal. But m is a parameter ideal, and the smallest number of generators of m is dimk (m/m2 ) by (10.11)(2). The assertion follows. □ Exercise (21.21). — Let A be a Noetherian local ring of dimension r, and x1 , . . . , xs ∈ A with s ≤ r. Set a := ⟨x1 , . . . , xs ⟩ and B := A/a. Prove these two conditions are equivalent: (1) A is regular, and there are xs+1 , . . . , xr ∈ A with x1 , . . . , xr a regular sop. (2) B is regular of dimension r − s. Solution: Assume (1). Then x1 , . . . , xr generate the maximal ideal m of A. So the residues of xs+1 , . . . , xr generate that n of B. Hence dim(B) ≥ r − s by (21.4). But dim(B) ≥ r − s by (21.5). So dim(B) = r − s. Thus (2) holds. Assume (2). Then n is generated by r − s elements, say by the residues of xs+1 , . . . , xr ∈ A. Hence m is generated by x1 , . . . , xr . Thus (1) holds. □ 22. Completion Exercise (22.3). — In the 2-adic integers, evaluate the sum 1 + 2 + 4 + 8 + · · · . Solution: In the 2-adic integers, 1 + 2 + 4 + 8 + · · · = 1/(1 − 2) = −1. □ Exercise (22.4). — Let R be a ring, a an ideal, and M a module. Prove the following three conditions are equivalent: ∩ c is injective; (1) κ : M → M (2) an M = ⟨0⟩; (3) M is separated. ∩ n Solution: Clearly, Ker(κ) = a M ; so (1) and (2) are equivalent. Moreover, (2) and (3) were proved equivalent in (22.1). □ Solutions: 22. Completion 191 Exercise (22.8). — Let A be a Noetherian semilocal ring, and m1 , . . . , mm all its b = ∏A bm . maximal ideals. Prove that A i Solution: Set m := rad(R). Fix n ≥ 0. Then A/mn is Noetherian of dimension 0; so it’s Artinian by (19.18). Hence (19.17) yields ∏ A/mn = i (A/mn )(mi /mn ) . However, (A/mn )(mi /mn ) is equal to (A/mn )mi by (11.22),∏so to Ami∩ /mn Ami n n by Exactness of Localization (12.16). Furthermore, m = ( mi ) = mni by n n (1.13). Now, mni is m ∏i -primaryn by (18.10). Hence m Ami = mi Ami by (18.23). n Therefore, A/m = i (Ami /mi Ami ). Taking inverse limits, we obtain the assertion, because inverse limit commutes with finite product by the construction of the limit. □ Exercise (22.9). — Let R be a ring, M a module, M = M0 ⊃ M1 ⊃ · · · a filtration, and N ⊂ M a submodule. Filter N by Nn := N ∩ Mn . Assume N ⊃ Mn b ⊂M c and M c/N b = M/N and G(M c ) = G(M ). for n ≥ n0 for some n0 . Prove N Solution: For each n ≥ n0 , form this commutative diagram with exact rows: → M/N − →0 0− → M/Mn+1 − → N/Mn+1 −       y y y 0 −−→ N/Mn −−−→ M/Mn −−→ M/N − →0 The left vertical map is surjective; the right is the identity. So the induced sequence b →M c → M/N → 0 0→N b ⊂M c and M c/N b = M/N . is exact by (22.6) and (22.7). Thus N c c cn /M cn+1 = Mn /Mn+1 . In particular, M /Mn = M/Mn for each n. Therefore, M c ) = G(M ). Thus G(M □ Exercise (22.10). — (1) Let R be a ring, a an ideal. If Ga (R) is a domain, show n b is an domain. If also ∩ R n≥0 a = 0, show R is a domain. (2) Use (1) to give an alternative proof that a regular local ring A is a domain. b be nonzero. Since R b is separated there Solution: Consider (1). Let x, y ∈ R r r+1 s s+1 b) are positive integers r and s with x ∈ b a −b a and y ∈ b a −b a . Let x′ ∈ Gbra (R b ) denote the images of x and y. Then x′ ̸= 0 and y ′ ̸= 0. Now, and y ′ ∈ Gbsa (R b Gba (R ) = Ga (R) by (22.9). Assume Ga (R) is a domain. Then x′ y ′ ̸= 0. Hence b is a domain. x′ y ′ ∈ Gbr+s is the image of xy ∈ b a r+s . Hence xy ̸= 0. Thus R ∩ a n b b is. Thus (1) holds. If n≥0 a = 0, then R ⊂ R by (22.4); so R is a domain if R ∩ As to (2), denote the maximal ideal of A by m. Then n≥0 mn = ⟨0⟩ by the Krull Intersection Theorem (18.28), and Gm (A) is a polynomial ring by (21.20), so a domain. Hence A is a domain, by (1). Thus (2) holds. □ Exercise (22.12). — Let A be a semilocal ring, m1 , . . . , mm all its maximal ideals, b is a semilocal ring, that m b 1, . . . , m b m are all its and set m := rad(A). Prove that A b ). b = rad(A maximal ideals, and that m 192 Solutions: 22. Completion b m b m b = A/m and A/ b i = A/mi . So m b is maxSolution: First, (22.9) yields A/ / ∩ ∏ ∏ bi b m b ⊂ (A/ b i ); so imal. By hypothesis, m = mi ; so A/m ⊂ (A/mi ). Hence A m ∩ b b b b= m b i . So m b ⊃ rad(A ). But m b ⊂ rad(A) by (22.2). Thus m b = rad(A ). m b Then m′ ⊃ rad(A b) = ∩ m b i . Hence Finally, let m′ be any maximal ideal of A. b i for some i by (2.2). But m b i is maximal. So m′ = m b i . Thus m b 1, . . . , m bm m′ ⊃ m b b are all the maximal ideals of A, and so A is semilocal. □ b its image. Exercise (22.15). — Let A be a Noetherian ring, x ∈ A, and x b∈A Prove x b is a nonzerodivisor if x is. Prove the converse holds if A is semilocal. Solution: Assume x is a nonzerodivisor. Then the multiplication map µx is injective on A. So by Exactness of Completion, the induced map µ bx is injective on b But µ A. bx = µxb. Thus x b is a nonzerodivisor. Conversely, assume x b is a nonzerodivisor and A is semilocal. Then µ bx is injective b So its restriction is injective on the image of the canonical map A → A. b on A. But this map is injective, as the completion is taken with respect to the Jacobson radical; further, µ bx induces µx . Thus x is a nonzerodivisor. □ Exercise (22.16). — Let p ∈ Z be prime. For n > 0, define a Z-linear map αn : Z/⟨p⟩ → Z/⟨pn ⟩ by αn (1) = pn−1 . ⊕ ⊕ ⊕ Set A := n≥1 Z/⟨p⟩ and B := n≥1 Z/⟨pn ⟩. Set α := αn ; so α : A → B. b (1) Show that the p-adic completion A is just A. (2) Show that, in the topology on A induced by the p-adic topology on B, the ∏∞ completion A is equal to n=1 Z/⟨p⟩. (3) Show that the natural sequence of p-adic completions α b b κ b b− 0→A →B − → (B/A) b b (Thus p-adic completion is not right-exact on ((Z-mod)).) is not exact at B. Solution: For (1), note pA = 0. So every Cauchy sequence is constant. Hence b = A. Thus (1) holds. A For (2), set Ak := α−1 (pk B). These Ak are the fundamental open neighborhoods of 0 in the topology induced from the p-adic topology of B. So ( ) ⊕ ⊕ Ak = α−1 0 ⊕ · · · ⊕ 0 ⊕ n>k ⟨pk ⟩/⟨pn ⟩ = (0 ⊕ · · · ⊕ 0 ⊕ n>k Z/⟨p⟩). ⊕k ∏k Hence A/Ak = i=1 Z/⟨p⟩ = n=1 Z/⟨p⟩. But by (22.7), in the induced topology, the completion A is equal to limk≥1 A/Ak . Thus ←− ∏k A = limk≥1 n=1 Z/⟨p⟩. ←− ∏k+1 ∏k In general, let M1 , M2 , . . . be modules, and πkk+1 : n=1 Mn → n=1 Mn the ∏k ∏∞ projections. Then (22.6) yields limk≥1 n=1 Mn = n=1 Mn . Thus (2) holds. ←− For (3), note that, by (2) and (22.6.2), the following sequence is exact: κ b b− 0→A→B → (B/A) b . b = A by (1), and A ̸= A as A is countable yet A is not. Thus Im(b But A α) ̸= Ker(b κ); that is, (3) holds. □ Solutions: 23. Discrete Valuation Rings 193 c preserves Exercise (22.18). — Let R be a ring, a an ideal. Show that M 7→ M b⊗M →M c is surjective if M is finitely generated. surjections, and that R c preserves Solution: The first part of the proof of (22.14) shows that M 7→ M surjections. So (8.16) yields the desired surjectivity. □ Exercise (22.21). — Let R be a Noetherian ring, and a and b ideals. Assume b is. a ⊂ rad(R), and use the a-adic toplogy. Prove b is principal if bR Solution: Since R is Noetherian, b is finitely generated. But a ⊂ rad(R). / Hence, b is principal if b/ab is cyclic by (10.11)(2). But b/ab = b b (ab)b by b by (22.19)(2). Hence, if bR b is principal, then b/ab is cyclic, (22.9), and b b = bR as desired. □ Exercise (22.24) (Nakayama’s Lemma for a complete ring). — Let R be a ring, a an ideal, and M a module. Assume R is complete, and M separated. Show m1 , . . . , mn ∈ M generate if their images in M/aM generate. Solution: Note that the images of m1 , . . . , mn in G(M ) generate over G(R). Therefore, m1 , . . . , mn ∈ M generate over R by the proof of (22.23). Alternatively, M is finitely generated over R and complete by the statement of c. Hence M is also an R-module. b (22.23). Since M is also separated, M = M Since b R is complete, κR : R → R is surjective. Now, a is closed by (22.1); so a is complete; whence, κa : a → b a is surjective too. Hence aM = b aM . Thus M/aM = M/b aM . So b by (22.2). So by Nakayama’s Lemma the mi generate M/b aM . But b a ⊂ rad(R) b so also over R as κR is surjective. (10.11)(2), the mi generate M over R, □ Exercise (22.28). — Let A be a Noetherian local ring, m the maximal ideal. b is a Noetherian local ring with m b as maximal ideal, (2) that Prove (1) that A b b is regular. dim(A) = dim(A ), and (3) that A is regular if and only if A b is Noetherian by (22.26), and local with m b as maximal ideal Solution: First, A by (22.8); thus (1) holds. b m b ) by (20.13). Thus (2) holds b n by (22.9). So d(A) = d(A Second, A/mn = A/ by (21.4). b m b 2 by (22.9). So m and m b have the same number of generators Third, m/m2 = m/ by (10.13). Thus (3) holds. □ 23. Discrete Valuation Rings Exercise (23.6). — Let R be a ring, M a module, and x, y ∈ R. (1) Prove that, if x, y form an M -sequence, then, given any m, n ∈ M such that xm = yn, there exists p ∈ M such that m = yp and n = xp. (2) Prove the converse of (1) if R is local, and x, y lie in its maximal ideal m, and M is Noetherian. 194 Solutions: 23. Discrete Valuation Rings Solution: Consider (1). Let n1 be the residue of n in M1 := M/xM . Then yn1 = 0, but y ∈ / z.div(M1 ). Hence n1 = 0. So there exists p ∈ M such that n = xp. So x(m − yp) = 0. But x ∈ / z.div(M ). Thus m = yp. Consider (2). Given m ∈ M such that xm = 0, take n := 0. Then xm = yn; so there exists p ∈ M such that m = yp and n = xp. Repeat with p in place of m, obtaining p1 ∈ M such that p = yp1 and 0 = xp1 . Induction yields pi ∈ M for i ≥ 2 such that pi−1 = ypi and 0 = xpi . Then Rp1 ⊂ Rp2 ⊂ · · · is an ascending chain. It stabilizes as M is Noetherian. Say Rpn = Rpn+1 . So pn+1 = zpn for some z ∈ R. Then pn = ypn+1 = yzpn . So (1 − yz)pn = 0. But y ∈ m. So 1 − yz is a unit. Hence pn = 0. But m = y n+1 pn . Thus m = 0. Thus x ∈ / z.div(M ). Given n1 ∈ M1 := M/xM such that yn1 = 0, take n ∈ M with n1 as residue. Then yn = xm for some m ∈ M . So there exists p ∈ M such that m = yp and n = xp. Thus n1 = 0. Thus y ∈ / z.div(M1 ). Thus x, y form an M -sequence. □ Exercise (23.7). — Let R be a local ring, m its maximal ideal, M a Noetherian module, x1 , . . . , xn ∈ m, and σ a permutation of 1, . . . , n. Assume x1 , . . . , xn form an M -sequence, and prove xσ1 , . . . , xσn do too; first, say σ transposes i and i + 1. Solution: Say σ transposes i and i + 1. Set Mj := M/⟨x1 , . . . , xj ⟩. Then xi , xi+1 form an Mi−1 -sequence; so xi+1 , xi do too owing to (23.6). So x1 , . . . , xi−1 , xi+1 , xi form an M -sequence. But M/⟨x1 , . . . , xi−1 , xi+1 , xi ⟩ = Mi+1 . Hence xσ1 , . . . , xσn form an M -sequence. In general, σ is a composition of transpositions of successive integers; hence, the general assertion follows. □ Exercise (23.8). — Prove that a Noetherian local ring A of dimension r ≥ 1 is regular if and only if its maximal ideal m is generated by an A-sequence. Solution: Assume A is regular. Given a regular sop x1 , . . . , xr , let’s show it’s an A-sequence. Set A1 := A/⟨x1 ⟩. Then A1 is regular of dimension r − 1 by (21.21). So x1 ̸= 0. But A is a domain by (21.22). So x1 ∈ / z.div(A). Further, if r ≥ 2, then the residues of x2 , . . . , xr form a regular sop of A1 ; so we may assume they form an A1 -sequence by induction on r. Thus x1 , . . . , xr is an A-sequence. Conversely, if m is generated by an A-sequence x1 , . . . , xn , then n ≤ depth(A) ≤ r by (23.4) and (23.5)(3), and n ≥ r by (21.17); thus n = r, and A is regular. □ Exercise (23.10). — Let A be a DVR with fraction field K, and f ∈ A a nonzero nonunit. Prove A is a maximal proper subring of K. Prove dim(A) ̸= dim(Af ). Solution: Let R be a ring, A ⫋ R ⊂ K. Then there’s an x ∈ R − A. Say x = utn where u ∈ A× and t is a uniformizing parameter. Then n < 0. Set y := u−1 t−n−1 . Then y ∈ A. So t−1 = xy ∈ R. Hence wtm ∈ R for any w ∈ A× and m ∈ Z. Thus R = K, as desired. Since f is a nonzero nonunit, A ⫋ Af ⊂ K. Hence Af = K by the above. So dim(Af ) = 0. But dim(A) = 1 by (23.9). □ Exercise (23.11). — Let k be a field, P := k[X, Y ] the polynomial ring in two variables, f ∈ P an irreducible polynomial. Say f = ℓ(X, Y ) + g(X, Y ) with ℓ(X, Y ) = aX + bY for a, b ∈ k and with g ∈ ⟨X, Y ⟩2 . Set R := P/⟨f ⟩ and p := ⟨X, Y ⟩/⟨f ⟩. Prove that Rp is a DVR if and only if ℓ ̸= 0. (Thus Rp is a DVR Solutions: 23. Discrete Valuation Rings 195 if and only if the plane curve C : f = 0 ⊂ k 2 is nonsingular at (0, 0).) Solution: Set A := Rp and m := pA. Then (12.18) and (12.4) yield A/m = (R/p)p = k and m/m2 = p/p2 . First, assume ℓ ̸= 0. Now, the k-vector space m/m2 is generated by the images x and y of X and Y in A. Clearly, the image of f is 0 in m/m2 . Also, g ∈ (X, Y )2 ; so its image in m/m2 is also 0. Hence, the image of ℓ is 0 in m/m2 ; that is, x and y are linearly dependent. Now, f cannot generate ⟨X, Y ⟩, so m ̸= 0; hence, m/m2 ̸= 0 by Nakayama’s Lemma, (10.10). Therefore, m/m2 is 1-dimensional over k; hence, m is principal by (10.11)(2). Now, since f is irreducible, A is a domain. Hence, A is a DVR by (23.9). Conversely, assume ℓ = 0. Then f = g ∈ (X, Y )2 . So m/m2 = p/p2 = ⟨X, Y ⟩/⟨X, Y ⟩2 . Hence, m/m2 is 2-dimensional. Therefore, A is not a DVR by (23.10). □ Exercise (23.12). — Let k be a field, A a ring intermediate between the polynomial ring and the formal power series ring in one variable: k[X] ⊂ A ⊂ k[[X]]. Suppose that A is local with maximal ideal ⟨X⟩. Prove that A is a DVR. (Such local rings arise as rings of power series with curious convergence conditions.) ∩ Solution: Let’s show that the ideal a := n≥0 ⟨X n ⟩ of A is zero. Clearly, a is a ∩ subset of the corresponding ideal n≥0 ⟨X n ⟩ of k[[X]], and the latter ideal is clearly zero. Hence (23.3) implies A is a DVR. □ Exercise (23.13). — Let L/K be an algebraic extension of fields, X1 , . . . , Xn variables, P and Q the polynomial rings over K and L in X1 , . . . , Xn . (1) Let q be a prime of Q, and p its contraction in P . Prove ht(p) = ht(q). (2) Let f, g ∈ P be two polynomials with no common prime factor in P . Prove that f and g have no common prime factor q ∈ Q. Solution: Since L/K is algebraic, Q/P is integral. Furthermore, P is normal, and Q is a domain. Hence we may apply the Going Down Theorem (14.9). So given any chain of primes p0 ⫋ · · · ⫋ pr = p, we can proceed by descending induction on i for 0 ≤ i ≤ r, and thus construct a chain of primes q0 ⫋ · · · ⫋ qr = q with qi ∩ P = pi . Thus ht p ≤ ht q. Conversely, any chain of primes q0 ⫋ · · · ⫋ qr = q contracts to a chain of primes p0 ⊂ · · · ⊂ pr = p, and pi ̸= pi+1 by Incomparability, (14.3); whence, ht p ≥ ht q. Hence ht p = ht q. Thus (1) holds. Alternatively, by (15.13), ht(p) + dim(P/p) = n and ht(q) + dim(Q/q) = n as both P and Q are polynomial rings in n variables over a field. However, by (15.12), dim P/p = tr. degK Frac(P/p) and dim Q/q = tr. degL Frac(Q/q), and these two transcendence degrees are equal as Q/P is an integral extension. Thus again, (1) holds. Suppose f and g have a common prime factor q ∈ Q, and set q := Qq. Then the maximal ideal qQq of Qq is principal and nonzero. Hence Qq is a DVR by (23.9). Thus ht(q) = 1. Set p := q ∩ P . Then p contains f ; whence, p contains some prime factor p of f . Then p ⊇ P p, and P p is a nonzero prime. Hence p = P p since ht p = 1 by (1). However, p contains g too. Therefore, p | g, contrary to the hypothesis. Thus (2) holds. (Caution: if f := X1 and g := X2 , then f and g have no common factor, yet there are no φ and ψ such that φf + ψg = 1.) □ 196 Solutions: 23. Discrete Valuation Rings Exercise (23.15). — Let R be a Noetherian ring. Show that R is reduced if and only if (R0 ) and (S1 ) hold. Solution: Assume ) hold. Consider an irredundant primary de∩ (R0 ) and (S √1 composition ⟨0⟩ = qi . Set pi := √ qi . Then pi is minimal by (S1 ), and pi = qi by ∩ (R0 ) and (18.22). So ⟨0⟩ = pi = ⟨0⟩. Thus R is reduced. Conversely, assume R is reduced. Then Rp is reduced for any prime ∩ p by (13.18). So if p is minimal, then Rp is a field. Thus (R0 ) holds. But ⟨0⟩ = p minimal p. So p is minimal whenever p ∈ Ass(R) by (18.19). Thus R satisfies (S1 ). □ Exercise (23.20). — Prove that a Noetherian domain R is normal if and only if, given any prime p associated to a principal ideal, pRp is principal. Solution: Assume R normal. Say p ∈ Ass(R/⟨x⟩). Then pRp ∈ Ass(Rp /⟨x/1⟩) by (17.9). So depth(Rp ) = 1. But Rp is normal by (11.31). Hence pRp is principal by (23.9). Conversely, assume that, given any prime p associated to a principal ideal, pRp is principal. Given any prime p of height 1, take a nonzero x ∈ p. Then p is minimal containing ⟨x⟩. So p ∈ Ass(R/⟨x⟩) by (17.17). So, by hypothesis, pRp is principal. So Rp is a DVR by (23.9). Thus R satisfies (R1 ). / Given any prime p with depth(Rp ) = 1, say pRp ∈ Ass(Rp ⟨x/s⟩) with x ̸= 0. Then ⟨x/s⟩ = ⟨x/1⟩ ⊂ Rp . So p ∈ Ass(R/⟨x⟩) by (17.9). So, by hypothesis, pRp is principal. So dim(Rp ) = 1 by (23.9). Thus R also satisfies (S2 ). So R is normal by Serre’s Criterion, (23.18). □ Exercise (23.21). — Let R be a Noetherian ring, K its total quotient ring, Set Φ := { p prime | ht(p) = 1 } and Σ := { p prime | depth(Rp ) = 1 }. Assuming (S1 ) holds in R, prove Φ ⊂ Σ, and prove Φ = Σ if and only if (S2 ) holds. Further, without assuming (S1 ) holds, prove this canonical sequence is exact: ∏ R → K → p∈Σ Kp /Rp . (23.21.1) Solution: Assume (S1 ) holds. Then, given p ∈ Φ, there exists a nonzerodivisor x ∈ p. Clearly, p is minimal containing ⟨x⟩. So p ∈ Ass(R/⟨x⟩) by (17.17). Hence depth(Rp ) = 1 by (23.5)(2). Thus Φ ⊂ Σ. However, as (S1 ) holds, (S2 ) holds if and only if Φ ⊃ Σ. Thus Φ = Σ if and only if R satisfies (S2 ). Further, without assuming (S1 ), consider (23.21.1). Trivially, the composition ∏ is zero. Conversely, take an x ∈ K that vanishes in p∈Σ Kp /Rp . Say x = a/b with a, b ∈ R and b a nonzerodivisor. Then a/1 ∈ bRp for all p ∈ Σ. But b/1 ∈ Rp is, clearly, a nonzerodivisor for any prime p. Hence, if p ∈ Ass(Rp /bRp ), then p ∈ Σ by (23.5)(2). Therefore, a ∈ bR by (18.25). Thus x ∈ R. Thus (23.21.1) is exact. □ Exercise (23.22). — Let R be a Noetherian ring, and K its total quotient ring. Set Φ := { p prime | ht(p) = 1 }. Prove these three conditions are equivalent: (1) R is normal. (2) (R1 ) and (S2 ) hold. ∏ (3) (R1 ) and (S1 ) hold, and R → K → p∈Φ Kp /Rp is exact. Solutions: 24. Dedekind Domains 197 Solution: Assume (1). Then R is reduced by (14.17). So (23.15) yields (R0 ) and (S1 ). But Rp is normal for any prime p by (14.16). Thus (2) holds by (23.9). Assume (2). Then (R1 ) and (S1 ) hold trivially. Thus (23.21) yields (3). Assume (3). Let x ∈ K be integral over R. Then x/1 ∈ K is integral over Rp for any prime p. Now, Rp is a DVR for all p of height 1 as R satisfies (R1 ). Hence, x/1 ∈ Rp for all p ∈ Φ. So x ∈ R by the exactness of the sequence in (3). But R is reduced by (23.15). Thus (14.17) yields (1). □ 24. Dedekind Domains Exercise (24.5). — Let R be a domain, S a multiplicative subset. (1) Assume dim(R) = 1. Prove dim(S −1 R) = 1 if and only if there is a nonzero prime p with p ∩ S = ∅. (2) Assume dim(R) ≥ 1. Prove dim(R) = 1 if and only if dim(Rp ) = 1 for every nonzero prime p. Solution: Consider (1). Suppose dim(S −1 R) = 1. Then there’s a chain of primes 0 ⫋ p′ ⊂ S −1 R. Set p := p′ ∩ R. Then p is as desired by (11.18)(2). Conversely, suppose there’s a nonzero p with p ∩ S = ∅. Then 0 ⫋ pS −1 R is a chain of primes by (11.18)(2); so dim(S −1 R) ≥ 1. Now, given a chain of primes 0 = p′0 ⫋ · · · ⫋ p′r ⊂ S −1 R, set pi := p′i ∩ R. Then 0 = p0 ⫋ · · · ⫋ pr ⊂ R is a chain of primes by (11.18)(2). So r ≤ 1 as dim(R) = 1. Thus dim(S −1 R) = 1. Consider (2). If dim(R) = 1, then (1) yields dim(Rp ) = 1 for every nonzero p. Conversely, let 0 = p0 ⫋ · · · ⫋ pr ⊂ R be a chain of primes. Set p′i := pi Rpi . Then 0 = p′0 ⫋ · · · ⫋ p′r is a chain of primes by (11.18)(2). So if dim(Rpi ) = 1, then r ≤ 1. Thus, if dim(Rp ) = 1 for every nonzero p, then dim(R) ≤ 1, as desired. □ Exercise (24.6). — Let R be a Dedekind domain, S a multiplicative subset. Prove S −1 R is a Dedekind domain if and only if there’s a nonzero prime p with p ∩ S = ∅. Solution: Suppose there’s a prime nonzero p with p ∩ S = ∅. Then 0 ∈ / S. So S −1 R is a domain by (11.4). And S −1 R is normal by (11.31). Further, S −1 R is Noetherian by (16.7). Also, dim(S −1 R) = 1 by (24.5)(1). Thus S −1 R is Dedekind. The converse results directly from (24.5)(1). □ Exercise (24.8). — Let R be a Dedekind domain, and a, b, c ideals. By first reducing to the case that R is local, prove that a ∩ (b + c) = (a ∩ b) + (a ∩ c), a + (b ∩ c) = (a + b) ∩ (a + c). Solution: By (13.17), it suffices to establish the two equations after localizing at each maximal ideal p. But localization commutes with sum and intersection by (12.15)(4), (5). So the localized equations look like the original ones, but with a, b, c replaced by ap , bp , cp . Thus we may replace R by Rp , and so assume R is a DVR. Referring to (23.1), take a uniformizing parameter t, and say a = ⟨ti ⟩ and b = ⟨tj ⟩ and c = ⟨tk ⟩. Then the two equations in questions are equivalent to these 198 two: Solutions: 24. Dedekind Domains { } { } max i, min{j, k} = min max{i, j}, max{i, k} , { } { } min i, max{j, k} = max min{i, j}, min{i, k} . However, these two equations are easy to check for any integers i, j, k. □ Exercise (24.12). — Prove that a semilocal Dedekind domain A is a PID. Begin by proving that each maximal ideal is principal. Solution: Let p1 , . . . , pr be the maximal ideals of A. Let’s prove they are principal, starting with p1 . By Nakayama’s lemma (10.10), p1 Ap1 ̸= p21 Ap1 ; so p1 ̸= p21 . Take y ∈ p1 −p21 . The ideals p21 , p2 , . . . , pr are pairwise comaximal because no two of them lie in the same maximal ideal. Hence, by the Chinese Remainder Theorem, (1.13), there is an x ∈ A with x ≡ y mod p21 and x ≡ 1 mod pi for i ≥ 2. The Main Theorem of Classical Ideal Theory, (24.10), yields ⟨x⟩ = pn1 1 pn2 2 · · · pnr r with ni ≥ 0. But x ̸∈ pi for i ≥ 2; so ni = 0 for i ≥ 2. Further, x ∈ p1 − p21 ; so n1 = 1. Thus p1 = ⟨x⟩. Similarly, all the other pi are principal. Finally, Main Theorem, (24.10), yields ∏ i let a be any nonzero ideal. Then the ∏ i a = pm for some m . Say p = ⟨x ⟩. Then a = xm □ i i i i i , as desired. Exercise (24.13). — Let R be a Dedekind domain, a and b two nonzero ideals. Prove (1) every ideal in R/a is principal, and (2) b is generated by two elements. Solution: To prove (1), let p1 , . . . , pr be the associated primes of a, and set ∩ S := i (R − pi ). Then S is multiplicative. Set R′ := S −1 R. Then R′ is Dedekind by (24.6). Let’s prove R′ is semilocal. Let q be a maximal ideal of R′ , and set p := q ∩ R. Then q = pR′ by (11.18). So p is nonzero, whence maximal since R has dimension 1. Suppose p is distinct from all the pi . Then p and the pi are pairwise comaximal. So, by the Chinese Remainder Theorem, (1.13), there is a u ∈ R that is congruent to 0 modulo p and to 1 modulo each pi . Hence, u ∈ p ∩ S, but q = pR′ , a contradiction. Thus p1 R′ , . . . , pr R′ are all the maximal ideals of R′ . So R′ is a PID by (24.12); so every ideal in R′ /aR′ is principal. But by (12.18), ′ R /aR′ = S −1 (R/a). Finally, S −1 (R/a) = R/a by (11.7) because every u ∈ S maps to a unit in R/a since the image lies in no maximal ideal of R/a. Thus (1) holds. Alternatively, we can prove (1) without using (24.12), as follows. The Main Theorem of Classical Ideal Theory, (24.10), yields a = pn1 1 · · · pnk k for distinct maximal ideals pi . The pni i are pairwise comaximal. So, by the Chinese Remainder Theorem, (1.13), there’s a canonical isomorphism: ∼ R/pn1 × · · · × R/pnk . R/a −→ 1 k Next, let’s prove each R/pni i is a Principal Ideal Ring (PIR); that is, every ideal is principal. Set S := R − pi . Then S −1 (R/pni i ) = Rpi /pni i Rpi , and the latter ring is a PIR because Rpi is a DVR. However, R/pni i = S −1 (R/pni i ) by (11.7), because every u ∈ S maps to a unit in R/pni i since p/pni i is the only prime in R/pni i . Finally, given finitely many PIRs R1 , . . . , Rk , we must prove their product is a PIR. Consider an ideal b ⊂ R1 × · · · × Rk . Then ⟨ b = b1 × ⟩· · · × bk where bi ⊂ Ri is an ideal by (1.15). Say bi = ⟨ai ⟩. Then b = (a1 , . . . , ak ) . Thus again, (1) holds. Consider (2). Let x ∈ b be nonzero. By (1), there is a y ∈ b whose residue Solutions: 25. Fractional Ideals 199 generates b/⟨x⟩. Then b = ⟨x, y⟩. □ 25. Fractional Ideals Exercise (25.2). — Let R be a domain, M and N nonzero fractional ideals. Prove that M is principal if and only if there exists some isomorphism M ≃ R. Construct the following canonical surjection and canonical isomorphism: π: M ⊗ N → → MN and ∼ Hom(N, M ). φ : (M : N ) −→ Solution: If M ≃ R, let x correspond to 1; then M = Rx. Conversely, assume M = Rx. Then x ̸= 0 as M ̸= 0. Form the map R → M with a 7→ ax. It’s surjective as M = Rx. It’s injective as x ̸= 0 and M ⊂ Frac(R). Form the canonical M × N → M N with (x, y) 7→ xy. It’s bilinear. So it induces a map π : M ⊗ N → M N , and clearly π is surjective. Define φ as follows: given z ∈ (M : N ), define φ(z) : N → M by φ(z)(y) := yz. Clearly, φ is R-linear. Say y ̸= 0. Then yz = 0 implies z = 0; thus, φ is injective. Finally, given θ : N → M , fix a nonzero n ∈ N , and set z := θ(n)/n. Given y ∈ N , say y = a/b and n = c/d with a, b, c, d ∈ R. Then bcy = adn. So bcθ(y) = adθ(n). Hence θ(y) = yz. Thus, φ is surjective, as desired. □ Exercise (25.6). — Let R be a domain, M and N fractional ideals. Prove that the map π : M ⊗ N → M N is an isomorphism if M is locally principal. Solution: By (13.20), it suffices to prove that, for each maximal ideal m, the localization πm : (M ⊗ N )m → (M N )m is bijective. But (M ⊗ N )m = Mm ⊗ Nm by (12.13), and (M N )m = Mm Nm by (25.4). By hypothesis, Mm = Rm x for some x. Clearly Rm x ≃ Rm . And Rm ⊗ Nm = Nm by (8.5)(2). Thus πm ≃ 1Nm . □ Exercise (25.11). — Let R be a UFD. Show that a fractional ideal M is invertible if and only if M is principal and nonzero. Solution: By (25.7), a nonzero principal ideal is always invertible. ∑ Conversely, assume M is invertible. Then trivially M ̸= 0. Say 1 = mi ni with −1 mi ∈ M and n ∈ M . Fix a nonzero m ∈ M . ∑i Then m = mi ni m. But ni m ∈ R as m ∈ M and ni ∈ M −1 . Set ∑ d := gcd{ni m} ∈ R and x := (ni m/d)mi ∈ M. Then m = dx. Given m′ ∈ M , write m′ /m = a/b where a, b ∈ R are relatively prime. Then d′ := gcd{ni m′ } = gcd{ni ma/b} = a gcd{ni m}/b = ad/b. So m′ = (a/b)m = (ad/b)x = d′ x. But d′ ∈ R. Thus M = Rx. □ Exercise (25.14). — Show that a ring is a PID if and only if it’s a Dedekind domain and a UFD. Solution: A PID is Dedekind by (24.2), and is a UFD by (2.23). Conversely, let R be a Dedekind UFD. Then every nonzero fractional ideal is invertible by (25.3) and (25.13), so is principal by (25.11). Thus R is a PID. Alternatively and more directly, every nonzero prime is of height 1 as dim R = 1, so is principle by (21.12). But, by (24.10), every nonzero ideal is a product of nonzero prime ideals. Thus again, R is a PID. □ 200 Solutions: 26. Arbitrary Valuation Rings Exercise (25.16). — Let R be an ring, M an invertible module. Prove that M is finitely generated, and that, if R is local, then M is free of rank 1. ∑ ∼ R and 1 = α( Solution: Say α : M ⊗ N −→ mi ⊗ ni ) with mi ∈ M and ni ∈ N . Given m ∈ M , set ai := α(m ⊗ ni ). Form this composition: ∼ M ⊗ M ⊗ N = M ⊗ N ⊗ M −→ ∼ R ⊗ M = M. β : M = M ⊗ R −→ ∑ Then β(m) = ai mi . But β is an isomorphism. Thus the mi generate M . Suppose R is local. Then R − R× is an ideal. So u := α(mi ⊗ ni ) ∈ R× for some i. Set m := u−1 mi and n := ni . Then α(m ⊗ n) = 1. Define ν : M → R by ν(m′ ) := α(m′ ⊗ n). Then ν(m) = 1; so ν is surjective. Define µ : R → M by µ(x) := xm. Then µν(m′ ) = ν(m′ )m = β(m′ ), or µν = β. But β is an isomorphism. So ν is injective. Thus ν is an isomorphism, as desired. □ Exercise (25.17). — Show these conditions on an R-module M are equivalent: (1) M is invertible. (2) M is finitely generated, and Mm ≃ Rm at each maximal ideal m. (3) M is locally free of rank 1. Assuming the conditions, show M is finitely presented and M ⊗ Hom(M, R) = R. Solution: Assume (1). Then M is finitely generated by (25.16). Further, say M ⊗ N ≃ R. Let m be a maximal ideal. Then Mm ⊗ Nm ≃ Rm . Hence Mm ≃ Rm again by (25.16). Thus (2) holds. Conditions (2) and (3) are equivalent by (13.28). Assume (3). Then (2) holds; so Mm ≃ Rm at any maximal ideal m. Also, M is finitely presented by (13.27); so HomR (M, R)m = HomRm (Mm , Rm ) by (12.21). Consider the evaluation map ev(M, R) : M ⊗ Hom(M, R) → R defined by ev(M, R)(m, α) := α(m). Clearly ev(M, R)m = ev(Mm , Rm ). Clearly ev(Rm , Rm ) is bijective. Hence ev(M, R) is bijective by (13.20). Thus the last assertions hold; in particular, (1) holds. □ 26. Arbitrary Valuation Rings Exercise (26.3). — Let V be a domain. Show that V is a valuation ring if and only if, given any two ideals a and b, either a lies in b or b lies in a. Solution: First, suppose V is a valuation ring. Suppose also a ̸⊂ b; say x ∈ a, but x ∈ / b. Take y ∈ b. Then x/y ∈ / V ; else x = (x/y)y ∈ b. So y/x ∈ V . Hence y = (y/x)x ∈ a. Thus b ⊂ a. Conversely, let x, y ∈ V − {0}, and suppose x/y ∈ / V . Then ⟨x⟩ ̸⊂ ⟨y⟩; else, x = wy with w ∈ V . Hence ⟨y⟩ ⊂ ⟨x⟩ by hypothesis. So y = zx for some z ∈ V ; in other words, y/x ∈ V . Thus V is a valuation ring. □ Exercise (26.4). — Let V be a valuation ring, m its maximal ideal, and p ⊂ m another prime ideal. Prove that Vp is a valuation ring, that its maximal ideal pVp is equal to p, and that V /p is a valuation ring of the field Vp /p. Solutions: 26. Arbitrary Valuation Rings 201 Solution: First, set K := Frac(Vp ). So K = Frac(V ). Let x ∈ K − Vp . Then 1/x ∈ V ⊂ Vp . Thus Vp is a valuation ring. Second, let r/s ∈ pVp where r ∈ p − {0} and s ∈ V − p. Then s/r ∈ / V , else s = (s/r)r ∈ p. Hence r/s ∈ V . Now, (r/s)s = r ∈ p, but s ∈ / p; since p is prime, r/s ∈ p. Thus pVp = p. Third, to prove V /p is a valuation ring of Vp /pVp , we need only show that, whenever x ∈ Vp −V , then x−1 ∈ V . But, V is a valuation ring; hence, x−1 ∈ V . □ Exercise (26.5). — Prove that a valuation ring V is normal. Solution: Set K := Frac(V ), and let m be the maximal ideal. Take x ∈ K integral over V , say xn + a1 xn−1 + · · · + an = 0 with ai ∈ V . Then −1 1 + a1 x−1 + · · · + an x−n = 0. (26.5.1) If x ∈ / V , then x ∈ m by (26.2). So (26.5.1) yields 1 ∈ m, a contradiction. Hence x ∈ V . Thus V is normal. □ Exercise (26.10). — Let K be a field, S the set of local subrings with fraction field K, ordered by domination. Show its maximal elements are the valuation rings. Solution: Let V be maximal in S. By (26.9), V is dominated by a valuation ring V ′ of K. By maximality, V = V ′ . Conversely, let V be a valuation ring of K. Then V lies in S by (26.2). Let V ′ ∈ S dominate V . Let m and m′ be the maximal ideals of V and V ′ . Take any nonzero x ∈ V ′ . Then 1/x ∈ / m′ as 1 ∈ / m′ ; so also 1/x ∈ / m. So x ∈ V by (26.2). ′ Hence, V = V . Thus V is maximal in S. □ Exercise (26.15). — Let V be a valuation ring, such as a DVR, whose value group Γ is Archimedean; that is, given any nonzero α, β ∈ Γ, there’s n ∈ Z such that nα > β. Show that V is a maximal proper subring of its fraction field K. Solution: Let R be a subring of K strictly containing V , and fix a ∈ R − V . Given b ∈ K, let α and β be the values of a and b. Then α < 0. So, as Γ is Archimedean, there’s n > 0 such that −nα > −β. Then v(b/an ) > 0. So b/an ∈ V . So b = (b/an )an ∈ R. Thus R = K. □ Exercise (26.16). — Let V be a valuation ring. Show that (1) every finitely generated ideal a is principal, and (2) V is Noetherian if and only if V is a DVR. Solution: To prove (1), say a = ⟨x1 , . . . , xn ⟩ with xi ̸= 0 for all i. Let v be the valuation. Suppose v(x1 ) ≤ v(xi ) for all i. Then xi /x1 ∈ V for all i. So xi ∈ ⟨x1 ⟩. Hence a = ⟨x1 ⟩. Thus (1) holds. To prove (2), first assume V is Noetherian. Then V is local by (26.2), and by (1) its maximal ideal m is principal. Hence V is a DVR by (23.9). Conversely, assume V is a DVR. Then V is a PID by (23.1), so Noetherian. Thus (2) holds. □ References [1] Artin, M., “Algebra,” Prentice-Hall, 1991. [2] Atiyah, M., and Macdonald, I., “Introduction to Commutative Algebra,” Addison-Wesley, 1969. [3] Eisenbud, D., “Commutative Algebra with a View Toward Algebraic Geomertry,” SpringerVerlag, 1999. [4] Judson, T., “Abstract Algebra: theory and Applications,” Open source, Electronic Book, [5] Lang, S., “Undergraduate Analysis,” Springer-Verlag, 1997. [6] Lang, S., “Algebra” Graduate Texts in Mathematics 211, Springer-Verlag, 2002. [7] Reid, M., “Undergraduate Commutative Algebra,” Cambridge University Press, 1995. [8] Stark H., “An Introduction to Number Theory,” MIT Press, 1978. 202 Index free: (4.10), 17 generators: (1.4), 2; (4.10), 17 homogeneous: (20.1), 101 homogeneous component: (20.21), 107 homogeneous of degree n: (20.21), 107; (20.28), 108 idempotent: (1.10), 5 initial component: (20.21), 107 integral over a ring: (10.16), 51 integrally dependent on a ring: (10.16), 51 irreducible: (2.6), 7 Kronecker delta function: (4.10), 18 limit: (22.1), 116 linear combination: (1.4), 2 linearly independent: (4.10), 17 lowest terms: (11.5), 164 multiplicative inverse: (1.1), 1 nilpotent: (3.18), 12; (13.10), 67 nonzerodivisor: (2.1), 6; (17.13), 88 orthogonal idempotents: (1.10), 5 p-adic integer: (22.1), 116 prime: (2.6), 7 reciprocal: (1.1), 1 relatively prime: (2.24), 8 residue of: (1.5), 3 restricted vectors: (4.10), 17; (4.13), 18 uniformizing parameter: (23.1), 122 unit: (1.1), 1 zerodivisor: (2.1), 6; (17.13), 88 algebra: (1.1), 1 algebra finite: (4.5), 16 algebra map: (1.1), 1 coproduct: (8.19), 42 faithfully flat: (9.4), 44 finitely generated: (4.5), 16 flat: (9.4), 44 group algebra: (26.13), 138 homomorphism: (1.1), 1 integral over a ring: (10.16), 52 localization: (11.22), 58 module finite: (10.16), 51 Rees Algebra: (20.16), 105 structure map: (1.1), 1 subalgebra: (4.5), 15 generated by: (4.5), 15 tensor product: (8.19), 41 canonical: (1.1), 1; (4.2), 14 category theory coequalizer: (6.8), 30 colimit: (6.6), 29 coproduct: (6.7), 30 direct limit: (6.6), 29 dually: (5.2), 20 filtered direct limit: (7.1), 33 identity: (6.1), 26 inclusion: (6.7), 30 initial object: (6.7), 30 insertion: (6.6), 29 inverse: (6.1), 26 isomorphism: (6.1), 26 map: (6.1), 26 morphism: (6.1), 26 object: (6.1), 26 pushout: (6.9), 30 transition map: (6.6), 29 category: (6.1), 26 directed set: (7.1), 33 discrete: (6.7), 30 functor: (6.6), 29 has direct limits: (6.6), 29 product: (6.1), 26 small: (6.6), 29 diagram chase: (5.12), 21 commutative: (1.5), 3 element annihilator: (4.1), 14 Cauchy sequence: (22.1), 116 complementary idempotents: (1.10), 5 equation of integral dependence: (10.16), 52 field: (2.3), 6 discrete valuation: (23.1), 122 fraction field: (2.3), 6 rational functions: (2.3), 6 Trace Pairing: (24.15), 129 trace: (24.15), 129 functor: (6.2), 26 additive: (8.17), 41 adjoint: (6.4), 27 adjoint pair: (6.4), 27 counit: (6.5), 28 unit: (6.5), 28 universal: (6.5), 28 cofinal: (7.13), 35 constant: (6.6), 29 contravariant: (6.1), 27 covariant: (6.2), 26 diagonal: (6.6), 29 direct system: (6.6), 29 exact: (9.2), 43 forgetful: (6.2), 26 isomorphic: (6.2), 27 left adjoint: (6.4), 27 left exact: (9.2), 43 203 204 linear: (8.4), 38; (9.2), 43 natural bijection: (6.4), 27 natural transformation: (6.2), 27 right adjoint: (6.4), 27 right exact: (9.2), 43 ideal: (1.4), 2 associated prime: (17.1), 87 chain stabilizes: (16.3), 82 comaximal: (1.13), 5 contraction: (1.4), 3 extension: (1.4), 3 fractional: (25.1), 131 integral: (25.1), 131 invertible: (25.7), 132 locally principal: (25.5), 132 principal: (25.1), 131 product: (25.1), 131 quotient: (25.1), 131 generated: (1.4), 2 height: (21.6), 111 idempotent: (1.16), 5 intersection: (1.4), 3 length of chain: (15.9), 78 lie over: (14.2), 71 maximal: (2.13), 7 nested: (1.8), 4 nilradical: (3.18), 12 parameter: (21.2), 109 prime: (2.1), 6 principal: (1.4), 2 product: (1.4), 3 proper: (1.4), 3 radical: (3.18), 12 saturated: (11.13), 57 saturation: (11.13), 57 sum: (1.4), 3 symbolic power: (18.26), 95 variety: (13.1), 66 Lemma Artin–Rees: (20.18), 105 E. Artin: (24.14), 129 Equational Criterion for Flatness: (9.19), 47 Equational Criterion for Vanishing: (8.18), 41 Five: (5.14), 22 Ideal Criterion for Flatness: (9.20), 47; (22.20), 120 Nakayama: (10.10), 50; (22.24), 120 Nine: (5.15), 22 Noether Normalization: (15.1), 75 Nonunit Criterion: (3.4), 10 Prime Avoidance: (3.15), 12; (21.4), 111; (23.5), 123 Schanuel: (5.23), 24 Snake: (5.12), 21 Zorn’s: (2.28), 9; (3.9), 11; (16.9), 83; Index (17.8), 88; (26.9), 137 map automorphism: (1.1), 1 bilinear: (8.1), 37 bimodule homomorphism: (8.7), 38 endomorphism: (1.1), 1; (4.4), 15 homogeneous: (20.21), 107 homomorphism: (1.1), 1; (4.2), 14 isomorphism: (1.1), 1; (4.2), 14 lift: (5.20), 23 Noether Isomorphisms: (4.8), 16 quotient map: (4.6), 16 retraction: (5.8), 21 section: (5.8), 21 trilinear: (8.9), 38 matrix of cofactors: (10.2), 49 module: (4.1), 14 a-dic topology: (22.1), 115 ascending chain condition (acc): (16.12), 84 annihilator: (4.1), 14 Artinian: (16.22), 86 associated graded: (20.11), 103 associated prime: (17.1), 87 bimodule: (8.7), 38 bimodule homomorphism: (8.7), 38 chain stabilizes: (16.12), 84; (16.22), 86 characteristic polynomial: (10.1), 49 closed: (4.1), 14 Cohen–Macaulay: (23.4), 123 coimage: (4.9), 17 cokernel: (4.9), 17 complete: (22.1), 116 composition series: (19.1), 97 cyclic: (4.7), 16 depth: (23.4), 123 descending chain condition (dcc): (16.22), 86 dimension: (21.1), 109 direct product: (4.13), 18 direct sum: (4.10), 18; (4.13), 18 discrete: (22.1), 115 embedded prime: (17.1), 87 endomorphism: (4.4), 15 extension of scalars: (8.7), 38 faithful: (4.4), 15; (10.18), 52; (12.22), 65 filtration: (20.11), 103 Hilbert–Samuel Function: (20.11), 103 Hilbert–Samuel Polynomial: (20.11), 103 Hilbert–Samuel Series: (20.11), 103 q-adic: (20.11), 103 q-filtration: (20.11), 103 stable q-filtration: (20.11), 103 topology: (22.1), 115 finitely generated: (4.10), 17 finitely presented: (5.18), 23 flat: (9.4), 44 free: (4.10), 17 Index free basis: (4.10), 17 free of rank ℓ: (4.10), 17 generated: (4.10), 17 graded: (20.1), 101 homogeneous component: (20.1), 101 Hilbert Function: (20.3), 102 Hilbert Polynomial: (20.3), 102 Hilbert Series: (20.3), 102 shifting (20.1), 101 homogeneous component: (20.21), 107 homomorphism: (4.2), 14 image: (4.2), 14 inverse limit: (22.6), 116 invertible: (25.15), 133 isomorphism: (4.2), 14 kernel: (4.2), 14 length: (19.1), 97 localization: (12.2), 61 localization at f : (12.2), 61 localizaton at p: (12.2), 61 locally finitely generated: (13.25), 69 locally finitely presented: (13.25), 69 locally free: (13.25), 69 maximal condition (maxc): (16.12), 84 minimal condition (minc): (16.22), 86 minimal generating set: (10.13), 51 minimal prime: (17.1), 87 modulo: (4.6), 16 M -sequence: (23.4), 123 Noetherian: (16.12), 84 presentation: (5.18), 23 projective (5.20), 23 quotient: (4.6), 16 quotient map: (4.6), 16 R-linear map: (4.2), 14 radical: (21.2), 109 regular sequence: (23.4), 123 residue: (4.6), 16 restriction of scalars left adjoint: (8.10), 39 restriction of scalars: (4.5), 15 restriction of scalars: (8.10), 39 saturated: (12.14), 63 saturation: (12.14), 63 scalar multiplication: (4.1), 14 semilocal: (21.2), 109 separated: (22.1), 115 separated completion: (22.1), 116 simple: (19.1), 97 standard basis: (4.10), 17 submodule: (4.1), 14 homogeneous: (20.6), 102 irredundant primary decomposition: (18.13), 92 minimal primary decomposition: (18.13), 92 p-primary: (18.1), 91 primary: (18.1), 91 205 primary decomposition: (18.13), 92 sum: (4.8), 17 support: (13.8), 67 system of parameters (sop): (21.2), 110 tensor product, see also torsion free: (9.22), 48 notation a + b: (1.4), 3 M = N : (4.2), 14 R = R′ : (1.1), 1 a ∩ b: (1.4), 3 p(n) : (18.26), 95 ∏ Mλ : (4.13), 18 R ≃ R′ : (1.1), 1 ab: (1.4), 3 M ≃ N : (4.2), 14 ((R-alg)): (6.1), 26 ((R-mod)): (6.1), 26 ((Rings)): (6.1), 26; (13.1), 66 ((Sets)): (6.1), 26 ((Top spaces)): (13.1), 66 aR′ : (1.4), 3 αR ⊗ α′ : (8.4), 38 aN : (4.1), 14 Ann(M ): (4.1), 14 Ann(m): (4.1), 14 aS : (11.13), 57 Ass(M ): (17.1), 87 BilR (M, M ′ ; N ): (8.1), 37 b/a: (1.8), 4 Coim(α): (4.9), 17 Coker(α): (4.9), 17 C: (2.3), 6 ⨿ Mλ : (6.7), 29 D(f ): (13.1), 66 δµλ : (4.10), 18 depth(a, M ): (23.4), 123 depth(M ): (23.4), 123 dim(M ): (21.1), 109 dim(R): (15.9), 78 lim Mλ : (6.6), 29 −→ d(M ): (21.2), 110 eµ : (4.10), 18 EndR (M ): (4.4), 15 F(R) : (25.21), 135 F2 : (1.1), 1 Fq : (15.2), 76 Frac(R): (2.3), 6 Γa (M ): (18.21), 94 G(M ): (20.11), 103 Gq (M ): (20.11), 103 Gq (R): (20.11), 103 h(M, n): (20.3), 102 H(M, t): (20.3), 102 Hom(M, N ): (4.2), 14 Im(α): (4.2), 14 lim Mλ : (22.6), 116 ←− 206 ικ : (4.13), 18 Ker(α): (4.2), 14 ⟨a1 , . . . , an ⟩: (1.4), 2 ℓ(M ): (19.1), 97 S −1 R: (11.1), 55; (11.22), 58 L + M : (4.8), 17 M (m): (20.1), 101 (M : N ) : (25.1), 131 c: (22.1), 116 M M −1 : (25.8), 132 Mf : (12.2), 61 Mp : (12.2), 61 M/N : (4.6), 16 M N : (25.1), 131 M ⊕ N : (4.13), 18 M ⊗R N : (8.2), 37 m ⊗ n: (8.2), 37 µR : (4.4), 15 µx : (4.4), 15 nil(M ): (13.10), 67 nil(R): (3.18), 12 1A : (6.1), 26 1M : (4.2), 15 p(M• , n): (20.11), 103 P (M• , t): (20.11), 103 P(R) : (25.21), 135 (ακ ): (4.13), 18 (mλ ): (4.13), 18 (xλ ): (4.10), 17 φp : (11.19), 58; (12.2), 61 φf : (11.10), 56; (12.2), 61 φS : (11.1), 55; (12.2), 61 πκ : (4.13), 18 Pic(R) : (25.21), 134 pq (M, n): (20.11), 103 Pq (M, t): (20.11), 103 Q: (2.3), 6 R/a: (1.5), 3 R× : (1.1), 1 R′ × R′′ : (1.11), 5 R[[X1 , . . . , Xn ]]: (3.7), 10 R[X1 , . . . , Xn ]: (1.3), 2 rad(R): (3.1), 10 rad(M ): (21.2), 109 R: (2.3), 6 Rf : (11.10), 56 Rp : (11.19), 58 Rℓ : (4.10), 17 R⊕Λ : (4.10), 17 R(M• ): (20.16), 105 (Rn ): (23.14), 125 R+ : (20.6), 102 R(q): (20.16), 105 R[x1 , . . . , xn ]: (4.5), 16 N S : (12.14), 63 S: (3.14), 12 s(M ): (21.2), 110 S − T : (1.1), 2 Index (Sn ): (23.14), 125 Spec(R): (13.1), 66 √ a: (3.18), 12 ⊕ M : (4.13), 18 ∑ λ β : (4.13), 19 ∑ κ aλ : (1.4), 2 Supp(M ): (13.8), 67 β: M → → N (5.20), 23 △: (1.1), 2 t: (24.15), 129 tr. deg: (15.8), 77 V(a): (13.1), 66 vp : (24.10), 128 x/s: (11.1), 55; (11.22), 58 Z: (1.1), 1 z.div(M ): (17.13), 88 z.div(R): (2.1), 6 ring: (1.1), 1 absolutely flat: (10.8), 50 algebra, see also Artinian: (16.22), 86; (16.26), 86; (19.8), 99; (19.11), 99; (19.13), 100 ascending chain condition (acc): (16.3), 82 associated graded: (20.11), 103 Boolean: (1.2), 2; (2.18), 8; (3.23), 13; (13.6), 67 catenary: (15.14), 78 coefficient field: (22.30), 121 Dedekind domain: (24.1), 127 dimension: (15.9), 78 Discrete Valuation Ring (DVR): (23.1), 122 domain: (2.3), 6 dominates: (26.8), 137 extension: (14.1), 71 factor ring: (1.5), 3 field, see also formal power series ring: (3.7), 10 graded: (20.1), 101 homomorphism: (1.1), 1 Ideal Class Group: (25.21), 135 integral closure: (10.25), 53 integrally closed: (10.25), 53 Jacobson: (15.19), 80 Jacobson radical: (3.1), 10 kernel: (1.5), 3 Laurent series ring: (3.8), 11 local: (3.3), 10 local homomorphism: (9.11), 44 localization: (11.1), 55 localization at f : (11.10), 56 localizaton at p: (11.19), 58 map: (1.1), 1 maximal condition (maxc): (16.3), 82 modulo: (1.5), 3 Noetherian: (16.1), 82 nonzerodivisor: (2.1), 6 normal: (10.25), 53; (10.28), 53; (11.31), Index 60; (14.8), 72; (14.16), 74; (23.17), 125; (23.20), 126; (24.1), 127; (24.16), 129; (26.5), 136 normalization: (10.25), 53 p-adic integers: (22.1), 116; (22.6), 117 Picard Group: (25.21), 134 polynomial ring: (1.3), 2 Principal Ideal Domain (PID): (2.26), 8; (2.23), 8; (3.8), 11; (9.22), 48; (16.1), 82; (23.1), 122; (24.2), 127; (24.12), 129; (25.14), 133 Principal Ideal Ring (PIR): (24.13), 198 product ring: (1.11), 5; (2.5), 6; (2.12), 7; (3.5), 10; (4.14), 19; (10.24), 53; (14.17), 74; (18.17), 93; (19.17), 100 quotient map: (1.5), 3 quotient ring: (1.5), 3 radical: (3.1), 10 reduced: (3.18), 12 regular local: (21.18), 113; (21.21), 113; (21.24), 114; (22.27), 121; (23.1), 122; (23.8), 124 regular system of parameters: (21.18), 113 residue ring: (1.5), 3 ring of fractions: (11.1), 55 semilocal: (3.3), 10 Serre’s Conditions: (23.14), 125 spectrum: (13.1), 66 compact: (13.6), 67 principal open set: (13.1), 66 quasi-compact: (13.4), 67 Zariski topology: (13.1), 66 subring: (1.1), 1 total quotient ring: (11.4), 55 Unique Factorization Domain (UFD): (2.6), 7; (2.23), 8; (10.28), 53; (16.1), 82; (23.1), 122; (25.11), 132; (25.14), 133; (25.21), 135 valuation: (26.1), 136 sequence Cauchy: (22.1), 116 exact: (5.1), 20 M -sequence: (23.4), 123 regular sequence: (23.4), 123 short exact: (5.3), 20 split exact: (5.8), 21 subset characteristic function: (1.2), 1 multiplicative: (2.1), 6 generated by: (11.5), 164 saturated: (3.12), 11 saturatation: (3.14), 12 symmetric difference: (1.2), 2 system of parameters (sop): (21.2), 110 regular (21.18), 113 tensor product: (8.2), 37 adjoint associativity: (8.9), 38 207 associative law: (8.9), 38 cancellation law: (8.10), 39 commutative law: (8.5), 38 unitary law: (8.5), 38 Theorem Additivity of Length: (19.9), 99 Akizuki: (19.11), 99 Artin’s Character: (24.14), 129 Characterization of DVRs: (23.9), 124 Cayley–Hamilton: (10.1), 49 Cohen: (16.9), 83 Cohen Structure: (22.30), 121 Determinant Trick: (10.2), 49 Dimension: (21.4), 110 Direct limits commute: (6.14), 32 Exactness of Localization: (12.16), 63 Exactness of Completion: (22.14), 119 Exactness of filtered direct limits: (7.10), 34 Finiteness of Integral Closure: (24.17), 129 First Uniqueness: (18.18), 94 Gauss: (10.28), 53 Generalized Hilbert Nullstellensatz: (15.24), 81 Going down for Flat Algebras: (14.11), 73 for integral extensions: (14.9), 72 Going up: (14.3), 71 Hilbert Basis: (16.11), 84 Hilbert Nullstellensatz: (15.6), 77 Hilbert–Serre: (20.7), 102 Incomparability: (14.3), 71 Jordan–Hölder: (19.3), 97 Krull Intersection: (18.28), 95; (20.19), 105 Krull Principal Ideal: (21.9), 112 Lasker–Noether: (18.20), 94 Lazard: (9.18), 46 Left Exactness of Hom: (5.17), 23 Lying over: (14.3), 71 Main of Classical Ideal Theory: (24.10), 128; (25.13), 133 Maximality: (14.3), 71 Noether: (24.20), 130 Scheinnullstellensatz: (3.22), 12 Second Uniqueness: (18.24), 95 Serre’s Criterion: (23.18), 125 Stone’s: (13.7), 67 Tower Law for Integrality: (10.22), 53 Watts: (8.15), 40 Weak Nullstellensatz: (15.4), 77 topological space closed point: (13.2), 66 Jacobson: (15.22), 80 locally closed subset: (15.22), 80 quasi-compact: (13.4), 67 very dense subset: (15.22), 80 topology a-adic: (22.1), 115 208 discrete: (1.2), 2 separated: (22.1), 115 Zariski: (13.1), 66 totally ordered group: (26.12), 137 Archimedean: (26.15), 139 value group: (26.12), 138 unitary: (6.1), 26 universal example: (1.3), 2 Universal Mapping Property (UMP) coequalizer: (6.8), 30 cokernel: (4.9), 17 colimit: (6.6), 29 coproduct: (6.7), 29 direct limit: (6.6), 29 direct product: (4.13), 18 direct sum: (4.13), 18 Index Formal Power Series: (22.29), 121 fraction field: (2.3), 6 free module: (4.10), 18 inverse limit: (22.6), 116 localization: (11.6), 55; (12.3), 61 polynomial ring: (1.3), 2 pushout: (6.9), 30 residue module: (4.6), 16 residue ring: (1.5), 3 tensor product: (8.3), 37 universal example: (1.3), 2 valuation discrete: (26.12), 122 general: (26.12), 138 p-adic: (23.2), 123