The Closed Support Problem over a Complete Intersection Ring

The Closed Support Problem over a Complete Intersection Ring by Monica Ann Lewis A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2021 Doctoral Committee: Professor Assistant Professor Professor Professor Melvin Hochster, Chair Professor Jennifer Kenkel Mircea Mustaţă Karen Smith James Tappenden Monica Ann Lewis [email protected] ORCID iD: 0000-0003-0589-2397 © Monica Ann Lewis 2021 ACKNOWLEDGEMENTS I owe a tremendous debt of gratitude to my advisor, Mel Hochster. His perspectives have fundamentally shaped the way I think about my research, and his support over the course of the many unforeseen challenges of the past few years has been absolutely invaluable. I cannot thank him enough for his mentorship. I would like to thank those within the Mathematics Department who have served as teachers, mentors, and collaborators. I am grateful to Eric Canton for a number of wonderful mathematical conversations. Many of the strongest results of this thesis represent joint work with Eric. Additionally, I would like to thank the members of my doctoral committee, Jennifer Kenkel, Mircea Mustaţă, Karen Smith, and James Tappenden for their reviewing of this document. I am particularly grateful for Karen’s very careful reading and thoughtful feedback. I am also thankful to the graduate students here at Michigan who I am happy to be able to call my friends, and who have each made a significant impact on my life in Ann Arbor in their own way. I would particularly like to thank Gilyoung Cheong, Drew Ellingson, Mark Greenfield, Patrick and Jasmine Kelley, Devlin Mallory, Erin Markiewitz, Jasmine Powell, Rebecca Sodervick, and Farrah Yhee. My time here has been made so much richer by their presence. There are far more members of the University community who have positively shape my experience at Michigan than I could possibly enumerate here. I will thank a few such people in particular. I have grown significantly as a teacher thanks to ii the advice and support of Hanna Bennett, Angela Kubena, and Beth Wolf. I would like to thank each of them for the enormous work they have put in towards creating an environment that has lead me to develop a love of teaching. Teresa Stokes has created a support structure in the mathematics graduate program that has only become more necessary with the start of the pandemic. Finally, I am enormously thankful to my family. My mom has been a vital source of love and support, and my pursuit of graduate study is in no small part inspired my dad’s encouragement from a young age. As he would often put it in terms of honorifics, he wanted me to ensure that the phrase “Mr. Lewis” refers exclusively to him, and not to me. Incidentally, I have already found a truly marvelous solution to that problem (which the margins are too narrow to contain), but it is likely that the present document is closer to what he meant. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v CHAPTER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. Background on Local Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 13 III. Frobenius Actions and F -Modules . . . . . . . . . . . . . . . . . . . . . . . . . 25 IV. The Hellus Isomorphism and Other Functorial Tools . . . . . . . . . . . . . 40 V. Parameter Ideals Following Hochster and Núñez-Betancourt . . . . . . . . 52 VI. Complete Intersection Rings as Annihilator Submodules . . . . . . . . . . . 68 VII. A Complex of Annihilator Submodules . . . . . . . . . . . . . . . . . . . . . . 82 VIII. Application to Closed Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 102 ABSTRACT The issue of closed support for the local cohomology of Noetherian modules and the related problem of finiteness of the set of associated primes of local cohomology have been intensely studied in the literature. Although it is known that the local cohomology of a hypersurface ring of characteristic p > 0 has closed support, it remains an open question whether this property holds for complete intersection rings of higher codimension. For an ideal J generated by a regular sequence of length c in a regular ring R of characteristic p > 0, the closed support property for the local cohomology of R/J would follow from known results in the literature if the local cohomology of J itself always had a finite set of associated primes. We give the first example of a module of the form HIi (J) with an infinite set of associated primes to demonstrate that this is not necessarily the case. In fact, for i < 4 (resp. i < 5), we show that if R/J is a domain (resp. factorial), then Ass HIi (J) is finite if and only if Ass HIi−1 (R/J) is finite. Our proof of this statement involves a novel generalization of an isomorphism of Hellus. To obtain positive results on closed support, in joint work of the author with Eric Canton, we construct a chain complex consisting of direct sums of Frobenius-stable annihilators in the local cohomology module HJc (R). We prove that this complex is exact, and using the exactness property, we show that ht(I/J)+c for an ideal I of R such that R/I is Cohen-Macaulay, the module HI/J has closed support. v (R/J) CHAPTER I Introduction Algebra is concerned with structure. We are interested in the similarities and differences between various instances of an abstract structure specified by some list of formal properties. For example, the abstract structure of a ring consists of a set R equipped with two binary operations + and ·, referred to as addition and multiplication, respectively, that together satisfy several properties. We require that addition is commutative and associative, that there exists an element 0 in R satisfying 0 + a = a for all a, and that for each a, there is an element −a satisfying −a + a = 0. Multiplication must be associative, must distribute over addition, and we require that there exists an element 1 in R such that 1 · a = a · 1 = a. If multiplication in R also satisfies the commutative property, a · b = b · a for all a and b, then R is called a commutative ring. Below are several instances of the abstract structure of a commutative ring. 1. Let Z denote the set of integers, where addition and multiplication are familiar. 2. Let Q[X] denote the set of polynomials in the variable X with coefficients belonging to the rational numbers Q. Addition and multiplication of polynomials is also quite familiar. 3. Let C denote the set of complex numbers a+ bi, where a and b are real numbers. 1 2 Addition and multiplication treat i as a variable in a polynomial expression with the additional requirement i · i = −1. 4. Let C12 denote the set {[1], [2], . . . , [12]} regarded as hour symbols on the face of a clock, with addition and multiplication defined accordingly – for example, [3] + [11] = [2] and [2] · [8] = [4]. 5. Let F2 denote the set of Boolean truth values {TRUE, FALSE}, with addition defined by logical XOR, and multiplication defined by logical AND. 6. Let F2 [z, w] denote the set of polynomials in the variables z and w, with coefficients belonging to F2 . Addition and multiplication are defined in a manner formally similar to in Q[X], but coefficients are manipulated using the operations of F2 . 7. Let S denote the set of continuous functions from the unit sphere to the real numbers, with addition and multiplication defined pointwise. While each is an example of a commutative ring, the algebraic differences between these instances are at least as interesting as their similarities. The statement “For each nonzero element a of the ring R, there is an element b such that ab = 1.” is true for C and F2 , but false for Z, Q[X], C12 , F2 [z, w], and S. The cancellation property “For any b and c in the ring R, if a 6= 0 and a · b = a · c, then b = c.” holds in every example given above except for the ring C12 – consider that [2] · [8] = [2] · [2] but [8] 6= [2] – and the ring S. A nonzero commutative ring that satisfies the cancellation property is called an integral domain or sometimes just a domain. This is equivalent to the condition that ab = 0 implies a = 0 or b = 0. 3 We may also investigate algebraic properties that have no analogue in familiar structures like the integers or real numbers. For example, the statement “For all elements a and b in the ring R, it holds that (a + b)2 = a2 + b2 .” is true for F2 and F2 [z, w], but false for all other examples given. This strange property has far-reaching implications for the rings in which it holds. The class of commutative rings is extraordinarily rich and, for this reason, it is extraordinarily difficult to prove nontrivial theorems that hold for all commutative rings. By imposing additional hypotheses, we may obtain interesting classes of commutative rings that are more tractable to work with. A subset I of a commutative ring R that is closed under addition and that satisfies ar ∈ I for all a ∈ I and all r ∈ R is called an ideal. We say that an ideal I is generated by a list of elements a1 , . . . , am ∈ I if for each b ∈ I, there exist r1 , . . . , rt ∈ R such that b = r1 a1 + . . . rm am . The ideal generated by a1 , . . . , am is sometimes denotes Ra1 + . . . + Ram or (a1 , . . . , am )R. A ring is called Noetherian – named after the algebraist Emmy Noether – if every ideal is generated by a finite list of elements. It is equivalent to require that every ascending chain of ideals I1 ⊆ I2 ⊆ I3 ⊆ · · · terminates in the sense that In = In+1 for all sufficiently large n. The study of commutative Noetherian rings holds a position of central importance in commutative algebra. The class is rich enough to capture most rings one will encounter in the related fields of algebraic number theory and algebraic geometry, while still imposing a sufficiently high level of control to enable the proof of some truly remarkable theorems. The rings Z, Q[X], C, C12 , F2 , and F2 [z, w] are Noetherian. The ring S is not. Almost all1 rings we study from this point on will be commutative and Noetherian. 1 with one notable exception, given in Chapter III 4 Some of the most striking differences between various special classes of Noetherian rings become apparent only when we consider how that ring acts on other algebraic objects. A module over a ring R, also called an R-module, is a set M equipped with an addition operation2 and a linear action of R, also called a scalar multiplication, r · m for r ∈ R and m ∈ M that distributes over addition: r · (m + n) = r · m + r · n for r ∈ R, m, n ∈ M . It is often easier to study finitely generated R-modules. We say that M is generated by a list of elements m1 , . . . , mt ∈ M if M = Rm1 + · · · + Rmt , where this notation is defined similarly to its use in the context of ideals. In fact, an ideal of R is precisely a subset that is also an R-module when equipped with the + and · operations inherited from R. The more general notion of a submodule of an R-module M is defined similarly. For commutative rings like C and F2 where every nonzero element has a multiplicative inverse – such rings are called fields – there is a particularly simple description of all finitely generated modules, at least up to isomorphism3 Theorem. Let K be a field. Every finitely generated R-module is isomorphic to a direct sum4 of the form K ⊕n . This number n is the rank of M , and two K-modules with different ranks are non-isomorphic. The module theory of a field – Q, C, F2 , or otherwise – is precisely the study of vector spaces over that field. For any commutative ring R, we can make sense of the direct sum R⊕n – called a free module of rank n – and it remains true that a free module is determined up to isomorphism by its rank. However, unless R is a field, 2 Namely, a commutative, associative operation + on M for which there exists an element 0 ∈ M satisfying 0 + m = m for all m ∈ M , and such that every element m of M has an additive inverse −m satisfying −m + m = 0 3 Two modules are called isomorphic if there is an invertible structure preserving map between them. That is, an invertible map f : M → N that L satisfies f (m + n) = f (m) + f (n) and f (r · m) = r · f (m) for r ∈ R, m ∈ M . 4 If M is an R-module, M n consists of n-tuples of elements of m with addition and scalar multiplication defined componentwise. The direct sum M1 ⊕ · · · ⊕ Mt is defined analogously. 5 the free modules alone do not paint a complete picture. The integers Z may, at first glance, seem somewhat simpler than the complex numbers C. Upon investigating their corresponding module theories, however, one will find that Z is structurally somewhat more complicated than C. Definition I.1. Let M be a Z-module and p be a prime integer. The p-torsion component of M , denoted Γp (M ), is the set of elements u in M such that pk · u = 0 for some natural number k. Theorem I.2. Every finitely generated Z-module M is isomorphic to a direct sum Z⊕n ⊕Γp1 (M )⊕· · ·⊕Γpt (M ) for some n ≥ 0 and some (possibly empty) list p1 , . . . , pt of distinct prime integers. It is possible to completely classify5 all finitely generated p-torsion Z-modules, but even a complete understanding of internal structure does not necessarily imply an understanding of mappings between structures. The situation over a field can be misleading in this respect. If K is a field, V is an n-dimensional vector space over K, and U is an m-dimensional subspace of V , then the quotient V /U is always an n − m dimensional vector space. Without knowing anything whatsoever about the manner in which U is embedded into V , we can immediately classify the quotient of V by U up to isomorphism. This is not the case over Z. For any prime integer p, the p-torsion components of Z and its submodules 2Z and 3Z are zero. The quotients Z/2Z and Z/3Z, however, both have nontrivial torsion. The quotient of Z⊕2 by the (free, rank 1) Z-module spanned by (1, −1) is a free module of rank 1 with no torsion components. The quotient of Z⊕2 by the (free, 5 An analogous classification exists for finitely generated Q[x]-modules, but both this classification and the analogue of Theorem I.2 fail even for a ring like F2 [z, w]. 6 rank 1) Z-span of (14, 0) has a free component of rank 1, a 2-torsion component, and a 7-torsion component. The nature of the embedding significantly affects the structure of the resulting quotient. That the p-torsion structure of M/N cannot be determined from the p-torsion components of M and N suggests that our understanding of the operation Γp (−) is incomplete. 1.1 The local cohomology of a commutative ring From this point onward, we will freely make use of terminology and basic tools of homological algebra. The reader may consult [Wei94] as a reference. For general definitions and results on commutative rings, see [Mat89]. To any commutative ring R and any ideal I of R, we may define a functor ΓI (−) that takes an R-module M to its I-torsion component ΓI (M ), defined as the set of elements u ∈ M such that I n u = 0 for some n > 0. Similar to what we observed in our discussion of p-torsion components over Z, one cannot determine ΓI (M/N ) given only the modules ΓI (M ) and ΓI (N ) or even given the map ΓI (M → N ). The functor ΓI (−) is left exact but not exact. For a short exact sequence 0 → N → M → M/N → 0 the corresponding sequence with the rightmost 0 omitted 0 → ΓI (N ) → ΓI (M ) → ΓI (M/N ) is exact, but the map ΓI (M ) → ΓI (M/N ) is typically not surjective. It is possible to construct a functor HI1 (−) such that the quotient of ΓI (M/N ) by the image of ΓI (M ) can be recovered as the kernel of HI1 (M → N ) – which is reassuring – but we encounter a similar problem if we try to determine HI1 (M/N ) using HI1 (M ) and 7 HI1 (N ) only. A new functor HI2 (−) can be built to describe the quotient of HI1 (M/N ) by the image of HI1 (M ) in terms of the modules HI2 (M ) and HI2 (N ). This process may continue (a priori) forever, requiring the construction of an infinite family of functors {HIi (−)} – the derived functors of ΓI (−), referred to as local cohomology functors – resulting in the following long exact sequence. 0 ΓI (N ) ΓI (M ) ΓI (M/N ) HI1 (N ) HI1 (M ) HI1 (M/N ) HI2 (N ) HI2 (M ) HI2 (M/N ) HI3 (N ) ··· It is striking that even if our primary interest is the study of finitely generated modules, we are required to consider non-finitely generated modules in order to fully understand the I-torsion components of quotients. This is even true over Z. While 12 12 the 2-torsion part of the quotient of Z − → Z cannot be recovered from Γ(2) (Z − → Z), 12 1 it does appear as the kernel of H(2) (Z − → Z). To understand how this embedding 1 works would require us to investigate the structure of H(2) (Z), which is not finitely generated. It is realized as the quotient Z[2−1 ]/Z and generated by the classes we will denote as {{1/2t }} for t ≥ 1. Multiplying {{1/2t }} by an element of Z can only ever decrease t. 1 The module H(x) (F2 [x]), isomorphic to F2 [x, x−1 ]/F2 [x], is generated over F2 [x] by 1 classes of fractions {{1/xt }} for t ≥ 1. Consider the effect of the map F : H(x) (F2 [x]) → 1 (F2 [x]) that sends {{a/b}} to {{a2 /b2 }}. Using the fact that 1 + 1 = 0 in F2 , one H(x) may show without too much difficulty that F is an additive map. While multiplying {{1/xt }} by an element of F2 [x] can only ever decrease t, repeated application of the  e map F can take {{1/x}} to any power 1/x2 for e ≥ 0. In a sense that we will make 8 1 precise in Chapter III, the module H(x) (F2 [x]) is finitely generated over a (necessarily noncommutative) augmentation of the ring F2 [x] by the F2 -linear operator F . It is far less obvious, but nonetheless true6 , that every module of the form HIi (Fp [x1 , . . . , xn ]) is finitely generated over the noncommutative augmentation of the ring Fp [x1 , . . . , xn ] by an Fp -linear operator F defined in terms of pth powers. Using a different sort of noncommutative enlargement of the ring C[x1 , . . . , xn ] involving differential operators, one can make a similar statement for modules of the form HIi (C[x1 , . . . , xn ]) [Lyu93]. Given the fact that at least some local cohomology modules over become finitely generated over various noncommutative enlargements of the base ring R, it is natural to wonder whether they may share any useful properties in common with finitely generated R-modules. Our primary focus shall be on two particular properties. Namely, the fact that any finitely generated R-module has a finite set of associated primes and a Zariski closed support in Spec(R). When R is a regular ring – such as Fp [x1 , . . . , xn ] or C[x1 , . . . , xn ] – there are a number of cases7 in which the local cohomology of R has a finite set of associated primes. The class of complete intersection rings – rings such as R[x, y, z]/(x2 + y 2 + z 2 − 1) or F2 [x, y, z, w]/(xz − yw, x3 + y 3 + z 3 + w3 ) – are generally more difficult to control. The finiteness of associated primes property, for example, is known to fail for the local cohomology of a number of complete intersection rings. It remains an open question, however, whether the local cohomology of a general Noetherian ring R must have closed support. We refer to this question as the closed support problem over R. The closed support problem is generally open for complete intersection rings. 6 See Theorem III.15, a result of Lyubeznik [Lyu97]. review what is known about finiteness of associated primes for the local cohomology of a regular ring in Section 2.4. 7 We 9 1.2 Overview of this thesis In Chapter II, we will state the basic properties of local cohomology modules that will be necessary in the sequel for ease of reference, and we will review what is known in the literature about the support and associated primes of local cohomology. The situation over a ring of prime characteristic p > 0 is generally the best understood. To make substantial statements about the local cohomology of a ring of prime characteristic p > 0, we require the notion of an RhF i-module [Bli01] and certain results from the closely related theory of F-modules [Lyu97]. The primary goal of Chapter III is to review a number of Lyubeznik’s finiteness results on the induced Frobenius action on local cohomology. These results make extensive use of the of the crucial hypothesis of regularity, and suggest the difficulties that we will encounter upon relaxing that hypothesis in subsequent chapters. Before beginning work on complete intersection rings proper, a few more functorial tools are necessary. For any containment of ideals I ⊆ I ′ , the natural inclusion ΓI ′ (−) → ΓI (−) induces a family of natural transformations HIi ′ (−) → HIi (−) for all i ≥ 0. In Chapter IV, we present the following original result of the author. Theorem (IV.4). Let R be a Noetherian ring and let I ⊆ R be any ideal. Fix i ≥ 0. There is an ideal I ′ ⊇ I (resp. I ′′ ⊇ I) of height ht(I ′ ) ≥ i − 1 (resp. of height ht(I ′′ ) ≥ i) such that the natural transformation HIi ′ (−) → HIi (−) (resp. HIi ′′ (−) → HIi (−)) is an isomorphism of functors (resp. a surjection of functors). This result generalizes an isomorphism theorem of Hellus [Hel01, Theorem 3], who gives an isomorphism of modules HIi (R) (rather than of functors HIi (−)) under the hypothesis that R is Cohen-Macaulay and local. In a different direction, for a ring map R → S and an ideal I of R, we study a family of natural transformations 10 hif (−) : S ⊗R HIi (−) → HIi (S ⊗R −) (see Definition III.12) and show that, in the following sense, these transformations are compatible the Frobenius homomorphism of S. Theorem (IV.10). Let R → S be a homomorphism between two Noetherian rings of prime characteristic p > 0, fix an ideal I ⊆ R and an index i ≥ 0, and let M be an RhF i-module. The natural map S ⊗R HIi (M ) → HIi (S ⊗R M ) is a morphism of ShF i-modules. Our investigation of complete intersections begins in Chapter V. We review a result of Hochster and Núñez-Betancourt stating that the local cohomology of a positive characteristic hypersurface ring has closed support. This result is a corollary of their theorem that if R is regular, J is an ideal, and Ass HIi (J) is a finite set, then Supp HIi−1 (R/J) is closed [HNB17, Theorem 4.12]. Their theorem raises the following question: If R is regular and J is an ideal generated by a regular sequence, must the set Ass HIi (J) be finite? We give the following positive answer in cohomological degree i = 2. Theorem I.3 (V.11). Let R be a regular ring, and I and J be ideals of R. The set Ass HI2 (J) is finite. In cohomological degree i ≥ 3, the situation is more complicated. We give the first example (Theorem V.5) in the literature of a module of the form HI3 (J) with an infinite set of associated primes, answering Hochster and Núñez-Betancourt’s question in the negative. We prove the following theorem giving conditions under which the finiteness of Ass HIi (J) is in fact equivalent to the finiteness of Ass HIi−1 (R/J). 11 Theorem (V.16). Let R be an LC-finite regular ring, let J ⊆ R be an ideal generated by a regular sequence of length c ≥ 2, and let S = R/J. For an ideal I ⊇ J, (ii) If the irreducible components of Spec(S) are disjoint (e.g. S is a domain), then Ass HI3 (J) is finite if and only if Ass HI2 (S) is finite. (iii) If S is normal and locally almost factorial (e.g. S is a UFD), then Ass HI4 (J) is finite if and only if Ass HI3 (S) is finite. This result presents a significant obstacle to the direct generalization of the methods of Hochster and Núñez-Betancourt to complete intersection rings of codimension c ≥ 2. Chapter VI and onward deal with joint work of the author and Eric Canton. To a Q regular sequence f = f1 , . . . , fc of length c in a Noetherian ring R, letting f = ci=1 fi , we study the Frobenius action f p−1 Fnat on the module HfcR (R), where Fnat denotes the natural action of the Frobenius. The primary goal of Chapter VI is to establish a number of basic properties of the action f p−1 Fnat and to motivate its relevance to the closed support problem for R/f . By sending 1 ∈ R/f to the Čech cohomology class {{1/f }}, one obtains an RhF i-linear embedding R/f R ֒→ HfcR (R). Our eventual is to use this embedding to construct an alternative complex (given in Chapter VII) to the RhF i-linear short exact sequence 0 → f R → R → R/f R → 0 that forms the basis in Hochster and Núñez-Betancourt’s approach to closed support. In codimension c ≥ 2, the embedding of R/f R into HfcR (R) leaves behind a cokernel whose local cohomology is somewhat too complicated to use directly. The purpose of Chapter VII is to arrange a family of annihilator submodules of HfcR (R) into a complex ∆ ∆•f (R) (see Definition VII.2) of length c whose terms are local cohomology modules with respect to various subsequences of f . These terms are somewhat more tractable to understand after applying local cohomology functors with respect to an 12 ideal I containing f . The main result of this section, representing joint work of the author and Eric Canton, is that the augmentation of the ∆ ∆•f (R) complex is exact (Theorem VII.7). Finally, in Chapter VIII, using the complex constructed in the previous chapter, we present our main result, concerning the support of the local cohomology of a positive characteristic complete intersection ring with respect to a Cohen-Macaulay ideal. Theorem (VIII.1). Let R be a regular ring of prime characteristic p > 0, let f be a permutable regular of length c, and let I be an ideal containing f such that R/I is Cohen-Macaulay. Let h denote the height of I/f R in the ring S = R/f R. Then h+c HI/f R (S) has closed support. Convention: Throughout this paper, we assume that all given rings are commutative and Noetherian unless stated otherwise. CHAPTER II Background on Local Cohomology Our goal in this chapter is to state some fundamental results in the theory of local cohomology for future reference. We will omit most proofs. The main reference for this material is Brodmann and Sharp [BS12], although we will also be using some terminology and properties of certain classes of rings for which the reader may wish to consult Bruns and Herzog [BH98]. General statements on homological algebra may be found in Weibel [Wei94]. Notational remark: If F and G are functors C → D, we will write natural transformations from F to G as φ(−) : F (−) → G(−), which consists of the data of a map denoted φ(A) : F (A) → G(A) for each object A of C, such that for any C-morphism f : A → B, φ(B) ◦ F (f ) = G(f ) ◦ φ(A). Let ModR denote the category of modules over a ring R (not necessarily of prime characteristic p > 0), and let KomR denote the category of cohomologically indexed complexes of R-modules. Let H i : KomR → ModR denote the functor that takes a complex C • to its ith cohomology module H i (C • ). 2.1 The I-torsion functor and local cohomology Definition II.1. Let R be a ring and let I be an ideal of R. The I-torsion functor, denoted ΓI (−), is the functor that takes an R-module M to the submodule consisting 13 14 of all elements u ∈ M annihilated by some sufficiently high power of I. For a map f : M → N , the map ΓI (f ) is the restriction of f to ΓI (M ). Since the annihilator of I n in M is precisely the set HomR (R/I n , M ), we may identify the functor ΓI (−) with the direct limit limn HomR (R/I n , −). −→ The I-torsion functor detects the presence of associated primes in the set V (I) = {P ∈ Spec(R) | P ⊇ I} in the following sense. Proposition II.2. Let R be a ring, I be an ideal, and M be an R-module. 1. M has an associated prime in V (I) if and only if ΓI (M ) 6= 0. 2. P is an associated prime of M if and only if ΓP RP (MP ) 6= 0 3. The support of M is contained in V (I) if and only if ΓI (M ) = M . 4. P is a minimal prime of M if and only if ΓP RP (MP ) = MP Given a quotient Q = M/N , one may be interested in describing the associated primes of Q in terms of data involving M and (the embedding map of) N only. Item (ii) of the proposition above suggests that we apply the functor ΓP RP ((−)P ) to a short exact sequence 0 → N → M → Q → 0, but the resulting sequence is only exact if the rightmost map to 0 is dropped: 0 → ΓP RP (NP ) → ΓP RP (MP ) → ΓP RP (QP ). In general, ΓI (−) is a left-exact functor but is not exact. If each module N , M , and Q in our short exact sequence were replaced with an injection resolution, the functor ΓI (−) would produce a short exact sequence of complexes that generally have nontrivial cohomology. Definition II.3. Let R be a ring, let I be an ideal, and let M be an R-module. Let M → E • be an injective resolution. The ith local cohomology module of M with respect to I is the module H i (ΓI (E • )) which does not depend on the choice of injective resolution E • . The functor that takes M to HIi (M ), that is, the ith 15 right derived functor of ΓI (−), is denoted HIi (−). To any short exact sequence 0 → N → M → Q → 0, there is a natural family of connecting homomorphisms δ i : HIi (Q) → HIi+1 (N ) resulting in a functorial long exact sequence in local cohomology with respect to I. 0 ΓI (N ) ΓI (M ) ΓI (Q) HI1 (N ) HI1 (M ) HI1 (Q) HI2 (N ) HI2 (M ) HI2 (Q) HI3 (N ) ··· Since ΓI (−) is unchanged if I is replaced by another ideal having the same radical, the same can be said of each functor HIi (−). A module of the form HIi (M ) is automatically I-torsion, and because an I-torsion module has an injective resolution by I-torsion modules [BS12, Corollary 2.1.6], it holds that HIi (M ) = 0 whenever M is I-torsion and i ≥ 1. From this, it is not hard to see that HIi (M ) ≃ HIi (M/ΓI (M )) for all i ≥ 1. There is another (very different) kind of functorial long exact sequence involving local cohomology modules that we will make extensive use of in the sequel. Theorem II.4 (The Mayer-Vietoris Sequence). Let R be a Noetherian ring and let I and J be ideals. There is a sequence of natural transformations, below, that is exact when (−) is replaced by any R-module M . 0 ΓI+J (−) ΓI (−) ⊕ ΓJ (−) ΓI∩J (−) 1 (−) HI+J HI1 (−) ⊕ HJ1 (−) 1 (−) HI∩J 2 (−) HI+J HI2 (−) ⊕ HJ2 (−) 2 HI∩J (−) 3 HI+J (−) ··· 16 For an R-module M , an M -regular sequence is a list of elements f = f1 , . . . , fc such that f M 6= M and such that fi is a nonzerodivisor on M/(f1 , . . . , fi−1 )M . We use the term regular sequence instead of R-regular sequence when M = R. Let R be Noetherian. The depth of a module M on an ideal I, denoted depthI (M ), is the length of a maximal M -regular sequence contained in I (the Noetherianity of R ensures that a regular sequence cannot be extended indefinitely). It is not a priori obvious that depth is well-defined – that is, that all maximal M -regular sequences in I have the same length – but this follows at once from a theorem of Rees. Theorem II.5 (Rees’s Theorem). Let R be a Noetherian ring, let I be an ideal, let N be a finitely generated I-torsion module, and let M be an arbitrary finitely generated R-module. If I contains a maximal M -regular sequence of length c, then ExtiR (N, M ) = 0 for i < c and ExtcR (N, M ) 6= 0. Since ΓI (−) may be identified with limn HomR (R/I n , −), it is a straightforward −→ exercise of homological algebra to show that HIi (−) may be identified with the direct limit limn ExtiR (R/I n , −) for all i. One may then extend Rees’s theorem to the −→ following statement. Theorem II.6. Let R be a Noetherian ring, let I be an ideal, and let M be a finitely depthI (M ) generated R-module. Then HIi (M ) = 0 for i < depthI (M ) and HI 2.2 (M ) 6= 0. I-transform functors For an ideal I ⊆ R, the I-transform functor is defined by DI (−) := lim HomR (I t , −) −→ t DI (−) is a left exact functor whose right derived functors satisfy R i DI (−) ≃ HIi+1 (−). There is a sense in which DI (−) forces depthI (−) ≥ 2 without modifying higher local 17 cohomology on I. Namely, for any R-module M , ΓI (DI (M )) = HI1 (DI (M )) = 0, and HIi (DI (M )) = HIi (M ) for all i ≥ 2. Lemma II.7. [BS12, Theorem 2.2.4(i)] Let R be a Noetherian ring and fix an ideal I ⊆ R. There is a natural transformation ηI (−) : Id → DI (−) such that, for any R-module M , there is an exact sequence ηI (M ) 0 → ΓI (M ) → M −−−→ DI (M ) → HI1 (M ) → 0 Lemma II.8. [BS12, Proposition 2.2.13] Let R be a Noetherian ring, and I ⊆ R be an ideal. Let e : M → M ′ be a homomorphism of R-modules such that Kere and Cokere are both I-torsion. Then (i) The map DI (e) : DI (M ) → DI (M ′ ) is an isomorphism. (ii) There is a unique R-module homomorphism ϕ : M ′ → DI (M ) such that the diagram M e M′ ϕ ηI (M ) DI (M ) commutes. In fact, ϕ = DI (e)−1 ◦ ηI (M ′ ). (iii) The map ϕ of (ii) is an isomorphism if and only if ηI (M ′ ) is an isomorphism, and this is the case if and only if ΓI (M ′ ) = HI1 (M ′ ) = 0. The main property of the ideal transform functor that we will require is the following. Proposition II.9. Let R be a Noetherian ring, y an element of R, and I0 ⊆ R an ideal. Let I = yR ∩ I0 . There is a natural isomorphism of functors DI0 (−)y ≃ DI (−) Proof. Precomposing ηI0 (−)y : (−)y → DI0 (−)y with Id → (−)y we obtain a natural transformation γ(−) : Id → DI0 (−)y . We claim that for any module M , both the kernel and cokernel of γ(M ) : M → DI0 (M )y are I = yR ∩ I0 -torsion: 18 • Ker(γ(M )) consists of those m ∈ M such that m/1 ∈ ΓI0 (M )y , or, equivalently, y t m ∈ ΓI0 (M ) for some t ≥ 0. Let s be such that I0s y t m = 0. Then √ √ (yI0 )max(s,t) m = 0, so m ∈ ΓyI0 (M ) = ΓyR∩I0 (M ) (since yI0 = yR ∩ I0 ). • An element of C = Coker(γ(M )) can be represented by c = f /y t for some f ∈ DI0 (M ), t ≥ 0. Coker(ηI0 (M )) is I0 -torsion, so there is some s such that I0s f ⊆ ImηI0 (M ). Since f = y t c, we have (yI0 )max(s,t) c ⊆ Imγ(M ). The element of C represented by c therefore belongs to ΓyI0 (C) = ΓyR∩I0 (C). Lemma II.8(ii) therefore gives a map ϕ(M ) : DI0 (M )y → DI (M ), specifically ϕ(M ) = DI (γ(M ))−1 ◦ ηI (DI0 (M )y ). Both of the composite maps come from natural transformations DI (γ(−))−1 and ηI (DI0 (−)y ), so the result is a natural transformation ϕ(−) : DI0 (−)y → DI (−). It remains to show that that ϕ(M ) is an isomorphism for each M , which is equivalent, by Lemma II.8(iii), to showing that ΓI (DI0 (M )y ) = HI1 (DI0 (M )y ) = 0. This can be done using the Mayer-Vietoris sequence associated with the intersection i (DI0 (M )y ) vanishes because y ∈ yR + I0 acts as a yR ∩ I0 . Each module HyR+I 0 1 unit on DI0 (M )y , and likewise for the modules ΓyR (DI0 (M )y ) and HyR (DI0 (M )y ). Note that depthI0 (DI0 (M )) ≥ 2, and localization can only make depth go up, so, ΓI0 (DI0 (M )y ) = HI10 (DI0 (M )y ) = 0. 0 0 0 ΓyR∩I0 (DI0 (M )y ) 0 0 1 HyR∩I (DI0 (M )y ) 0 0 0 ⊕ HI20 (DI0 (M )y ) 2 (DI0 (M )y ) HyR∩I 0 We can now see that ΓI (DI0 (M )y ) = HI1 (DI0 (M )y ) = 0, as desired. Corollary II.10. Let R be a Noetherian ring, y an element of R, and I0 ⊆ R an ideal. Let I = yR ∩ I0 . Then for all i ≥ 2, there is a natural isomorphism of functors 19 HIi (−) ≃ HIi0 (−)y . Proof. It is equivalent to show that R i−1 DI (−) ≃ (R i−1 DI0 (−))y . We can calculate R i−1 DI (M ) as H i−1 (DI (E • )) where M → E • is an injective resolution, but by Proposition II.9, DI (−) ≃ DI0 (−)y where (−)y commutes with the formation of cohomology. Thus, H i−1 (DI (E • )) ≃ H i−1 (DI0 (E • ))y = R i−1 DI0 (M ) 2.3
y . The Čech complex If f is an element of R and M is an R-module, the sequence 0 → M → Mf → 0 = (0 → R → Rf → 0) ⊗R M is called the Čech complex on M with respect to f , denoted Č • (f ; M ). Let f = f1 , . . . , ft be a sequence of elements of arbitrary length. The Čech complex on R with respect to f , denoted Č • (f ; R), is the total complex of the tensor product Č • (f1 ; R) ⊗ · · · ⊗ Č • (ft ; R) and Čech complex on M with respect to f is M ⊗R Č • (f ; R), denote Č • (f ; M ). This complex has the form 0→M → M 1≤i≤t Mfi → M 1≤i<j≤t Mfi fj → · · · → Mf1 ···ft → 0 Theorem II.11. [BS12, Theorem 5.1.19] Let R be a Noetherian ring, let I be an ideal, and let f = f1 , . . . , ft be any sequence of elements generating I. The functor that takes M to H i (Č • (f ; M )) is naturally isomorphic to the ith local cohomology functor HIi (−). 20 A straightforward but important application of Čech cohomology is the following result. Theorem II.12. [BS12, Theorem 4.2.1] Let R → S be a homomorphism between two Noetherian rings, let I be an ideal of R, and let N be an S-module. The module HIi (N ) is obtained by restricting scalars to regard N as an R-module, and we may s regard HIi (N ) as an S-module by letting s ∈ S act as HIi (N → − N ). There is a natural i isomorphism of S-modules HIi (N ) ≃ HIS (N ). The arithmetic rank of an ideal I, denoted ara(I) is the least number of generators of an ideal having the same radical as I. Since the local cohomology functors with respect to I are unchanged if I is replaced by an ideal having the same radical, since local cohomology may be computed using the Čech complex on any set of generators, and since the Čech complex has no nonzero terms in cohomological degree greater than the length of the generating sequence chosen, the following result is clear. Corollary II.13. [BS12, Corollary 3.3.3] Let R be a Noetherian ring and I be an ideal. The functor HIi (−) is equal to the zero functor for all i > ara(I). This is not the only functorial vanishing theorem that we will make use of in the sequel. Theorem II.14. [BS12, Theorem 6.1.2] Let (R, m) be a local ring of dimension n. The functor HIi (−) is equal to the zero functor for i > dim(R). Another final key application of the Čech complex is the following manipulation obtained by forming the cohomology of the total complex of a tensor product of two Čech complexes, one of which has a large amoung of vanishing in its cohomology. Theorem II.15. Let R be a Noetherian ring, let f = f1 , . . . , fc be a regular sequence, 21 and let I be an ideal containing f . There is a natural isomorphism HIi (HfcR (R)) ≃ HIi+c (R). Proof. Let g = g1 , . . . , gt be a sequence of elements generating I. This is [Wei94, Theorem 5.5.10] (see also [Wei94, Definitions 5.6.1, 5.6.2]) applied to the double complex Č • (g; R) ⊗R Č • (f ; R) using the vanishing Hfi (Č • (f ; R)) = 0 for i 6= c. 2.4 A brief review of finiteness properties In what follows, it will be helpful to refer to the following property. Definition II.16. Let R be a Noetherian ring. Call an R-module M LC-finite if, for any ideal I of R and any i ≥ 0, the module HIi (M ) has a finite set of associated primes. For example, over any Noetherian ring R, the indecomposable injective module ER (R/P ) is LC-finite. Since Ass ER (R/P ) = {P } [BH98, Lemma 3.2.7], we have ΓI (ER (R/P )) = 0 if and only if I ⊇ P , and since HIi (ER (R/P )) = 0 for i > 0 (ER (R/P ) is injective) it holds that HIi (ER (R/P )) has as its set of associated primes either {P } (if i = 0 and P ⊇ I) or ∅ (otherwise). As another example over any Noetherian ring R since Supp(HIi (M )) ⊆ Supp(M ) for any R-module M , a module with finite support is also trivially LC-finite. Any module over a semilocal ring of dimension at most 1 is trivially LC-finite. We shall call a ring LC-finite if it is LCfinite as a module over itself. As we shall soon see, it is not typically the case that all (finitely generated, nor even cyclic) modules over an LC-finite ring are LC-finite. The class of LC-finite rings is closed under localization. If there is a finite set of maximal ideals m1 , · · · , mt of R such that Spec(R) − {m1 , · · · , mt } can be covered by finitely many charts Spec(Rf ), each of which is LC-finite, then R is LC-finite. If R is LC-finite and A → R is pure (e.g., if A is a direct summand of R), then A is 22 LC-finite [HNB17, Theorem 3.1(d)]. While there are interesting classes of not-necessarily-regular rings known to have the property of LC-finiteness – for example, F -finite rings of finite F -representation type (FFRT) [HNB17, Theorem 5.7] – much of the existing finiteness literature is concerned primarily with the class of regular rings. When R is regular of prime characteristic p > 0, a celebrated theorem of Huneke and Sharp states that R is LC-finite1 [HS93, Corollary 2.3]. Lyubeznik proved that LC-finiteness holds for smooth K-algebras when K is a field of characteristic 0 [Lyu93, Remark 3.7(i)] and for any regular local ring containing Q [Lyu93, Theorem 3.4]. Concerning regular rings of mixed characteristic, unramified regular local rings [Lyu00, Theorem 1], smooth Z-algebras [BBL+ 14, Theorem 1.2], and regular local rings of dimension ≤ 4 [Mar01, Theorem 2.9] are LC-finite. It is an open question whether there exist non-LC-finite regular rings [Hoc19]. The property of LC-finiteness can fail over a hypersurface ring. The first example of this phenomenon is due to Singh [Sin00], who describes a hypersurface ring S finitely generated over Z, S= Z[u, v, w, x, y, z] ux + vy + wz 3 such that for all prime integers p, the module H(x,y,z) (S) has a nonzero p-torsion element. Katzman [Kat02] showed that the property is not necessarily recovered by restricting to (graded or local) rings containing a field. The hypersurface ring S= K[[u, v, w, x, y, z]] − (w + z)uxvy + zv 2 y 2 wu2 x2 2 has a local cohomology module, H(x,y) (S), with an infinite set of associated primes. Katzman’s hypersurface is not a domain, but Singh and Swanson [SS04] construct ex1 This result was later generalized by Lyubeznik’s theory of F -modules [Lyu97]. Lyubeznik in fact shows that the local cohomology modules HIi (R) – and in fact, any iterated local cohomology module HIi11 (· · · (HIitt (R)) · · · ) is itself LC-finite. See Chapter III. 23 amples of equicharacteristic local hypersurface rings to demonstrate that Ass HI3 (S) can be infinite even if S is a UFD that is simultaneously F -regular ring (in characteristic p > 0) or has rational singularities (in equal characteristic 0). In this sense, the presence of even relatively mild singularities can obstruct LC-finiteness. Controlling the support is sometimes more tractable than controlling the full set of associated primes. If M finitely generated over S, then the set Supp HIi (M ) is known to be closed – equivalently, HIi (M ) is known to have finitely many minimal primes – whenever (i) S has prime characteristic p > 0, I is generated by i elements, and M = S [Kat05, Theorem 2.10], (ii) S is standard graded, M is graded, I is the irrelevant ideal, and i is the cohomological dimension of I on M [RŞ05, Theorem 1], (iii) S is local of dimension at most 4 [HKM09, Proposition 3.4], (iv) I has cohomological dimension at most 2 [HKM09, Theorem 2.4], or (v) M = S and S = R/f R for some regular ring R of prime characteristic p > 0 and some nonzerodivisor f ∈ R [HNB17, KZ17]. It is not known how far these results generalize. It is an open question whether HIi (M ) must be closed for all i ≥ 0 and all ideals I, where S is Noetherian and M is finitely generated [Hoc19, Question 2]. We restrict our attention to this question of closed support in the case M = S, which we will refer to as the closed support problem for S. The closed support problem has a trivially positive answer over an LC-finite ring, so assume S is not LC-finite. We shall focus in particular on the results of Katzman and Zhang [KZ17] or Hochster and Núñez-Betancourt [HNB17] on the local cohomology of positive characteristic hypersurface2 rings. It is not known whether this result generalizes to hypersurface rings of characteristic 0, or to positive characteristic complete intersection rings of codimension ≥ 2. Our primary interest shall be in the latter ques2A hypersurface ring is a complete intersection ring of codimension 1. 24 tion, though in Chapter V we will investigate the prospect of applying Hochster and Núñez-Betancourt’s methods to a ring R/J where R is an arbitrary LC-finite regular ring and J is a complete intersection ideal of codimension c ≥ 2. We will require some supplementary results on finiteness of associated primes in the sequel. The following two statements are both well known, and a suitable reference is Hellus [Hel01]. Theorem II.17. Let R be a Noetherian ring and let M be a finitely generated module. For any ideal I, the module HI1 (M ) has a finite set of associated primes. Theorem II.18. Let R be a Noetherian ring, let I be an ideal, and let M be an depthI (M ) R-module. The module HI (M ) has a finite set of associated primes. A stronger version of the latter is given by Brodmann and Lashgari Faghani. Theorem II.19 (Brodmann, Lashgari Faghani [BLF00]). Let R be a Noetherian ring, let I be an ideal, and let M be an R-module. Let t be the least integer such that HIt (M ) is not finitely generated. Then HIt (M ) has a finite set of associated primes. CHAPTER III Frobenius Actions and F -Modules Introduction Throughout this chapter, R shall denote a commutative Noetherian ring of prime characteristic p > 0, that is, a ring for which the kernel of the canonical map Z → R is exactly pZ, for some prime integer p > 0. Since p divides each binomial coeffi
cient pi for 0 < i < p, the Frobenius map r 7→ rp of R, denoted FR : R → R, is a ring homomorphism. Some R-modules can be naturally equipped with an addi- tive self-map that formally shares certain properties in common with the Frobenius homomorphism of R. A Frobenius action on an R-module M is an additive map β : M → M that satisfies β(rm) = rp β(m) for all r ∈ R, m ∈ M . There are at least two useful equivalent perspectives one may use to study Frobenius actions. The first is the notion of an RhF i-module, where RhF i denotes the noncommutative ring R{F }/(rp F −F r | r ∈ R) presented as a quotient of the ring R{F } obtained by freely adjoining a single noncommutative variable F to R. We will be particularly concerned with the property of being finitely generated over RhF i. The second perspective involves the structure morphism 1 of a Frobenius action. When the structure morphism of an action is an isomorphism, the corresponding RhF imodule is called unit, following [Bli01]. Unit RhF i-modules are precisely the subject 1 Definition III.2 25 26 of Lyubeznik’s theory of F -modules [Lyu97], although Lyubeznik did not use this terminology in his paper. Of particular importance are F -finite F -modules, namely, those unit RhF i-modules that are finitely generated over RhF i. We will highlight some of the remarkable finiteness properties of finitely generated unit RhF i-modules. In Section 3.3 we will describe the natural action of the Frobenius induced by a local cohomology functor HIi (−). If R is regular, Lyubeznik proved that if M is unit and finitely generated over RhF i, then so is HIi (M ), when equipped with the natural action induced from M . This result strengthens an earlier result of Huneke and Sharp [HS93] on the associated primes of the local cohomology of a regular ring, and is of fundamental importance to a number of results we will prove in the sequel. The main reference for this section is Blickle [Bli01] in the setting where R is not necessarily regular and M is not necessarily unit, and Lyubeznik [Lyu97] in the unit setting over a regular ring. 3.1 Notation and teminology It can sometimes be helpful to have notation that distinguishes the domain and codomain of the Frobenius homomorphism. Let R1/p denote the set of formal symbols r1/p for r ∈ R, with addition r1/p + s1/p = (r + s)1/p and multiplication r1/p s1/p = (rs)1/p . It is clear that R1/p is isomorphic to R as an abstract ring, but we regard R1/p as an R-algebra via the Frobenius homomorphism FR : R → R1/p , sending r 7→ (rp )1/p . The resulting R-module structure on R1/p has the form r ·s1/p = (rp s)1/p for r ∈ R, s1/p ∈ R1/p . The Frobenius map is injective if and only if R is reduced. In this case, we do no harm in identifying r ∈ R with its image (rp )1/p in R1/p . For an R-module M , we define the R1/p -module M 1/p of formal symbols m1/p for m ∈ M with addition m1/p + n1/p = (m + n)1/p and R1/p multiplication r1/p m1/p = 27 (rm)1/p , for r ∈ R, m, n ∈ M . By restriction of scalars along FR : R → R1/p , we can also make M 1/p into an R-module, with rm1/p = (rp m)1/p . There is an isomorphism of abelian groups M 1/p → M that sends m1/p 7→ m for all m ∈ M , and we refer to this as the formal p-th power map. For any power q = pe , the ring R1/q , the module M 1/q , and the formal q-th power map M 1/q → M may be defined in an analogous manner. Depending on whether we are denoting the target copy of FR as R or R1/p in a given context, we may regard base change along the Frobenius homomorphism as either a functor ModR → ModR , in which case we use the notation M 7→ FR (M ), or as a functor ModR → ModR1/p , where we will use the notation M 7→ R1/p ⊗R M . 3.2 RhF i-modules and their structure morphisms Let RhF i denote the ring RhF i = (rp F R{F } − F r | r ∈ R) where we use R{F } to denote the algebra obtained by adjoining a free noncommutative variable F to R, and the denominator is the two-sided ideal of R{F } with generators rp F −F r for r ∈ R. There is a natural ring homomorphism RhF i → HomZ (R, R) sending a ∈ R to the multiplication map r 7→ ar and sending F to the Frobenius homomorphism of R. In general, the homomorphism RhF i → HomZ (R, R) is not injective – for example, F − 1 ∈ Fp hF i acts as zero on the field Fp , and F − F 2 acts as zero on the ring Fp [t]/t2 . As a left R-module, RhF i is free on the generators 1, F, F 2 , . . .. As a right RL 1/pe module, RhF i is isomorphic to ∞ . The tensor product RhF i ⊗R M with an e=0 R R-module M may be understood accordingly. An RhF i-module (by which we always mean a left RhF i-module) is precisely the 28 data of an R-module M equipped with an additive map β : M → M satisfying β(rm) = rp β(m) for r ∈ R and m ∈ M , describing the action of F ∈ RhF i. We refer to β as a Frobenius action on M . When M = R, we refer to the Frobenius homomorphism FR : R → R as the natural action of R. If M and N are RhF imodules with Frobenius actions α : M → M and β : N → N , respectively, then an RhF i-linear map h : M → N is precisely the data of an R-linear map that satisfies h◦β = α◦h – we will call a map Frobenius stable if this is the case. Since both Ker(h) and Coker(h) are themselves RhF i-modules, we may refer the induced actions of the Frobenius inherited from M and N , respectively. If W is a multiplicative subset of R, and M is an RhF i-module, then there is a unique Frobenius action on W −1 M that makes the natural map M → W −1 M Frobenius stable. This action is described by F (m/w) = F (m)/wp for m ∈ M , w ∈ W . We may also regard W −1 M as an (W −1 R)hF i-module in an obvious way. Of primary importance to our applications is the fact that some non-finitely generated R-modules, when equipped with a suitable action of the Frobenius, become finitely generated when regarded as modules over the ring RhF i. For example, if f is a nonunit of R, then Rf is not finitely generated over R. As an RhF i-module, however, Rf is cyclic with generator 1/f . When R is a Noetherian ring, finite generation over RhF i implies closed support as an R-module2 . Theorem III.1. [HNB17, Lemma 4.5] Let R be a Noetherian ring of prime characteristic p > 0, let M be an RhF i-module, and let N be an R-submodule of M that generates M over RhF i. Then the support of M , regarded as an R-module, is equal to the support of N . In particular, if M is finitely generated over RhF i and N is 2 When we speak of the support of an RhF i-module, we always mean the support of the underlying R-module upon applying the forgetful functor ModRhF i → ModR . 29 the R-span of a finite generating set, then Supp (M ) = Supp (N ) is a Zariski closed subset of Spec(R). An RhF i-linear homomorphic image of a finitely generated RhF i module is still finitely generated, and thus, still has closed support. However, we caution that RhF i is generally neither left nor right Noetherian – consider, for example, R = Fp [x] and the (left, right, or two-sided) ideal of RhF i generated by xF, xF 2 , xF 3 , . . .. There are additional conditions we may impose on an RhF i-module to gain better control over submodules of finitely generated modules. These conditions involve the structure morphism of the corresponding Frobenius action. Definition III.2. Let R be a ring of prime characteristic p > 0 and let M be an RhF i-module with Frobenius action β : M → M . The map M → M 1/p that sends m 7→ (β(m))1/p is an R-linear map from M to an R1/p -module, and therefore induces an R1/p -linear map θ : R1/p ⊗R M → M 1/p , with r1/p ⊗m 7→ (rβ(m))1/p for r ∈ R and m ∈ M . If we are regarding Frobenius as a map R → R (rather than R → R1/p ) with base change functor FR : ModR → ModR , then θ may be regarded as an R-linear map θ : FR (M ) → M . This map is the structure morphism of the RhF i-module M (or of the Frobenius action β, depending on context). Given an arbitrary R1/p -linear map θ : R1/p ⊗R M → M 1/p , we can define a corresponding Frobenius action on M by first mapping M → R1/p ⊗R M via m 7→ 1 ⊗ m, and letting F (m) be the image of θ(1 ⊗ m) under the formal pth power map M 1/p → M . An RhF i-module structure on an R-module M is in this manner equivalent data to specifying an R1/p -linear (resp. R-linear) map R1/p ⊗R M → M 1/p (resp. FR (M ) → M ). The action of higher powers of the Frobenius, F e for e ≥ 0, can be recovered from the structure morphism θ : FR (M ) → M in the following way. Construct a map 30 t+1 t e t (M ) → FR (M ) for 0 ≤ t < e, as shown θe : FR (M ) → M by composing FR (θ) : FR below. e θe : FR (M ) e−1 FR (θ) e−1 (M ) FR ··· FR (θ) 2 FR (M ) FR (M ) θ M By convention, we let θ0 denote the identity M → M . The composition of the e e natural map M → FR (M ) with θe : FR (M ) → M is precisely the action of F e , and the module R · F e (M ) is exactly the image of θe . In R1/p notation, we may take a direct sum over all such maps θe to obtain an R-linear map (3.1) Θ: ∞ M e=0 e FR (M ) → M Recalling the structure of RhF i as a right R-module, Θ may be understood as a map RhF i ⊗R M → M . For any R-submodule N of M , the RhF i-span of N is the Θ image of the composition RhF i ⊗R N → RhF i ⊗R M − → M , with N generating M over RhF i if and only if RhF i ⊗R N → M is surjective. To say that N is RhF i-stable is precisely to say that the image of RhF i ⊗R N → M is contained in N . It is clearly θ sufficient to ensure that the image of FR (N ) → FR (M ) → − M is contained in N . We gain a particularly fine level of control when the structure morphism of an RhF i-module is an isomorphism. Definition III.3. Let R be a ring of prime characteristic p > 0. Call an RhF imodule M unit 3 if the corresponding structure morphism θ : FR (M ) → M of M is an isomorphism. If FR (R) is identified with R in the natural way4 , then the structure morphism 1 FR (R) → R of the natural action on R is the identity map R → − R, so the natural 3 Lyubeznik [Lyu97] uses the term “F -module” for what we refer to here as a “unit RhF i-module” and he proves a number of powerful results within the category of F -modules. Our choice of terminology follows Blickle [Bli01] (also [BB05, EK04]) because we will need to consider morphisms between both unit and non-unit RhF i-modules in the sequel. There are places in the literature where the term “level” is used in place of “unit” [HS77]. 4 As S ⊗ R may be identified with S for any R-algebra S. R 31 action of R is trivially unit. For an ideal I ⊆ R, the Frobenius homomorphism of R/I can be understood as an action either over R or over R/I. In the former case, the structure morphism FR (R/I) = R/I [p] → R/I is the quotient map by I/I [p] . In the latter case, the structure morphism FR/I (R/I) = R/I → R/I is the identity. In other words, if we regard R/I equipped with its natural action as an (R/I)hF imodule, it is unit, but unless I = I [p] , the RhF i-module R/I is never unit. To avoid ambiguity, we will refer to the former as the RhF i structure morphism and the latter as the (R/I)hF i structure morphism. In general, given a map R → S and an RhF i-module M , there is a natural way to endow S ⊗R M with an ShF i-module structure in such a way that M being an RhF i unit implies that S ⊗R M is an ShF i unit. We will use R1/p notation for clarity. Note that the commutative square of maps R1/p S 1/p R S gives a canonical identification of the functors S 1/p ⊗R1/p (R1/p ⊗R −) and S 1/p ⊗S (S ⊗R −) that take R-modules to S 1/p -modules. Definition III.4. Let R → S be a homomorphism between two rings of prime characteristic p > 0 and let M be an RhF i-module with structure morphism θR : R1/p ⊗R M → M 1/p . Define an S 1/p -linear map, the base-changed structure morphism of M , as follows. 1⊗θ R S 1/p ⊗R1/p M 1/p = (S ⊗R M )1/p θS : S 1/p ⊗S (S ⊗R M ) = S 1/p ⊗R1/p (R1/p ⊗R M ) −−−→ When we refer to S ⊗R M as an ShF i-module, it is understood that this refers to the structure morphism θS . If θR is an isomorphism, it is clear that the same is true of 1 ⊗ θR . If u1 , . . . , ut 32 generate M over RhF i, then 1 ⊗ u1 , . . . , 1 ⊗ ut generate S ⊗R M over ShF i. The following is now clear. Proposition III.5. Let R → S be a map between two rings of prime characteristic p > 0, and let M be an RhF i-module. 1. ([Bli01, pp. 20]) If M is unit over RhF i, then S ⊗R M is unit over ShF i. 2. ([Bli01, Proposition 2.22]) If M is finitely generated over RhF i, then S ⊗R M is finitely generated over ShF i. A fundamental observation is that the category of unit RhF i-modules, as a (full) subcategory of the category of RhF i-modules, is abelian [Lyu97, pp. 72]. This follows quickly from the proposition below. It is an observation of Blickle [Bli01, pp. 18] that the unit property of the cokernel requires no extra hypotheses on R, but the unit property of the kernel requires the Frobenius homomorphism to be flat. This will be the case for many results that follow. Due to a classic result of Kunz [Kun69], for a Noetherian ring of prime characteristic p > 0, flatness of the Frobenius homomorphism is equivalent to the assumption that R is regular. Proposition III.6. (see [Lyu97, pp. 72], [Bli01, pp. 18]) Let R be a Noetherian ring of prime characteristic p > 0, and let h : M → N be an RhF i-linear map between two unit RhF i-modules. Then Coker(h), equipped with the RhF i-module structure induced from N , is unit. If R is regular, then Ker(h), equipped with the RhF i-module structure induced from M , is also unit. Proof. Express h : M → N as the composition of two RhF i-linear maps M ։ V and V ֒→ N . Let α, β, γ, δ, and ε denote the structure morphisms of M , N , V , Ker(h), and Coker(h), respectively. Since FR (−) is exact, we have the following two commutative diagrams whose rows are exact. 33 FR (Ker(h)) FR (M ) Ker(h) 0 γ α δ 0 FR (V ) M V 0 and FR (V ) FR (N ) γ 0 V FR (Coker(h)) 0 ε β Coker(h) N 0 The surjectivity of α and β imply the surjectivity of γ and ε, respectively. The snake lemma together with the fact that β is an isomorphism implies that Ker(ε) is isomorphic to Coker(γ), which vanishes, so ε is an isomorphism. If R is regular, then the flatness of FR (−) implies that the kernels of δ and γ embed into the kernels of α and β, respectively, and therefore vanish. The snake lemma implies that Coker(δ) ≃ Ker(γ) = 0, so δ is an isomorphism. Of particularly importance is Lyubeznik’s result that finitely generated unit RhF imodules also form an abelian category. The issue of passing finite generation to unit submodules of finitely generated unit modules is the main source of difficulty. The following theorem of Lyubeznik grants an incredible degree of control over unit the submodules of unit RhF i-modules. Theorem III.7. [Lyu97, Proposition 2.5(b)] Let R be a regular ring and M be an RhF i-module generated over RhF i by the R-submodule N . Let U be a unit RhF isubmodule of M . Then U ∩ N generates U over RhF i. Proof. Since M = S∞ e=0 R · F e (N ) and U = S∞ e=0 U ∩ (R · F e (N )), it suffices to show that for each e, R · F (U ∩ (R · F e (N ))) = U ∩ (R · F e+1 (N )). e Let θ denote the structure morphism of M and let θe : FR (M ) → M be defined as in Diagram (3.2). Note that if R is regular, then for any R-submodule V of M , 34 e e the exactness of FR (−) allows us to identify FR (V ) with a submodule of FR (M ). We e can therefore make sense of the statement R · F e (V ) = θe (FR (V )). Crucially, given two submodules V1 and V2 of M , the exactness of FR (−) also gives FR (V1 ∩ V2 ) = FR (V1 )∩FR (V2 ). Since M is unit, θ is also compatible with intersections. We proceed to compute e R · F (U ∩ (R · F e (N ))) = θ(FR (U ∩ θe (FR (N ))) e = θ(FR (U )) ∩ θ(FR (θe (FR (N )))) Since U is unit under the action restricted from M , θ(FR (U )) = U . Finally, we have e (N )))) = R · F (R · F e (N )) = R · F e+1 (N ), as desired. θ(FR (θe (FR Once this method of controlling unit submodules has been established, Lyubeznik obtains the following two statements as essentially as a corollary of Theorem III.7. Corollary III.8. Let R be a regular ring and let M be a finitely generated unit RhF i-module. 1. [Lyu97, Proposition 2.7] The set of unit submodules of M satisfies the ascending chain condition. 2. [Lyu97, Theorem 2.8] Every unit submodule of M is finitely generated over RhF i. The ascending chain condition in particular is the main ingredient of Lyubeznik’s finiteness theorem on the set of associated primes for a unit RhF i-module. We shall sketch his argument to illustrate how this is the case. For an ideal I of R, the I-torsion submodule ΓI (M ) of an RhF i-module M is clearly Frobenius stable. Using the exactness of FR (−), one can directly show that M is unit, then ΓI (M ) is unit5 . If P is 5 This statement is actually a particular case of a more general phenomenon concerning the preservation of the unit property under the application of local cohomology functors, which we will discuss in more detail in the next section. 35 a maximal associated prime of M , we obtain produce a unit submodule, M1 = ΓP (M ) with only one associated prime. Repeating the argument on M/M1 and taking preimages in M , we obtain another unit submodule M2 ⊇ M1 such that M2 /M1 has a single associated prime. This procedure results in a chain M1 ⊆ M2 ⊆ M3 ⊆ · · · of unit submodules such that each quotient Mi /Mi−1 has a single associated prime. The chain terminates after finitely many steps, so Ass(M ) is contained in the union of the sets of associated primes of only finitely many factors in this filtration. Theorem III.9. [Lyu97, Theorem 2.12(a)] Let R be a regular ring and let M be a finitely generated unit RhF i-module. The set Ass(M ) is finite. 3.3 The Natural Action on Local Cohomology If R is a Noetherian ring and M is an RhF i-module, then for any ideal I and any i ≥ 0, the local cohomology modules HIi (M ) inherit an RhF i-module structure from M . We describe its structure morphism in terms only of maps induced by the functor HIi (−) to show that there is no dependence on the choice of generators for I, but once this independence has been established, we will typically prefer to work in terms of the Čech complex. Definition III.10. Let R be a Noetherian ring of prime characteristic p > 0. Let jM : M → R1/p ⊗R M denote the natural R-linear map u 7→ 1⊗u for u ∈ M . Let I be an ideal of R and fix i ≥ 0. The R-linear map HIi (jM ) : HIi (M ) → HIi (R1/p ⊗R M ) = i 1/p HIR ⊗R M ) has as its target an R1/p -module, and therefore induces an R1/p 1/p (R i linear map jI,M : R1/p ⊗R HIi (M ) → HIi 1/p (R1/p ⊗R M ), where we have identified i i 1/p HIR = (I [p] R)1/p and I 1/p of R1/p have the 1/p (−) = HI 1/p (−) since the ideals IR same radical. 36 i Define the RhF i-structure morphism θI,M of HIi (M ) as follows. (3.2) i θI,M :R 1/p ⊗R HIi (M ) i jI,M −−→ HIi 1/p (R1/p H i 1/p (θ) ⊗R M ) −−I−−−→ HIi 1/p (M 1/p ) = (HIi (M ))1/p The corresponding Frobenius action on HIi (M ) is called the natural action induced by M . i If R1/p is flat over R, the map jI,M is readily seen to be an isomorphism [BS12, Theorem 4.3.2]. If M is unit, HIi 1/p (θ) is an isomorphism. The following is now clear. Proposition III.11. [Lyu97, Example 1.2 (b)] Let R be a regular ring of prime characteristic p > 0 and let M be a unit RhF i-module. Then for any ideal I of R and any i ≥ 0, the natural action on HIi (M ) is unit. Let f = f1 , . . . , ft be a choice of generators for I, let C • denote the Čech complex • = C • ⊗R M . The homomorphisms jM : M → R1/p ⊗R M Č • (f ; R), and let CM • • and θ : R1/p ⊗R M → M 1/p induce maps of complexes6 JM : CM → R1/p ⊗R CM • • 1/p and Θ : R1/p ⊗R CM → (CM ) . The induced maps on the cohomology of these complexes, H i (JM ) and H i (Θ), are precisely the maps HIi (jM ) and HIi (θ) induced by the local cohomology functor HIi (−), and therefore, do not depend in any way on the choice of generators for I. We may therefore use these maps H i (JM ) and H i (Θ) to describe the structure morphism of HIi (M ) in terms of the Čech complex without worrying the result may change given a different choice of generators. We will make use of the following family of natural transformations both here and in the next chapter. Definition III.12. If f : R → S is a ring homomorphism and C • is an R-complex, then the natural (R-linear) map C • → S ⊗R C • induces H i (C • ) → H i (S ⊗R C • ), 6 We • )1/p . have used the identification M 1/p ⊗ C • = Č(f [p] ; M )1/p = Č(f ; M )1/p = (CM 37 which factors uniquely through the natural map H i (C • ) → S ⊗R H i (C • ) to an Slinear map S ⊗R H i (C • ) → H i (S ⊗R C • ). Call this map hif (C • ), and let hif denote the corresponding natural transformation hif (−) : S ⊗R H i (−) −→ H i (S ⊗R −) of functors KomR → ModS . If the homomorphism f : R → S is understood from context, we will write hiS/R (−) instead of hif (−). The latter, more precise, notation is reserved for ambiguous cases, such as when R = S has prime characteristic p and f is the Frobenius homomorphism. Note also that S is flat over R if and only if hiS/R (−) is an isomorphism of functors. Let f = f1 , · · · , ft be a sequence of elements of R, let C • == Č • (f ; R), and for M • • an R-module, let CM := C • ⊗R M . In this context, the map hiR1/p /R (CM ) : R1/p ⊗R • • i H i (CM ) → H i (R1/p ⊗R CM ) is precisely jI,M : R1/p ⊗R HIi (M ) → HIi 1/p (R1/p ⊗R M )1/p from diagram 3.2. In FR (−) notation, note that the complex FR (C • ) is canonically identified7 with • C • itself, and likewise, for any R-module M , FR (CM ) is canonically identified with • CF•R (M ) . We can therefore understand C • ⊗R − applied to θ as a map Θ : FR (CM )= • C • ⊗R FR (M ) → CM . The Čech construction of the structure morphism of HIi (M ) is as follows. (3.3) i • θI,M : FR (H i (CM )) • ) hiF (CM R • H i (FR (CM )) H i (Θ) • H i (CM ) • Since CM = C • ⊗R M , the definition is completely functorial in M , so that the map HIi (M ) → HIi (N ) induced by N → M with the structure morphisms above is readily seen to be RhF i-linear. We may therefore regard HIi (−) as a functor from 7 For any element g ∈ R, there is no difference between localization with respect to the multiplicative system {1, g, g 2 , g 3 , · · · } and the multiplicative system {1, g p , g 2p , g 3p }. 38 RhF i-modules to RhF i-modules. By applying C • ⊗R − to a short exact sequence 0 → N → M → Q → 0 of RhF i-modules, the resulting short exact sequence of complexes • 0 → CN• → CM → CQ• → 0 makes it clear that all connecting homomorphisms in the corresponding long exact sequence in local cohomology are also RhF i-linear. Proposition III.13. [Lyu97, Example 1.2 (b), (b’)8 ] Let R be a Noetherian ring of prime characteristic p > 0. Let I be an ideal of R and fix i ≥ 0. Given an RhF imodule M , if HIi (M ) is regarded as an RhF i-module via the structure morphism (3.2) induced from M , then the association M 7→ HIi (M ) is a functor from the category of RhF i-modules to itself. Moreover, for any short exact sequence of RhF i-modules 0 → N → M → Q → 0, the connecting homomorphisms in the long exact sequence induced by ΓI (−) are RhF i-linear. The Čech complex definition of the RhF i structure of HIi (M ) is given in terms of the structure morphism of the complex C • ⊗R M , and the structure morphism of C • ⊗R M is precisely the data of an RhF i-module structure on each term C i ⊗R M such that the differentials of the complex are RhF i-linear. In the case where M = R, each term of C • is a direct sum of localizations Rf equipped with their natural actions. Proposition III.14. [Bli01, Lemma 2.2.49 ] Let R be a Noetherian ring and let f = f1 , . . . , fc be a sequence of elements. The RhF i-module HfcR (R), equipped with its natural action, is unit and finitely generated. Proof. Let f = Qc i=1 fi . In terms of the complex C • = Č • (f ; R), Hfc (R) is the cokernel of the map C c−1 → C c where both C c−1 and C c are unit and finitely generated. By 8 Lyubeznik’s argument uses an injective resolution equipped with compatible Frobenius actions on every term, and the construction of this resolution requires that R is regular and M is unit. These hypotheses can be relaxed if the role of the injective resolution in his argument is replaced by the Čech complex, making use of [Wei94, Proposition • → C • → C • → 0. 1.3.4] on the short exact sequence of complexes 0 → CN M Q 9 We are only using the case M = R, and the preservation of the unit property and finite generation under homomorphic image, which as Blickle discusses on pp. 18, does not require R to be regular. 39 III.6, the cokernel of a map between unit RhF i-modules is unit10 . Since C c = Rf is finitely generated (e.g., by 1/f ), so is its RhF i-homomorphic image, Hfc (R). A much stronger statement can be made if R is regular. In this case, if M is a finitely generated unit RhF i-module, Lyubeznik shows that any localization Mg for g ∈ R is also unit and finitely generated [Lyu97, Proposition 2.9(b)]. Since Čech complex Č • (f ; M ) is a complex of finitely generated unit RhF i-modules, the kernels of the differentials are themselves unit and finitely generated by Proposition III.6 and Corollary III.8. The same can be said for any RhF i-homomorphic image of those kernels, and the result below follows at once. Theorem III.15 (Lyubeznik; Proposition 2.10 [Lyu97]). Let R be a regular ring of prime characteristic p > 0. Let M be a finitely generated unit RhF i-module, let I be an ideal of R, and fix i ≥ 0. Then HIi (M ), regarded as an RhF i-module via the structure induced from M , is finitely generated and unit over RhF i. The following is an immediate consequence of Theorems III.15 and III.9. Theorem III.16. Let R be a regular ring of prime characteristic p > 0 and let M be a finitely generated unit RhF i-module. For any ideal I of R and any i ≥ 0, the module HIi (M ) has finitely many associated primes. 10 The hypothesis of regularity is only necessary to ensure the unit property of kernels. CHAPTER IV The Hellus Isomorphism and Other Functorial Tools Let R be a Noetherian ring. To a given local cohomology functor HIi (−) with respect to some ideal I of R, we may associate a pair (i, h) consisting of the cohomological degree i and the height of the defining ideal h = ht(I). In this section we will show that those local cohomology functors for which the pair (i, h) satisfies i ≤ h + 1 are fully general in the sense of the following theorem (consider k = i − h + 1 if i > h + 1). Note that the height of the unit ideal is inf(∅) = +∞. Theorem (IV.4). Let R be a Noetherian ring, let I be an ideal of height h, and fix k ≥ 0. There is an ideal Ik ⊇ I such that ht(Ik ) ≥ h + k and such that the natural transformation HIik (−) → HIi (−) is an isomorphism (resp. a surjection) of functor for all i > h + k (resp. i = h + k). This statement is an result of the author, written up for publication in [Lew19]. Reduction to the case i ≤ h + 1 will drastically reduce the difficulty of certain key proofs in subsequent chapters. It is a generalization of an isomorphism theorem of Hellus that we restate below. Theorem IV.1. [Hel01, Theorem 3] Let R be a Cohen-Macaulay local ring, let I be an ideal of height h, and fix k ≥ 0. There is an ideal Ik ⊇ I such that ht(Ik ) ≥ h + k and such that the natural map HIh+k+1 (R) → HIh+k+1 (R) is an isomorphism. If k 40 41 HIh+k+1 (R) 6= 0, then Ik can be chosen such that ht(Ik ) = h + k. While it is interesting that the Cohen-Macaulay hypothesis can be eliminated, the Cohen-Macaulay case of this isomorphism remains particularly interesting. The modules HIi (R) such that (i, ht(I)) satisfies i ≤ ht(I) + 1 vanish unless i = ht(I) or i = ht(I) + 1. The former is fairly straightforward to deal with as far as finiteness questions are concerned, see e.g. Theorem II.18. Effectively, we need only consider the case i = ht(I) + 1 – a drastic simplification. After proving our generalization of Hellus’s isomorphism, we move on to a separate issue involving the compatibility (Theorem IV.9) of the natural transformations hiI;S/R (−) : S ⊗R HIi (−) → HIi (S ⊗R −) with the Frobenius functors of R and S. This leads to the following statement Theorem (IV.10). Let R → S be a map between two Noetherian rings of prime characteristic p > 0, and let M be an RhF i-module. Then the natural map hiI;S/R (M ) : S ⊗R HIi (M ) → HIi (S ⊗R M ) is a morphism of ShF i-modules. This will be particularly useful in Section 5.4 where R and S are regular and M is unit. These results of the author are also written up for publication in [Lew19]. 4.1 A generalized isomorphism of Hellus As this situation may arise in a number of proofs in this section, note that, by convention, the intersection of prime ideals of R indexed by the empty set is taken T to be i∈∅ Pi = R. Recall also that an ideal is said to have pure height h if all of its minimal primes have height exactly h. 42 We require a notion of parameters in a global ring to proceed, and the following lemma provides one suitable for use in our main proofs. Lemma IV.2. Let R be a Noetherian ring, let I be a proper ideal of height h ≥ 0, and let J ⊆ I be an ideal of height j ≥ 0. (a) If an ideal of the form (x1 , · · · , xh )R has height h, then it has pure height h. (b) Any sequence x1 , · · · , xj ∈ J generating an ideal of height j (including the empty sequence if j = 0) can be extended to a sequence x1 , · · · , xh ∈ I generating an ideal of height h. (c) There is a sequence x1 , · · · , xh ∈ I such that (x1 , · · · , xh )R has (necessarily pure) height h. Proof. (a) Every minimal prime of a height h ideal has height at least h by definition, and every minimal prime of an h-generated ideal has height at most h by Krull’s height theorem [Mat89, Theorem 13.5]. (b) If j = h, there is nothing to do, so assume j < h. By induction, it is enough to show that we can extend the sequence by one element. Since j < h, I is not contained in any minimal prime of (x1 , · · · , xj )R (all of which have height j), and so we may choose x ∈ I avoiding all such primes. A height j prime containing (x1 , · · · , xj )R therefore cannot also contain xR. Thus, the minimal primes of (x, x1 , · · · , xj )R have height at least j + 1. By Krull’s height theorem, they also have height at most j + 1. (c) This follows at once from (b) by taking J = (0). Our method of enlarging an ideal to obtain the desired functorial isomorphism will proceed inductively replacing I with I + yR for some suitable choice of y ∈ R. The lemma below describes how we will choose the element y. Note that some of our applications require the resulting isomorphism to have special properties with 43 respect to a sequence of j elements generating an ideal J of height j contained in I, and this needs to be taken into consideration in our choice of the element y. Lemma IV.3. Let R be a Noetherian ring, let I be a proper ideal of height h, and let J ⊆ I be an ideal of height j ≤ h generated by j elements. There is an element y ∈ R that satisfies the following properties. (i) araR (yR ∩ I) = h (ii) araR/J (y(R/J) ∩ (I/J)) = h − j (iii) ht(yR + I) ≥ h + 1 Proof. Write J = (x1 , · · · , xj )R, and extend this sequence to x1 , · · · , xh ∈ I generating an ideal of height h. I is contained in at least one minimal prime of (x1 , · · · , xh )R. Let P1 , · · · , Pt be the minimal primes of (x1 , · · · , xh )R containing I, and Q1 , · · · , Qs be the minimal primes of (x1 , · · · , xh )R that do not. We may have s = 0. Since these primes are pairwise incomparable, there exist elements y ∈ Q1 ∩ · · · ∩ Qs that avoid the union P1 ∪ · · · ∪ Pt (if s = 0, we can take y = 1). For any such y, it holds that yR ∩ I ⊆ P1 ∩ · · · ∩ Pt ∩ Q1 ∩ · · · ∩ Qs = and thus yR ∩ I ⊆ yR ∩ p (x1 , · · · , xh )R p (x1 , · · · , xh )R. Since (x1 , · · · , xh ) ⊆ I, we see that yR ∩ (x1 , · · · , xh )R ⊆ yR ∩ I ⊆ yR ∩ p (x1 , · · · , xh )R It follows at once that that p yR ∩ I = p p yR ∩ (x1 , · · · , xh )R = (yx1 , · · · , yxh )R producing an upper bound on arithmetic rank: araR (yR ∩ I) ≤ h. To obtain the lower bound araR (yR ∩ I) ≥ h, suppose that for some t < h we had a sequence of elements z1 , · · · , zt generating an ideal with the same radical as yR ∩ I. Then 44 p p (z1 , · · · , zt )R = (yx1 , · · · , yxh )R, and upon localizing at P1 , we would have p p (z1 , · · · , zt )RP1 = (x1 , · · · , xh )RP1 since y is a unit in RP1 . It would follow that p (x1 , · · · , xh )RP1 has height no more than t, which is a contradiction. p Since yR/J ∩ I/J ⊆ (xj+1 · · · , xh )R/J, an identical argument to the above shows that q p yR/J ∩ I/J = (yxj+1 , · · · , yxh )R/J so araR/J (yR/J ∩ I/J) ≤ h − j. We have established (i) and (ii). Concerning (iii), note that that all primes containing yR + I also contain (x1 , · · · , xh )R, and thus, to show that ht(yR + I) ≥ h + 1, it is enough to show that none of the height h primes containing (x1 , · · · , xh )R appear in V (yR + I). But this is clear, since {P ⊇ (x1 , · · · , xh )R | ht(P ) = h} = {P1 , · · · , Pt , Q1 , · · · , Qs }. None of the primes Pi contain yR, and none of the primes Qj contain I. Theorem IV.4. Let R be a Noetherian ring, let I be an ideal of height h, and let J ⊆ I be an ideal of height j ≥ 0 generated by j elements. For any k ≥ 0, there is an ideal Ik,J ⊇ I such that • ht(Ik,J ) ≥ ht(I) + k • The natural transformation HIik,J (−) → HIi (−) is an isomorphism on R-modules for all i > h + k, and an isomorphism on R/J-modules for all i > h − j + k. If i = h + k (resp. i = h − j + k) this natural transformation is a surjection on R-modules (resp. R/J-modules). Proof. If k = 0, choose I0,J = I. Fix k ≥ 1, and suppose that we’ve chosen the ideal Ik−1,J by induction. We must choose Ik,J . For brevity, we will suppress J from our notation, and write Ik−1 and Ik for Ik−1,J and Ik,J , respectively. 45 If ht(Ik−1 ) > h + k − 1 we can simply pick Ik = Ik−1 , so assume that ht(Ik−1 ) = h + k − 1. By Lemma IV.3 there is an element y ∈ R such that ht(yR + Ik−1 ) ≥ (h + k − 1) + 1, with araR (yR ∩ Ik−1 ) ≤ h + k − 1 and araR/J (y(R/J) ∩ Ik−1 /J) ≤ (h + k − 1) − j Consider the Mayer-Vietoris sequence on the intersection yR ∩ Ik−1 . We use (−) in our notation to mean that the sequence is exact when − is replaced by any R-module M , and that all maps in the sequence are given by natural transformations. i HyR+I (−) k−1 ··· i−1 HyR∩I (−) k−1 i HyR (−) ⊕ HIik−1 (−) i HyR∩I (−) k−1 i−1 Let i > h + k. Since i − 1 > araR (yR ∩ Ik−1 ), we get vanishing HyR∩I (−) = k−1 i i HyR∩I (−) = 0. Since i ≥ h + k + 1 ≥ 2, we also have HyR (−) = 0, and therefore k−1 ∼ i obtain a natural isomorphism HyR+I (−) − → HIik−1 (−). Notice that if i = h + k, k−1 i then we still have HyR∩I (−) = 0, so k−1 i i HyR+I (−) → HyR (−) ⊕ HIik−1 (−) → 0 k−1 i is exact, implying that the component map HyR+I (−) → HIik (−) is surjective. k−1 Working with R/J-modules, an identical argument using the fact that araR/J (y(R/J) ∩ Ik−1 /J) ≤ (h + k − 1) − j shows that ∼ i Hy(R/J)+I (−) − → HIik−1 /J (−) k−1 /J when i > h + k − j and i Hy(R/J)+I (−) ։ HIik−1 /J (−) k−1 /J 46 when i = h + k − j. Finally, ht(yR + Ik−1 ) ≥ h + k, so we may in fact choose Ik = yR + Ik−1 , which completes the induction. Corollary IV.5. Let R be a Noetherian ring and let I ⊆ R be any ideal. Fix i ≥ 0. There is an ideal I ′ ⊇ I (resp. I ′′ ⊇ I) such that • ht(I ′ ) ≥ i − 1 (resp. ht(I ′′ ) ≥ i) ∼ → HIi (−) (resp. HIi ′′ (−) ։ HIi (−)) • HIi ′ (−) − Proof. Let h = ht(I). If h ≥ i − 1 (resp. h ≥ i) simply choose I ′ = I (resp. I ′′ = I). Otherwise, h < i − 1 (resp. h < i). Apply Theorem IV.4 in the case k = i − 1 − h (resp. k ′ = i − h) to obtain an ideal I ′ ⊇ I (resp. I ′′ ⊇ I) satisfying ∼ → HIi (−), since ht(I ′ ) ≥ h + k = i − 1 (resp. ht(I ′′ ) ≥ h + k ′ = i) and HIi ′ (−) − i > h + k (resp. HIi ′′ (−) ։ HIi (−), since i = h + k ′ ). An immediate application of this theorem is to generalize a corollary of Hellus [Hel01, Corollary 2]. This generalization provides a new proof of a result of Marley [Mar01, Proposition 2.3], namely, for any Noetherian ring R, any ideal I ⊆ R, and any R-module M , {P ∈ Supp HIi (M ) | ht(P ) = i} is a finite set. Since our result comes from a surjection of functors, we will describe it in terms of the “support” of HIi (−). By the support of a functor F : ModR → ModR , we mean the set of primes P ∈ Spec(R) such that F (−)P is not the zero functor. That is to say, Supp(F ) := {P ∈ Spec(R) | ∃M ∈ ModR such that F (M )P 6= 0} For example, if I ⊆ R is an ideal and i ≥ 0, then Supp HIi (−) ⊆ V (I). It is clear (see, e.g., Theorem II.13 or Theorem II.14) that this inclusion need not be sharp. For any i > ht(I), the following Corollary shows us how to find a closed set containing Supp HIi (−) strictly smaller than V (I). 47 Corollary IV.6. (cf. Marley [Mar01, Proposition 2.3]) Let R be a Noetherian ring and I be an ideal. Then for all i ≥ 0, there is an ideal I ′′ ⊇ I with ht(I ′′ ) ≥ i such that Supp HIi (−) ⊆ V (I ′′ ). In particular, for any R-module M , the set {P ∈ Supp HIi (M ) | ht(P ) = i} is a subset of MinR (R/I ′′ ), and is therefore finite. If R is semilocal and i ≥ dim(R)− 1, then Supp HIi (M ) is a finite set. Proof. Fix i ≥ 0 and write h = ht(I). If i < h, then because Supp HIi (−) ⊆ V (I), we already have ht(P ) ≥ h > i for all P ∈ Supp HIi (−) and there is nothing to prove. So assume that i ≥ h. By Corollary IV.5, there is an ideal I ′′ ⊇ I such that ht(I ′′ ) ≥ i and HIi ′′ (−) ։ HIi (−). In particular, for any R-module M , HIi (M ) is I ′′ -torsion, and thus Supp HIi (M ) ⊆ V (I ′′ ). All primes in V (I ′′ ) have height at least i. Any primes of height exactly i must be among the minimal primes of I ′′ , of which there are only finitely many. 4.2 Compatibility and simultaneous base change Recall the natural transformations hif (−) of Definition III.12, also denoted hiS/R (−), associated with a ring homomorphism f : R → S. In this section, we will prove a number of compatibility properties for these transformations prior to proving our main result on the ShF i-linearity of certain base changed maps. Proposition IV.7. Let R → S → T be ring homomorphisms. The following diagram of functors KomR → ModT commutes1 . 1 The equalities shown in this diagram come from identifying the functor T ⊗S (S ⊗R −) with T ⊗R − 48 T ⊗R H i (−) T ⊗S (S ⊗R H i (−)) idT ⊗S hiS/R (−) T ⊗S H i (S ⊗R −) hiT /S (S⊗R −) H i (T ⊗S (S ⊗R −)) hiT /R (−) H i (T ⊗R −) Proposition IV.8. Fix a commutative square of ring homomorphisms R S R′ S′ There is a commutative square of functors KomR → ModS ′ idS ′ ⊗S hiS/R (−) S ′ ⊗S (S ⊗R H i (−)) S ′ ⊗R′ H i (R′ ⊗R −) hiS ′ /R′ (R′ ⊗R −) S ′ ⊗S H i (S ⊗R −) H i (S ′ ⊗R′ (R′ ⊗R −)) Proof. Apply Proposition IV.7 to R → S → S ′ in order to see that the upper right corner of the below diagram commutes, and then to R → R′ → S ′ to see that the lower left corner commutes as well: S ′ ⊗S (S ⊗R H i (−)) idS ′ ⊗S hiS/R (−) S ′ ⊗S H i (S ⊗R −) hiS ′ /S (S⊗R −) S ′ ⊗R H i (−) H i (S ′ ⊗S (S ⊗R −)) hiS ′ /R (−) S ′ ⊗R′ (R′ ⊗R H i (−)) H i (S ′ ⊗R −) idS ′ ⊗R′ hiR′ /R (−) S ′ ⊗R′ H i (R′ ⊗R −) hiS ′ /R′ (R′ ⊗R −) H i (S ′ ⊗R′ (R′ ⊗R −)) 49 The main application of the above compatibility statement is when R → S is a map between two rings of prime characteristic p > 0, and R → R′ , S → S ′ are the Frobenius homomorphisms of R and S, respectively. If M is an RhF i-module with structure morphism θ : FR (M ) → M , note that S ⊗R M can be regarded as an ShF i-module via the structure isomorphism idS ⊗ θ : S ⊗R FR (M ) → S ⊗R M , where S ⊗R FR (M ) and FS (S ⊗R M ) are identified in the canonical way2 . The relevant version of Proposition IV.8 in this setting is as follows. Corollary IV.9. Let R → S be a homomorphism between two rings of prime characteristic p > 0. The following diagram commutes, FS (S ⊗R H i (−)) e S ⊗R H i (FR (−)) FS (hiS/R (−)) hiS/R (FR (−)) FS (H i (S ⊗R −)) e H i (S ⊗R FR (−)) For a Noetherian ring R of prime characteristic p > 0, M an RhF i-module, I an ideal of R, and i ≥ 0, we recall below the construction of the structure morphism of HIi (M ) as an RhF i-module in terms of the structure morphism of M in terms of the Čech complex. Let f = f1 , · · · , ft be a sequence of elements of R, let C • = Č • (f ; R), • and let CM = C • ⊗R M . Let θ : FR (M ) → M be the structure morphism of M • • and let Θ : FR (CM ) → CM be the corresponding map of complexes. From diagram (3.3), the structure morphism of the natural action on HIi (M ) induced by M is the composition shown below. (4.1) 2 Using • FR (H i (CM )) • ) hiF (CM R • H i (FR (CM )) H i (Θ) • H i (CM ) different notation, this is simply identifying S 1/p ⊗R1/p (R1/p ⊗R M ) with S 1/p ⊗S (S ⊗R M ) using the equality of the composite maps R → S → S 1/p and R → R1/p → S 1/p 50 Theorem IV.10. Let R → S be a homomorphism between two Noetherian rings of prime characteristic p > 0, fix an ideal I ⊆ R and an index i ≥ 0, and let M be an RhF i-module with structure morphism θ : FR (M ) → M . The natural map S ⊗R HIi (M ) → HIi (S ⊗R M ) is a morphism of ShF i-modules. • Proof. Let CM = Č • (f ; M ) be the Čech complex on M associated with a sequence of elements f = f1 , · · · , ft generating I. It is enough to show that the diagram • FS (S ⊗R H i (CM )) • S ⊗R H i (CM ) • )) FS (hiS/R (CM • ) hiS/R (CM • FS (H i (S ⊗R CM )) • H i (S ⊗R CM ) commutes, where the vertical arrows are the structure morphisms of S ⊗R HIi (M ) • • and HIi (S ⊗R M ) as ShF i-modules, respectively. Let Θ : F(CM ) → CM denote the morphism of complexes induced by θ. Using the decomposition (3.3) of the structure morphism of HIi (M ), the stated result is equivalent to showing that the following diagram commutes. FS (S ⊗R H i • (CM )) • )) FS (hiS/R (CM • FS (H i (S ⊗R CM )) • ) hiF (S⊗R CM S S ⊗R FR (H i • (CM )) i • H (FS (S ⊗R CM )) • ) idS ⊗hiF (CM R • S ⊗R H i (FR (CM )) • )) hiS/R (FR (CM idS ⊗H i (Θ) • ) S ⊗R H i (CM • H i (S ⊗R FR (CM )) H i (idS ⊗Θ) • ) hiS/R (CM • ) H i (S ⊗R CM The commutativity of the rectangle of maps in the top three rows is precisely the • content of Corollary IV.9 applied to the complex CM . The square of maps in the bottom two rows is induced from the diagram that results from applying H i (−) to 51 • FR (CM ) nat (idS ⊗Θ) Θ • CM • S ⊗R FR (CM ) nat • S ⊗ R CM • • Recall that CM = C • ⊗R M and FR (CM ) = C • ⊗R FR (M ), where C • = Č • (f ; R), so that the above diagram is C • ⊗R − applied to the diagram below, which obviously commutes. FR (M ) nat (idS ⊗θ) θ M S ⊗R FR (M ) nat S ⊗R M CHAPTER V Parameter Ideals Following Hochster and Núñez-Betancourt A complete intersection ring is a Noetherian ring S such that, for all prime ideals cP of the local ring (SP , P SP ) is the quotient of a regular local P of S, the completion S ring by an ideal generated by a regular sequence. Our interest in this chapter and those that follow is in those complete intersection rings that are globally presentated as homomorphic images of a regular rings. Namely, given a regular ring R, we are interested in studying the local cohomology of complete intersection rings of the form S = R/f R where f = f1 , . . . , fc a regular sequence in R. The length c of this regular sequence is the codimension of S. If c = 1, we refer to S as a hypersurface ring. When f is a regular sequence, we refer to the ideal f R generated by f as a parameter ideal regardless of whether R is local. We will further restrict our focus to the setting in which the regular ring R has the property that Ass HIi (R) is finite for all i ≥ 0 and all ideals I, which we refer to as LC-finiteness for shorthand (Definition II.16). This property holds, for example, when R has prime characteristic p > 0 [Lyu97, HS93], or if R is a smooth algebra over a field of characteristic 0 [Lyu93] or over the integers [BBL+ 14]. Let R be an LC-finite regular ring, let f = f1 , . . . , fc be a regular sequence of R, and let S = R/f R. That is to say, there is a short exact sequence 0 → fR → R → S → 0 52 53 that presents S as the homomorphic image of the LC-finite module R. Let I be an ideal of R containing f (corresponding to an arbitrary ideal of S) and fix i ≥ 0. There is an exact sequence (5.1) · · · → HIi (f R) → HIi (R) → HIi (S) → HIi+1 (f R) → · · · If we would like to investigate the question of closed support for HIi (S), we are naturally lead to ask the following two questions. 1. Does the cokernel of the map HIi (f R) → HIi (R) have closed support? 2. Does the kernel of the map HIi+1 (f R) → HIi+1 (R) have closed support? A key insight of Hochster and Núñez-Betancourt [HNB17] is that in prime characteristic p > 0, we can give an affirmative answer to Question 1 in the following manner. When R and S are equipped with their natural Frobenius actions and f R is equipped with the Frobenius action restricted from that of R, the short exact 0 → f R → R → S → 0 can be understood as a short exact sequence of RhF imodules. By Proposition III.13, all morphisms in the long exact sequence (5.1) are RhF i-linear. Since R is unit and finitely generated, Theorem III.15 shows that HIi (R) remains unit and finitely generated. Thus, the cokernel of HIi (f R) → HIi (R) is an RhF i-linear homomorphic image of a finitely generated RhF i-module, and hence, remains finitely generated. By Theorem III.1, that homomorphic image has closed support. To show that the kernel of HIi+1 (f R) → HIi+1 (R) has closed support, it would clearly suffice to show that HIi+1 (f R) has finitely many associated primes. We are therefore lead to the following theorem. Note that while our interest is primarily in the case where J is generated by a regular sequence, this hypothesis was not required in the preceding argument – nor was the hypothesis that I ⊇ J. 54 Theorem V.1 (Hochster, Núñez-Betancourt; Theorem 4.12). Let R be a regular ring of prime characteristic p > 0, let J be an ideal of R, and let S = R/J. Let I be an ideal of R and fix i ≥ 0. If Ass HIi+1 (J) is finite, then Supp HIi (S) is closed. If J is the principal ideal generated by some nonzerodivisor f ∈ R – which is to say, S = R/J is the hypersurface ring R/f R – then J is isomorphic to R as an abstract R-module, and in particular, is LC-finite. Hochster and Núñez-Betancourt thereby obtain the following result on hypersurfaces as essentially a corollary of Theorem V.1. Theorem V.2 (Hochster, Núñez-Betancourt; Corollary 4.13). Let R be a regular ring of prime characteristic p > 0, let f ∈ R be a nonzerodivisor, and let S = R/f R. Let I be an ideal of R and fix i ≥ 0. The support of HIi (S) is Zariski closed in Spec(S). It remains an open question whether their theorem on the support of the local cohomology of positive characteristic hypersurface rings generalizes to complete intersection rings of arbitrary codimension, or even whether it generalizes to hypersurface rings of characteristic 0. At least in the positive characteristic setting, by Theorem V.1, it would suffice to show that for any ideal J generated by a regular sequence f of R, the module HIi (J) has finitely many associated primes. Despite only having immediate applications in prime characteristic p > 0, one may pose the following question for LC-finite regular rings in any characteristic. Question V.3. Let R be an LC-finite regular ring and f = f1 , . . . , fc be a regular sequence of R. Is the R-module f R LC-finite? In other words, for I an ideal of R, and i ≥ 0, must Ass HIi (f R) be a finite set? We prove the following positive result in cohomological degree i = 2. Our main 55 case of interest for this theorem is when R is regular and J is generated by a regular sequence, but these hypotheses are not necessary for the theorem. Theorem V.4 (V.11). Let R be a locally almost factorial (Definition V.6) Noetherian normal ring, and I and J be ideals of R. The set Ass HI2 (J) is finite. In cohomological degree i ≥ 3, we show that Question V.3 has a negative answer at the level of generality in which it’s stated. In Theorem V.5 we present an example of the author, written up for publication in [Lew19], showing that Ass HI3 (f R) can be an infinite set when f = f, g is a regular sequence of length 2. The counterexample presented in Section 5.1 crucially requires that HI2 (R/f ) has an infinite set of associated primes – in fact, R/f in this example is Katzman’s hypersurface ring [Kat02]. It is natural to ask whether one can avoid choosing a sequence f with HIi−1 (R/f ) already having an infinite set of associated primes, but this may not be possible to do. In Section 5.3 we prove the following. Theorem (V.16). Let R be an LC-finite regular ring, let J ⊆ R be an ideal generated by a regular sequence of length c ≥ 2, and let S = R/J. For an ideal I ⊇ J, (ii) If the irreducible components of Spec(S) are disjoint (e.g. S is a domain), then Ass HI3 (J) is finite if and only if Ass HI2 (S) is finite. (iii) If S is normal and locally almost factorial (e.g. S is a UFD), then Ass HI4 (J) is finite if and only if Ass HI3 (S) is finite. Since we are only interested in Question V.3 in the case where Ass HIi (S) is infinite, the above theorem presents a significant challenge to the prospect of applying Theorem V.1 to the closed support problem in the codimension c ≥ 2 setting. We will address this issue further in the next chapter. At the end of the present chapter, in Section 5.4, we show that in the positive 56 characteristic setting, it is possible to impose sufficiently restrictive hypotheses on R/f so as to recover a positive answer to Question V.3. We actually prove a somewhat more general statement. Theorem (V.18). Let R be a regular ring of prime characteristic p > 0, let M be a finitely generated unit RhF i-module M , and let f = f1 , . . . , fc be a regular sequence such that R/f is regular. The module f M is LC-finite. 5.1 An example in which Ass HI3 ((f, g)R) is an infinite set The following example demonstrates that Question V.3 has a negative answer. Theorem V.5. Let K be a field, let R = K[u, v, w, x, y, z, t], and let f = wv 2 x2 − (w + z)vxuy + zu2 y 2 be the defining equation of Katzman’s hypersurface ring [Kat02]. 3 ((t, f )R) is infinite. The set Ass H(t,f,x,y) Proof. Let f = t, f , a codimension 2 regular sequence in R, and let A = K[u, v, w, x, y, z]. Note that R/f R = A/f A. Let I = (t, f, x, y)R, and observe that HIi (R/f R) = i i (A/f ) for all i. Since depthI (R) = 3 (the sequence t, f, x ∈ I H(x,y) (R/f R) = H(x,y) is R-regular), the long exact sequence from applying ΓI (−) to 0 → f R → R → R/f R → 0 begins with 0 0 0 0 HI1 (f R) 0 1 H(x,y) (A/f A) HI2 (f R) 0 2 H(x,y) (A/f A) HI3 (f R) HI3 (R) 0 HI4 (f R) HI4 (R) 0 2 From this, we see that H(x,y) (A/f A) embeds into HI3 (f R). By [Kat02, Theorem 1.2], 2 Ass H(x,y) (A/f A) is infinite, so Ass HI3 (f R) is infinite as well. 57 Katzman’s hypersurface could be replaced with any globally presented complete intersection ring S known to have an ideal I and index i ≥ 0 such that HIi (S) has infinitely many associated primes. Let S = A/gA where A is a regular ring and g = g1 , . . . , gt is a regular sequence. Let R = A[z1 , · · · , zn ] for n ≫ 0, and let f = z1 , · · · , zn . We have S ≃ R/(f , g)R. Let I ′ = (f , g)R + IR and observe that HIi (S) ≃ HIi ′ (S) for all i ≥ 0. By choosing n large enough, we can ensure that depthI ′ (R) > dim(S) + 1. Using the long exact sequence from applying ΓI ′ (−) i to 0 → (f , g)R → R → S → 0, it follows at once that HIi+1 ′ ((f , g)R) ≃ HI (S). In sufficiently large cohomological degrees, Ass HIj ((f , g)R) is isomorphic to HIj′ (R), and in degrees 1 ≤ j ≤ dim(S) + 1, the local cohomology of (f , g)R is identical to that of S in one degree lower. 5.2 The Finiteness of Ass HI2 (J) While our main interest for the results in this section is the case in which R is regular and LC-finite, we do not require the full strength of those hypotheses here. We need only assume that R is normal and satisfies the following condition. Definition V.6. A normal domain R is called almost factorial if the class group of R is torsion. A normal ring R is called locally almost factorial if RP is almost factorial for all P ∈ Spec(R). For example, if RP is a UFD for all P ∈ Spec(R), then R is locally almost factorial. A regular ring is therefore locally almost factorial. Hellus shows that an almost factorial Cohen-Macaulay local ring of dimension at most four is LC-finite [Hel01, Theorem 5]. Our usage of the almost factorial hypothesis is motivated by its use in [Hel01], although our motivating setting is ultimately the regular case. Our goal is to show that Ass HI2 (J) is finite for any ideals I and J of R. The 58 results of this section no hypotheses on the ideal J of R – we even permit the case where J is the unit ideal. When R is a domain, the lemma below shows that the main case is depthI (R) = 1. Lemma V.7. Let R be a Noetherian domain, and let J ⊆ R be an ideal. If I ⊆ R is an ideal such that depthI (R) 6= 1, then Ass HI2 (J) is finite. Proof. If I = (0) or I = R, there is nothing to do, so we assume that I is a nonzero proper ideal. Since R is a domain, this implies that both J and R are I-torsionfree, giving depthI (R) > 0 and by hypothesis depthI (R) 6= 1, so we have depthI (R) ≥ 2. The following sequence is exact. 0 0 0 ΓI (R/J) HI1 (J) 0 HI1 (R/J) HI2 (J) HI2 (R) HI2 (R/J) Note that HI1 (J) ≃ ΓI (R/J) is finitely generated, meaning that HI2 (J) is either finitely generated or the first non-finitely-generated local cohomology module of J on I, and the stated result follows at once from Brodmann and Lashgari Faghani [BLF00, Theorem 2.2]. To deal with the case depthI (R) = 1, our goal is to locally decompose I (up to radicals) as an ideal of the form yR ∩ I0 for y ∈ R a nonzerodivisor and I0 an ideal such that depthI0 (R) ≥ 2. Such a decomposition would enable us to rewrite HI2 (J) as HI20 (J)y using the functorial isomorphism of Corollary II.10. √ The first step in decomposing I is to express I as the intersection of a depth ≥ 2 component with a component of pure height 1. 59 Lemma V.8. Let R be a Noetherian normal ring, and I ⊆ R be an ideal such that √ depthI (R) = 1. Then I = L∩I0 for some ideal L given by the intersection of height one primes, and some ideal I0 ⊆ R with depthI0 (R) ≥ 2. Proof. First note that for an ideal a in a normal ring R, deptha (R) = 1 if and only if ht(a) = 1. Indeed, if ht(a) = 1, then certainly deptha (R) ≤ 1. R has no embedded primes, so if a is not contained in any minimal prime of R, a contains a nonzerodivisor, and thus deptha (R) ≥ 1. If on the other hand, we assume deptha (R) = 1, then clearly ht(a) ≥ 1. Take x ∈ a a nonzerodivisor. Since deptha (R/xR) = 0, a is contained in an associated prime of xR, all of which have height 1, giving ht(a) ≤ 1. Since ht(I) = 1, the radical of I can be written as L ∩ I0 where L has pure height one and I0 is an intersection of primes of height ≥ 2. Since ht(I0 ) ≥ 2, it must be the case that depthI0 (R) ≥ 2. To proceed, we require the hypothesis of “almost factoriality.” The main property we require is that every height 1 prime of an almost factorial ring is principal up to taking radicals. An ideal of pure height 1 can be expressed up to radicals as the product of height 1 primes, so in an almost factorial ring, any pure height 1 ideal is principal up to radicals. In a locally almost factorial ring, we can cover Spec(R) with finitely many charts in which this is the case. To show that 2 Ass HI2 (J) is finite, it would certainly suffice to show that Ass HIR (JRf ) is finite on f each chart Spec(Rf ) of a finite cover of Spec(R). Lemma V.9. Let R be a locally almost factorial Noetherian normal ring, and L be an ideal of pure height 1. There is a finite cover of Spec(R) by open charts Spec(Rf1 ), · · · , Spec(Rft ) such that for each i, the expanded ideal LRfi has the same radical as a principal ideal. 60 Proof. We do no harm in replacing L with √ L, so assume L is radical. Consider a √ single point P ∈ Spec(R). Since RP is almost factorial, we can write LRP = yRP for some y ∈ RP . Up to multiplying by units of RP , we may assume that y is an element of R. Since y ∈ LRP ∩ R, there is some u ∈ R − P such that uy ∈ L. Also, since R is Noetherian, there is some n > 0 such that Ln RP ⊆ yRP , hence Ln ⊆ yRP ∩ R, and there is some v ∈ R − P such that vLn ⊆ yR. If f = uv, then p we see that y ∈ LRf and Ln ⊆ yRf , giving LRf = yRf . Our choice of f depends on P . Varying over all P ∈ Spec(R), we obtain a collec- tion of open charts {Spec(RfP )}P ∈Spec(R) which cover Spec(R) such that (the expansion of) L is principal up to radicals on each chart. Since Spec(R) is quasicompact, finitely many of these charts cover the whole space. Corollary V.10. Let R be a locally almost factorial Noetherian normal ring, and I ⊆ R be an ideal such that depthI (R) = 1. Then there is an ideal I0 ⊆ R with depthI0 (R) ≥ 2, and a finite cover of Spec(R) by open charts Spec(Rf1 ), · · · , Spec(Rft ) p p such that for each i, IRfi = yi Rfi ∩ I0 for some yi ∈ R. The main result of this section now follows. Theorem V.11. Let R be a locally almost factorial Noetherian normal ring, and I, J be ideals of R. The set Ass HI2 (J) is finite. Proof. R is a product of normal domains R1 × · · · × Rk , and J is a product of ideals 2 (Ji ) is finite for all i, so J1 × · · · × Jk with Ji ⊆ Ri . It is enough to show that Ass HIR i assume that R is a domain. By Lemma V.7, we need only deal with the case in which 2 depthI (R) = 1. We will show that Ass HIR (Jf ) is finite for each chart Spec(Rf ) in f a finite cover of Spec(R). By Corollary V.10, working with one chart at a time, and replacing R by Rf and I by an ideal with the same radical, we may assume that I has 61 the form I = yR ∩ I0 where depthI0 (R) ≥ 2. By Corollary II.10, this decomposition gives HI2 (J) ≃ HI20 (J)y . It is therefore enough to show that Ass HI20 (J) is finite. But depthI0 (R) ≥ 2, so this follows from Lemma V.7. 5.3 Finiteness of Ass HIi (J) vs finiteness of Ass HIi−1 (R/J) In this section, we concern ourselves with the following question. Question V.12. Let R be an LC-finite regular ring, f = f1 , . . . , fc be a regular sequence, and I be an ideal of R containing f . Does the finiteness of Ass HIi−1 (R/f R) imply the finiteness of Ass HIi (f R)? If c = 1, then f R ≃ R as an R-module, and thus Ass HIi (J) is finite by hypothesis, and the question has a trivially positive answer. We therefore restrict our attention to the case c ≥ 2. We think of i as being fixed with I varying. The case i = 2 has a positive answer, since Ass HI1 (R/f R) is finite – as is true of HI1 (M ) for any finitely generated module M , due to Theorem II.17 – and Ass HI2 (f R) is finite by Theorem V.11. Our goal is to give a partial positive answer to this question when i = 3 and when i = 4. As i gets larger, our results require increasingly restrictive hypotheses on the ring R/f R. To begin, notice that we can very easily ignore ideals I where the depth of R on I is too large. Lemma V.13. Let R be a Noetherian ring, let I and J be ideals of R, and let S = R/J. Fix i ≥ 1 and assume Ass HIi (R) is finite. If depthI (R) > i − 1, then Ass HIi (J) is finite if and only if Ass HIi−1 (R/J) is finite. Proof. There is a short exact sequence 0 → HIi−1 (R/J) → HIi (J) → N → 0 62 where N ⊆ HIi (R), so Ass N is finite. We may therefore restrict our focus to the case where depthI (R) ≤ i − 1. Using the isomorphism of Theorem IV.4, we may further restrict ourselves to the case depthI (R) = i − 1, as described in the following proposition. Proposition V.14. Let R be a Cohen-Macaulay ring, and let f = f1 , . . . , fc be a regular sequence of length j ≥ 1. Fix a nonnegative integer i, and let I be an ideal containing f such that depthI (R) ≤ i − 1. Then there is an ideal I ′ ⊇ I such that i−1 (R/f R). depthI ′ (R) ≥ i−1 and such that HIi ′ (f R) ≃ HIi (f R) and HIi−1 ′ (R/f R) ≃ HI Proof. Write h = ht(I). Applying Theorem IV.4 with k = i−1−h, we obtain an ideal I ′ ⊇ I such that depthI ′ (R) = ht(I ′ ) ≥ i − 1 such that the natural transformation HIℓ′ (−) → HIℓ (−) is an isomorphism on R-modules whenever ℓ > i − 1 and on R/f Rmodules whenever ℓ > i − 1 − c. In particular, we see that HIi ′ (−) → HIi (−) is i−1 (−) is an isomorphism on R/f Ran isomorphism on R-modules HIi−1 ′ (−) → HI modules. Assume that R is Cohen-Macaulay and f = f1 , . . . , fc is a regular sequence of codimension c ≥ 2. Fix i ≥ 0 and let I be an ideal of R containing f . Write a = depthI (R/f R) = depthI (R) − c. If a + c ≤ i − 1, then by Corollary V.14, we can replace I with a possibly larger ideal I ′ in order to assume that a + c ≥ i − 1, without affecting HIi (f R) and HIi−1 (R/f R). Lemma V.13 gives a positive answer to Question 3 if a + c > i − 1, so we may assume that a + c = i − 1. Note in particular that this allows us to ignore all values of i and c for which c > i − 1. Below is a table illustrating the relevant values of a to consider for various small values of i and c. 63 i=3 i=4 i=5 i=6 i=7 c=2 a=0 a=1 a=2 a=3 a=4 c=3 ∅ a=0 a=1 a=2 a=3 c=4 ∅ ∅ a=0 a=1 a=2 c=5 ∅ ∅ ∅ a=0 a=1 c=6 ∅ ∅ ∅ ∅ a=0 c=7 ∅ ∅ ∅ ∅ ∅ We will attack the cases a = 0 and a = 1 directly in order to deal with cohomological degrees i = 3 and i = 4. The next lemma is our main tool in doing so. Lemma V.15. Fix a ≥ 0. Let R be a Noetherian ring, let f = f1 , . . . , fc be a regular sequence of length c ≥ 2 − a, let I be an ideal containing f , and let S = R/f R. Suppose that IS can be decomposed (up to radicals) as yS ∩ I0 with depthI0 (S) > a. Suppose further that Ass HIc+a+1 (R) is finite. Then Ass HIc+a+1 (f R) is finite if and only if Ass HIc+a (S) is finite. Proof. By Corollary II.10, there is a natural isomorphism HIi0 (S)y ≃ HIi (S) for all i ≥ 2, so that in particular, HIc+a (S) is an Sy -module. The natural map ψ : HIc+a (R) → HIc+a (S) therefore factors through HIc+a (R) → Sy ⊗R HIc+a (R) to give an Sy -linear map Sy ⊗R HIc+a (R) → HIc+a (S). HIc+a (R) ψ HIc+a (S) Sy ⊗R HIc+a (R) We claim that ψ = 0, and for this it suffices to show that Sy ⊗R HIc+a (R) = 0. Consider the decomposition of I up to radicals as yS ∩ I0 in S. We can replace y by some lift mod f R to assume that y ∈ R, and since I0 is expanded from R, 64 we can write I0 = I0′ S for some ideal I0′ of R containing f . We therefore have I = (y, f )R ∩ I0′ in R (after possibly replacing I by an ideal with the same radical). c+a Note that depthI0′ (R) > c+a. We can write Sy ⊗R HIc+a (R) = Sy ⊗Ry HIR (Ry ) where y c+a (R)y . Since depthI0′ (R) > (Ry ) = HIc+a IRy = (y, f )Ry ∩ I0′ Ry = I0′ Ry , and thus HIR ′ y 0 c + a, we have (R) HIc+a ′ 0 = 0 and consequently, ψ = 0. We therefore have an exact sequence 0 → HIc+a (S) → HIc+a+1 (J) → HIc+a+1 (R). Since Ass HIc+a+1 (R) is finite, the claim follows at once. We can now prove the main result of this section. Theorem V.16. Let R be an LC-finite regular ring, let f = f1 , . . . , fc be a regular sequence of length c ≥ 2, and let S = R/f R. For any ideal I of R containing f , (i) Ass HIi (f R) and Ass HIi−1 (S) are always finite for i ≤ 2. (ii) If the irreducible components of Spec(S) are disjoint, then Ass HI3 (f R) is finite if and only if Ass HI2 (S) is finite. (iii) If S is normal and locally almost factorial, then Ass HI4 (f R) is finite if and only if Ass HI3 (S) is finite. Proof. Concerning (i), it holds that for any finitely generated R-module M , Ass HIi (M ) is finite whenever i ≤ 1 by Theorem II.17. The finiteness of Ass HI2 (f R) is the subject of Theorem V.11. For (ii), we may use Corollary V.14 to replace I with a possibly larger ideal in order to assume that depthI (R) ≥ 2. By Lemma V.13, (ii) is immediate if depthI (R) > 2, so assume depthI (R) = 2. Since f ⊆ I and c ≥ 2, it follows that c = 2 and depthI (S) = ht(IS) = 0. Let e1 , · · · , et ∈ S be a complete set of orthogonal idemp p potents. The minimal primes of S are (1 − e1 )S, · · · , (1 − et )S, and thus, every 65 pure height 0 ideal of S has arithmetic rank at most 1. Up to radicals, we can therefore write IS as yS ∩ I0 where ht(I0 ) = depthI0 (S) > 0. Since c ≥ 2 − 0, the claim follows from Lemma V.15 in the case where a = 0. For (iii), again using Corollary V.14 and Lemma V.13, we may assume that depthI (R) = depthI (S) + c = 3. Since c ≥ 2, this means depthI (S) ≤ 1. If c = 3, giving depthI (S) = 0, then we may argue as in the proof of (ii) (note that S is a product of domains). If c = 2, giving depthI (S) = 1, then by Corollary V.10 there is a finite cover of Spec(S) by charts Spec(Sf1 ), · · · , Spec(Sft ) such that for each i, we can write (up to radicals) ISfi = yi Si ∩I0,i with depthI0,i (Sfi ) > 1. Replace f1 , · · · , ft with lifts from R/f R to R in order to assume f1 , · · · , ft ∈ R. Lemma V.15 in the case a = 1 shows that for each i, Ass HI4 (f R)fi is finite if and only if Ass HI3 (S)fi is finite. The charts Spec(Rf1 ), · · · , Spec(Rft ) do not necessarily cover Spec(R), but they do cover the subset V (f R). Since I ⊇ f , Supp HIℓ (−) ⊆ V (I) ⊆ V (f R) for all ℓ, so showing that Ass HI4 (f R) is finite is equivalent to showing that Ass HI4 (f R)fi is finite for each i. The result we proved on each chart therefore implies Ass HI4 (f R) is finite if and only if Ass HI3 (S) is finite. Under the hypotheses of (iii), we can give the following partial answer to Question 3 for local rings of sufficiently small dimension. Corollary V.17. Let (R, m, K) be an LC-finite regular local ring of dimension at most 7, let f = f1 , . . . , fc be a regular sequence of length c ≥ 2 such that S = R/J is normal and almost factorial. Let I be any ideal of R containing f . Then for all i ≥ 1, Ass HIi (f R) is finite if and only if Ass HIi−1 (S) is finite. Proof. The case i ≤ 4 is the subject of Theorem V.16. We must have dim(S) ≤ 5 since c ≥ 2, so by Corollary IV.6, Supp HIi−1 (S) (and hence Ass HIi−1 (S)) is a finite 66 set if i − 1 ≥ 4. Likewise, for any homomorphic image HIi−1 (S) ։ N , the set Supp (N ) is finite. There is an exact sequence 0 → N → HIi (f R) → M → 0 where N is a homomorphic image of HIi−1 (S) and M is a submodule of HIi (R). If i ≥ 5, both Ass (N ) (a subset of Supp (N )) and Ass (M ) (a subset of Ass HIi (R)) are finite, so Ass HIi (f R) is finite as well. 5.4 Regular parameter ideals in characteristic p > 0 Let (R, m, K) be a regular local ring. Recall that a parameter ideal J ⊆ R is called regular if it is generated by an R-regular sequence whose images in m/m2 are linearly independent over K. Every ideal J such that R/J is regular has this form. If R is complete and contains a field, then by the Cohen Structure Theorem, all examples of regular parameter ideals are isomorphic to an example of the form R = K[[x1 , · · · , xm , z1 , · · · , zn ]] and J = (x1 , · · · , xm )R for some m, n ≥ 0. In this section, we will show that if R is a regular ring of prime characteristic p > 0 and J is an ideal such that R/J is regular, then for any ideal I ⊆ R and any i ≥ 0, the set Ass HIi (J) is finite. This result is a corollary of a stronger result, taking M = R in the theorem below. Theorem V.18. Let R be a regular ring of prime characteristic p > 0, let J ⊆ R be an ideal such that R/J is regular, and let M be a finitely generated unit RhF imodule. Then JM is LC-finite. That is, for any ideal I ⊆ R and any i ≥ 0, the module HIi (JM ) has finitely many associated primes. Proof. By Theorem III.15, HIi (M ) is unit and finitely generated over RhF i. By Proposition III.5, (R/J) ⊗R HIi (M ) is finitely generated and unit over R/J. Proposition III.5 also shows that (R/J) ⊗R M is a finitely generated unit (R/J)hF i-module, so because R/J is regular, Theorem III.15 shows that HIi ((R/J) ⊗R M ) is finitely 67 generated and unit over (R/J)hF i as well. The natural map HIi (M ) → HIi (M/JM ) factors through the map HIi (M ) → HIi (M )/JHIi (M ), and since R ։ R/J is surjective, the images of HIi (M ) and HIi (M )/JHIi (M ) inside HIi (M/JM ) are equal. Thus,

Coker HIi (M ) → HIi (M/JM ) = Coker HIi (M )/JHIi (M ) → HIi (M/JM ) By Theorem IV.10, HIi (M )/JHIi (M ) → HIi (M/JM ), which we may write as (R/J) ⊗R HIi (M ) → HIi ((R/J) ⊗R M ), is an (R/J)hF i-linear map. We have already recognized both the source and target as finitely generated and unit over (R/J)hF i. The cokernel of HIi (M )/JHIi (M ) → HIi (M/JM ) is therefore itself finitely generated – as is any quotient of a finitely generated ShF i-module by an ShF i-submodule – and is unit by Proposition III.6. By Theorem III.9, it must therefore have a finite set of associated primes. Regarding the claim about the associated primes of HIi (JM ), apply ΓI (−) to 0 → JM → M → M/JM → 0 to obtain the exact sequence ··· HIi−1 (M ) HIi−1 (M/JM ) HIi (JM ) HIi (M ) ··· We have a short exact sequence
0 → Coker HIi−1 (M ) → HIi−1 (M/JM ) → HIi (JM ) → N → 0 for some submodule N ⊆ HIi (M ), and the stated result now follows at once. CHAPTER VI Complete Intersection Rings as Annihilator Submodules Throughout this chapter and those that follow, we will sometimes write f as shorthand for the ideal generated by f , for example, in the notation Hfc (R) or R/f . In context, this should not cause any confusion. Let R be a regular ring of prime characteristic p > 0, let f = f1 , . . . , fc be a regular sequence in R, and let S = R/f denote the corresponding complete intersection ring of codimension c. For an ideal I of R and an index i ≥ 0, we return to the question of whether the support of HIi (S) is Zariski closed. We recall briefly the approach to the closed support problem discussed in the previous chapter. The corresponding short exact sequence 0 → f R → R → R/f → 0 induces a long exact sequence αi+1 α i → HIi (R) → HIi (S) → HIi+1 (f R) −−→ · · · · · · → HIi (f R) − and at least in positive characteristic setting, Hochster and Núñez-Betancourt (see Theorem V.1) prove closed support for the cokernel of αi , thereby reducing the main problem to a matter of controlling the associated primes of the kernel of αi+1 – if Ass HIi+1 (f R) is finite, Supp HIi (S) must be closed. The original question of whether 68 69 HIi (S) is closed – equivalently, whether Min HIi (S) is a finite set – is nontrivial only in the setting where Ass HIi (R/f ) is presumed to be infinite. In codimension c ≥ 2, Theorem V.16 shows that certain hypotheses on the ring R/f (e.g., that it is a domain) guarantee a negative answer to the following question even in cohomological degree less than 4. Question VI.1. In the notation established above, suppose that Ass HIi (R/f ) is infinite. Can the set Ass HIi+1 (f R) be finite? Because this question can have a negative answer, a proof of closed support for complete intersection rings of codimension 2 and higher is therefore unlikely to straightforwardly arise from attempting to control (submodules of) HIi+1 (f R). From the viewpoint of RhF i-modules, there are at least two difficulties that stand out. The first is that f R is not unit, so despite being finitely generated, we cannot expect finite generation over RhF i to be preserved after applying a local cohomology functor HIi+1 (−). The second difficulty is that, even if HIi+1 (f R) did happen to be finitely generated over RhF i, the property of finite generation does not generally pass to RhF i submodules1 . Suppose that we were able to embed R/f into an LC-finite module M , say via 0 → R/f → M → Q → 0 for some quotient Q. Suppose further that all of these modules are finitely generated over RhF i and that all maps in the short exact sequence are RhF i-linear. In this case, we would consider the long exact sequence β − HIi−1 (Q) → HIi (R/f ) → HIi (M ) → · · · · · · → HIi−1 (M ) → Since M is LC-finite by hypothesis, every submodule of HIi (M ) has finitely many associated primes. The question of whether HIi (R/f ) has closed support is reduced, 1 An exception is when both the original module and the submodule in question happened to be unit. 70 in this case, to the question of whether the cokernel of β has closed support. Since all maps in the long exact sequence are RhF i-linear, it would suffice to show that HIi−1 (Q) is finitely generated. Of course, if HIi (R/f ) were finitely generated over RhF i, we would be finished, so assume that it is not. The analogue of Question VI.1 in this setting is as follows. Question VI.2. In the notation established above, suppose that the RhF i-module HIi (R/f ) is not finitely generated. Can the RhF i-module HIi−1 (Q) be finitely generated? In Chapter VIII, we will describe a short exact sequence 0 → R/f → M → Q where M is LC-finite and, under appropriate vanishing conditions on the local cohomology of R, the RhF i-mdodule HIi−1 (Q) is indeed finitely generated, yielding a novel closed support result on the local cohomology of S. This short exact sequence is part of a longer (exact) complex of RhF i-modules, whose construction is the main concern of Chapter VII. Our present goal in the chapter is to motivate and construct the RhF i-linear embeddings that will be used in the sequel. This and the next two chapters represent joint work of the author and Eric Canton, originally appearing in [CL20]. 6.1 The Fedder action We continue the notation established in the introduction of this chapter. Regarding the closed support problem over a positive characteristic hypersurface ring, Katzman and Zhang [KZ17] give a proof of Theorem V.2 independent of the methods of Hochster and Núñez-Betancourt, based on explicitly describing the supf − HIi (R) for each i ≥ 0. The ports of the kernel and cokernel of the map HIi (R) → 71 short exact sequence 0 → f R → R → S → 0 in the case c = 1 is isomorphic to f 0→R→ − R → R/f → 0, giving the long exact sequence below. (6.1) f f · · · → HIi (R) → − HIi (R) → HIi (R/f ) → HIi+1 (R) → − ··· Recall from Theorem II.15 that if f = f1 , . . . , fc is a regular sequence of codimension c, then for any ideal I ⊇ f and any i ≥ 0, there is a natural isomorphism f − HIi (R) is precisely the HIi (Hfc (R)) ∼ = HIi+c (R). The multiplication map HIi (R) → f map induced by HIi−1 (−) on Hf1 (R) → − Hf1 (R). Thus, one may consider the long exact sequence 6.1 as instead being induced by applying ΓI (−) to the short exact sequence below. (6.2) f 0 → R/f R → Hf1 (R) → − Hf1 (R) → 0 In this sequence, 1 ∈ R/f R is sent to the Čech cohomology class {{1/f }} ∈ Hf1 (R). If one does not wish to directly compute the supports of the kernel and cokernel of the multiplication-by-f map on HIi (R), then we can still obtain the conclusion that Supp HIi (R/f ) is closed by an RhF i-module argument analogous to Hochster and Núñez-Betancourt, so long as the short exact sequence (6.2) can be made RhF i-linear. Indeed, if the right-most copy of Hf1 (R) is equipped with the natural Frobenius action Fnat and the middle copy of Hf1 (R) is equipped with the Frobenius action f p−1 Fnat , then one can readily verify that this is the case. In the short exact sequence 0 → R/f → M → Q → 0 of (6.2), now regarded as a sequence of RhF i-modules, one might observe that Q is unit and finitely generated by Proposition III.14, answering Question VI.2 in the affirmative. An alternate proof of Theorem V.2 follows immediately from these observations. The observation that R/f is isomorphic to the annihilator of the ideal f R in the module Hf1 (R), and that the inclusion map of that annihilator submodule can be 72 made RhF i-linear, generalizes to the higher codimension setting. Fix c ≥ 2 and let f = f1 · · · fc . The Čech cohomology class {{1/f }} is precisely the annihilator submodule (0 :Hfc (R) f ) in Hfc (R). If Hfc (R) is equipped with the Frobenius action f p−1 Fnat , one may directly verify that the R-submodule spanned by {{1/f }} is RhF istable. We will prove a more slightly general statement in Proposition VI.6. Letting Qf denote the cokernel of R/f ֒→ Hfc (R), we obtain the following short exact sequence of RhF i-modules whose middle term, Hfc (R), is LC-finite2 . (6.3) 0 → R/f → Hfc (R) → Qf → 0 The RhF i-module Hfc (R) equipped with the Frobenius action f p−1 Fnat is finitely generated – in fact, it is cyclic, generated by the Čech cohomology class {{1/f 2 }}. However, Hfc (R)fed is never unit and Qf is not unit for c ≥ 2. In both cases, the structure morphism is a surjective map with a nontrivial kernel that we compute explicitly in Section 6.4. Thus, we cannot necessarily expect finite generation over RhF i for HIi (Hfc (R)fed ) or HIi (Qf ). For this reason, we will eventually require a somewhat more elaborate construction than the short exact sequence 6.3, and this construction is the subject of Chapter VII. In what follows, we will refer to the f p−1 Fnat as the Fedder action on Hfc (R). The terminology is motivated by the relationship between this action and a result of Fedder [Fed83] concerning Gorenstein local rings. 6.2 The Fedder action associated with a Gorenstein local ring In this section, the term F -finite 3 to refer to a ring R of prime characteristic p > 0 with the property that R1/p is a finitely generated R-module. 2 Recall that LC-finiteness is a property of R-modules, not RhF i-modules, so Theorem III.16 using the natural action is sufficient to prove this statement. 3 This is not to be confused with Lyubeznik’s definition of F -finite, discussed in previous chapters, and referring to the property of finite generation for unit RhF i-modules. 73 In the following lemma, we refer to the canonical module ωA of a Cohen-Macaulay local ring A. See Chapter 3 of [BH98] for proofs of the assertions made below. Lemma VI.3. Let (S, m) be an F -finite Gorenstein local ring of prime characteristic p > 0, and let q = pe . For some non-zero map T : S 1/q → S, we have an isomorphism of S 1/q -modules HomS (S 1/q , S) ≃ T · S 1/q When S = R/J is Gorenstein for R an F -finite regular local ring, the fact that HomS (S 1/q , S) is cyclic leads to the following description of (J [p] : J) [Fed83]. Lemma VI.4. Let R be a regular local ring of prime characteristic p > 0 and let J ⊂ R is an ideal such that R/J is Gorenstein. For some g ∈ R, we have e (J [p] : J) = gR + J [p] , and (J [p ] : J [p e−1 ] ) = gp e−1 e R + J [p ] for all e ≥ 1. e Proof. Since R is regular, FR (−) is exact, and thus, (J [p ] :R J [p e−1 ] ) = (J [p] : J)[p e−1 ] for all e ≥ 1. Thus, it suffices to prove the case in which e = 1. If we fix a generator T for HomR (R1/p , R) over R1/p , then the map ∼ → HomR/J ((R/J)1/p , (R/J)) (J [p] : J)/J [p] − sending (r + J [p] ) to (T · r) + J is an isomorphism [Fed83]. By VI.3, we know HomR/J ((R/J)1/p , R/J) is cyclic over (R/J)1/p , say via (T · g) + J for g ∈ (J [p] : J). We conclude (J [p] : J) = gR + J [p] . Let R/J be a Gorenstein quotient with R regular, and fix a generator g for (J [p] : J). Lemma VI.4 allows us to define a directed system (6.4) 0 R/J g R/J whose transition maps are injective. [p] gp R/J [p2 ] gp 2 R/J [p 3] ··· 74 e e Let M = lime (R/J [p ] , g p ) denote the direct limit of the system. The embedding −→ R/J ֒→ M can be made Frobenius stable with respect to the natural action of R/J. Specifically, let M be equipped with the action β : M → M described by gF : e R/J [p ] → R/J [p e+1 ] at the unit of its defining directed system. The compatibility with the natural action F : R/J → R/J is shown below. 0 F 0 R/J g R/J g R/J [p] R/J [p gF gF R/J [p] gp gp R/J [p 2] gp 2 ··· β gF 2] 2 gp R/J [p 3] M 3 gp ··· M We refer to the resulting action β : M → M on M as the Fedder action. 6.3 The Fedder action associated with a regular sequence For a regular sequence f = f1 , . . . , fc , by an abuse of notation, we will use f [t] to denote the sequence f1t , . . . , fct – which is still regular [BH98, Exercise 1.1.10] – Q regardless of whether t is a power of the characteristic. Let f = ci=1 fi . When the ideal J in the directed system (6.4) is generated by a regular sequence f = f1 , . . . , fc , the Fedder socle of R/f [p] has a clear choice of generator: f p−1 . Moreover, the direct limit (6.4) is identifiable as HfcR (R). For q = pe , the action sending r + f [q] ∈ R/f [q] to f p−1 rp + f [qp] ∈ R/f [qp] at the unit of the directed system (R/f [q] , f qp−q )∞ e=0 has the form  r fq  7→  rp f p−1 f qp  in terms of Čech cohomology classes in Hfc (R). While our primary motivation is the case in which R is a regular ring, the colon properties (f [b] : f b−a ) = f [a] and (f [b] : f [a] ) = f b−a R + f [b] 75 for two positive integers b > a hold for an arbitrary regular sequence f in a Noetherian ring. Many properties of the constructions that follow therefore work in greater generality than the motivating setting of the previous section. Definition VI.5. Let R be a Noetherian ring of prime characteristic p > 0, let Q f = f1 , . . . , fc ∈ R be a regular sequence of codimension c, and let f = ci=1 fi . Let Fnat denote the natural action of the Frobenius on Hfc (R) (see Definition III.10). The Fedder action with respect to f on Hfc (R) – or simply the Fedder action, if the sequence f is understood – is defined by Ffed := f p−1 Fnat . When there is risk of confusion, we write Hfc (R)nat and Hfc (R)fed to distinguish the (non-equivalent) RhF i-modules obtained when Hfc (R) is equipped with the natural action and the Fedder action, respectively. The RhF i-module Hfc (R)nat is cyclic with generator {{1/f }}. Concerning the Fedder action, Hfc (R)fed is still cyclic, but we require a different generator. The element 1 7→ {{1/f }} is preserved by the Fedder action, but notice that for each q = pe we have Ffed :  r f q+1  7→  rp f qp+1  for all r ∈ R, and thus, for example, the class {{1/f 2 }} generates. Perhaps the most useful property of the Fedder action is its compatibility with embeddings of annihilators of subsequences of f , in the sense of the following proposition. This compatibility was observed in [Can16], essentially as the result of applying Hf∗ (−) to the Koszul complex K • (g; R). Proposition VI.6. Let R be a Noetherian ring of prime characteristic p > 0, let g1 , . . . , gt , f1 , . . . , fc ∈ R be a regular sequence, and write g = g1 , . . . , gt , f = Q Q t+c f1 , . . . , fc ,, g = ti=1 gi and f = ci=1 fi . Consider Hfc (R/g) and Hg,f (R) as RhF i- 76 modules via the Fedder actions with respect to f and g, f , respectively. There is an RhF i-linear injection t+c Hfc (R/g) ֒→ Hg,f (R) whose image is the annihilator (0 :H t+c (R) g). g,f Proof. Since (g[a] , f [a] ) : g a−1 = (g, f [a] ) for all a ≥ 1, multiplication by g a−1 induces g a−1 a well-defined injection φa : R/(g, f [a] ) −−→ R/(g[a] , f [a] ), and since (g[a] , f [a] ) : g = g a−1 R + (g[a] , f [a] ), the image of φa is precisely (0 :R/(g[a] ,f [a] ) g). The maps φa form a ∞ [a] [a] map of directed systems (R/(g, f [a] ), f )∞ a=1 to (R/(g , f ), gf )a=1 via R/(g, f [a] ) g a−1 (gf )b−a f b−a R/(g, f [b] ) R/(g[a] , f [a] ) g b−1 R/(g[b] , f [b] ) t+c On the direct limits, this produces an injection φ : Hfc (R/g) ֒→ Hg,f (R), whose image is precisely (0 :H t+c (R) g). For each q = pe , the Fedder action with respect to g,f f sends the class r + (g, f [q] ) ∈ R/(g, f [q] ) to f p−1 rp + (g, f [qp] ) ∈ R/(g, f [qp] ), which is sent by φqp to g qp−1 (f p−1 rp + (g[qp] , f [qp] ) ∈ R/(g[qp] , f [qp] ). On the other hand, φq sends r + (g, f [q] ) to g q−1 r + (g[q] , f [q] ), and the Fedder action with respect to g, f sends g q−1 r + (g[q] , f [q] ) to (gf )p−1 g q−1 r as desired. 6.4
p + (g[qp] , f [qp] ) = g qp−1 (f p−1 rp ) + (g[qp] , f [qp] ), The structure morphism of the Fedder action Let R be regular, let f = f1 , . . . , fc ∈ R be a regular sequence of codimension c, and let Qf denote the cokernel of the Frobenius stable embedding R/f ֒→ Hfc (R)fed that sends 1 7→ {{1/f }}, equipped with its induced action. (6.5) 0 → R/f → Hfc (R)fed → Qf → 0 77 To give a Frobenius action on a complex A• is precisely to choose a Frobenius action on each term Ai such that the differentials di : Ai → Ai+1 are Frobenius stable. Analogous to the situation with modules, the data of a Frobenius action on A• is equivalent to specifying an R-linear map of complexes Θ : FR (A• ) → A• – the structure morphism of the complex. In this section, we will describe the structure morphism of the three-term complex (6.5). Theorem VI.7. Let R be a Noetherian ring of prime characteristic p > 0, let f = Q f1 , . . . , fc ∈ R be a regular sequence, let f = ci=1 fi , and let A• denote the complex described in (6.5) corresponding to the Frobenius stable embedding R/f → Hfc (R)fed . Let θR/f , θfed , and θQ denote the RhF i structure morphisms of R/f , Hfc (R)fed , and Qf , respectively. Let Θ denote the structure morphism of A• . There is an exact sequence of complexes Θ 0 → K • → FR (A• ) − → A• → 0 described term-by-term in the diagram below with exact4 rows and columns, where FR (Hfc (R)) is identified with Hfc (R) in the natural way5 . 0 A• : 0 0 0 0 R/f Hfc (R) Qf θR/f Θ 0 θQ θfed FR (A• ) : 0 R/f [p] Hfc (R) FR (Qf ) 0 K• : 0 f /f [p] (0 :Hfc (R) f p−1 ) Vf 0 0 0 0 0 4 Note that exactness of the second row is a property of regular sequences in any Noetherian ring, and does not require FR (−) to be exact. 5 That is, using the structure isomorphism of the natural action; see Theorem III.14. 78 The module Vf := Ker(θQ ) can be described by the direct limit f q+1 f qp+1 f qp−q ··· → → · · · → Vf −−−→ f q R + f [q+p] f qp R + f [qp+p] Proof. We first describe A• as a direct limit of complexes lime (A•e , ψe ), −→ A• : 0 R/f Hfc (R) Qf .. . .. . .. . R/f [qp+1] R/(f qp R + f [qp+1] ) .. . A•e+1 : 0 f qp R/f f qp−q 1 ψe A•e : 0 .. . fq R/f 0 f qp−q R/f [q+1] R/(f q R + f [q+1] ) .. . .. . .. . 0 0 In the direct limit, an element r+f [q+1] ∈ R/f [q+1] maps to the Čech cohomology class {{r/f q+1 }}. To describe the structure morphism Θ : FR (A• ) → A• , we specify a map Θe−1 : FR (A•e−1 ) → A•e for each e by taking the obvious quotient maps term-by-term. A•e : 0 R/f 0 R/f [p] fq R/f [q+1] R/(f q R + f [q+1] ) 0 fq R/f [q+p] R/(f q R + f [q+p] ) 0 Θe−1 FR (A•e−1 ) : The exactness of the second row of the above diagram is a property of regular sequences that holds irrespective of whether FR (−) is exact. One may verify that the following diagram of complexes commutes FR (A•e ) Θe FR (ψe−1 ) FR (A•e−1 ) A•e+1 ψe Θe−1 A•e • • so that the (Θe )∞ e=1 induce a well-defined map on the direct limit FR (A ) → A . To describe the Frobenius action on the class r + f [q+1] , the inclusion R/f [q+1] → 79 F∗ R ⊗R (R/f [q+1] ) sends r + f [q+1] to rp + f [qp+p] once the codomain is identified with R/f [qp+p] . The quotient R/f [qp+p] ։ R/f [qp+1] sends that class to rp + f [qp+1] . On the corresponding Čech cohomology classes in the direct limit, we have  r f q+1  7→  rp f qp+1  =  f p−1 rp f qp+p  = Ffed  r f q+1  as desired. Concerning the kernel of the structure map, we have for each e a commutative diagram with exact rows and columns. 0 0 0 0 : 0 R f fq R f [q+1] R (f q R+f [q+1] ) 0 FR (A•e−1 ) : 0 R fq R f [q+p] R (f q R+f [q+p] ) 0 Ke• : 0 fq f [q+1] f [q+p] (f q R+f [q+1] ) (f q R+f [q+p] ) 0 0 0 A•e Θe−1 f [p] f [p] f 0 0 • where FR (ψe−1 ) induces a map Ke• → Ke+1 • Ke+1 : 0 f f f qp [p] f qp−q 1 Ke• : 0 f f f [qp+1] f [qp+p] [p] fq f [q+1] f [q+p] f [qp+1] (f qp R+f [qp+p] ) 0 f qp−q f [q+1] (f q R+f [q+p] ) 0 compatible with the rest of the directed system. Note that f [q+1] /f [q+p] = (f [q+p] : f p−1 )/f [q+p] for each q = pe , so that in the direct limit, Ker(θfed ) = (0 :Hfc (R) f p−1 ). Note that the kernel (0 :Hfc (R) f p−1 ) of θfed can be explicitly described as a local cohomology module, namely, it is isomorphic to Hfc−1 (R/f p−1 ). Proposition VI.8. Let R be a Noetherian ring, let f = f1 , . . . , fc ∈ R be a regular 80 sequence of codimension c, and let h ∈ R be a nonzerodivisor. Then (0 :Hfc (R) h) ∼ = Hfc−1 (R/h). h Proof. Consider the double complex (0 → R → − R → 0) ⊗R Č • (f ; R). 0 Č 1 (f ; R) 0 ··· h Č 1 (f ; R) 0 0 0 Č c−1 (f ; R) Č c (f ; R) h ··· 0 0 h Č c−1 (f ; R) Č c (f ; R) 0 0 0 If we compute cohomology first horizontally and then vertically, we see that the cohomology of the total complex in degree c is HomR (R/h, Hfc (R)), since E2 = E∞ and the first c − 1 columns vanish. On the other hand, if we first take cohomology vertically and then horizontally, then E2 = E∞ and the first row vanishes, so the cohomology of the total complex in degree c is also isomorphic to H c−1 (R/h ⊗R Č • (f ; R)) = Hfc−1 (R/h), as desired. The kernel of θQ has a particularly nice description in codimension 2, where it decomposes as a direct sum of two local cohomology modules, Vf,g ∼ = Hf1 (R/g p−1 ) ⊕ Hg1 (R/f p−1 ). Proposition VI.9. Let R be a Noetherian ring of prime characteristic p > 0, let f, g ∈ R be a regular sequence, and let Vf,g denote the following direct limit over all q = pe ··· → (f q+1 , g q+1 ) (f qp+1 , g qp+1 ) f qp−q → · · · → Vf,g − − − → ((f g)q , f q+p , g q+p ) ((f g)qp , f qp+p , g qp+p ) Then Vf,g ∼ = Hg1 (R/f p−1 ) ⊕ Hf1 (R/g p−1 ) 81 Proof. Note that if af q+1 = bg q+1 mod ((f g)q , f q+p , g q+p ) for some a, b ∈ R, then
a ∈ ((f g)q , f q+p , g q+1 ) : f q+1 = (f q+p , g q+1 ) : (f p , g) : f q+1
= (f q+p , g q+1 ) : f q+1 : (f p , g)
= f p−1 , g q+1 : (f p , g) = f p−1 , g q
so that af q+1 ∈ (f q+p , f q+1 g q ), which is zero mod ((f g)q , f q+p , g q+p ). Thus, the generators u1,e := f q+1 and u2,e := g q+1 of (f q+1 , g q+1 )/((f g)q , f q+p , g q+p ) have Rspans with an intersection of 0, yielding a direct sum Ru2,e Ru1,e (f q+1 , g q+1 ) ∼ ⊕ q = q q q+p q+p p−1 ((f g) , f , g ) g u1,e R + f u1,e R f u2,e R + g p−1 u2,e R The transition map (f g)qp−q sends f q+1 = u1,e to g qp−q f qp+1 = g qp−q u1,e+1 , and likewise, u2,e 7→ f qp−q u2,e+1 , breaking into the direct sum of transition maps on the u1 and u2 components, ··· → gq u Ru1,e Ru1,e+1 g qp−q → · · · → Hg1 (R/f p−1 ) −−−→ qp p−1 p−1 u R + f u R g u R + f R 1,e 1,e 1,e+1 1,e+1 and ··· → f qu as desired. Ru2,e+1 Ru2,e f qp−q → · · · → Hf1 (R/g p−1 ) −−−→ qp p−1 p−1 u R + g u R f u R + g R 2,e 2,e 2,e+1 2,e+1 CHAPTER VII A Complex of Annihilator Submodules Throughout this chapter, R denotes a Noetherian ring of prime characteristic p > 0. For a regular sequence f = f1 , . . . , fc in R, let f = f1 · · · fc , and let Hfc (R)nat and Hfc (R)fed denote the RhF i-modules obtained when Hfc (R) is equipped with the natural action, Fnat , or with the Fedder action, f p−1 Fnat , respectively. In codimension c = 1, for f ∈ R a nonzerodivisor, Proposition VI.6 gives an RhF ilinear map R/f → Hf1 (R)fed whose image is the annihilator submodule (0 :Hf1 (R) f ). Since multiplication by f is surjective, the cokernel of the inclusion (0 :Hf1 (R) f ) ֒→ Hf1 (R) is isomorphic to Hf1 (R). If this copy of Hf1 (R) is equipped with the natural action, we obtain the short exact sequence of RhF i-linear maps below. (7.1) f 0 → R/f → Hf1 (R)fed → − Hf1 (R)nat → 0 1 The unitness of the cokernel of the embedding R/f → H(f,g) (R)fed is a significant advantage in this setting. Consider the case of codimension c = 2. Let f, g be a regular sequence of R with the property that g is a nonzerodivisor1 . Proposition VI.6 provides an RhF i2 linear embedding R/(f, g) → Hf,g (R)fed whose image is the annihilator submodule 2 (R) (f, g)), but the cokernel of this embedding, is not unit – see Theorem VI.7 (0 :Hf,g in general and Proposition VI.9 regarding codimension c = 2 in particular. 1 This turns out to be equivalent to requiring that both f, g and g, f are regular sequences. 82 83 It is not impossible to obtain a unit RhF i-module by taking suitable RhF i2 linear quotients of Hf,g (R)fed . For example, consider the annihilator (RhF i-stable) 2 (R) f g), which is generated over R by those Čech cohomology submodule (0 :Hf,g  classes f −a g −b such that either a = 1 or b = 1. The cokernel of the embed- 2 2 (R) f g) → H ding (0 :Hf,g f,g (R)fed is spanned over R by the images of those classes  −a −b f g with a ≥ 2 or b ≥ 2, and the map 2 2 2 (R) f g) → H Hf,g (R)fed /(0 :Hf,g f,g (R)nat that sends  f −a g −b 2 2 (R) f g) to the class in Hf,g (R)fed /(0 :Hf,g  f −a+1 g −b+1 in 2 Hf,g (R)nat is readily verified to be RhF i-linear. The surjectivity of multiplication by 2 f g on Hf,g (R) now gives the following exact sequence of RhF i-modules. (7.2) fg 2 2 (R) f g) → H 0 → (0 :Hf,g → Hf2g (R)nat → 0 f,g (R)fed − Let us extend this sequence further to the left. Our complete intersection rings 2 (R) (f, g)), which is the intersection of two other annihiR/(f, g) appears as (0 :Hf,g 2 (R) f ) and (0 :H 2 (R) g), whose sum lator submodules of RhF i-submodules, (0 :Hf,g f,g 2 (R) f g). The submodule (0 :H 2 (R) f ) is spanned over R by Čech is all of (0 :Hf,g f,g  −1 −b 2 (R) g) is spanned by cohomology classes of the form f g for b ∈ N, and (0 :Hf,g   those of the form {{f −a g −1 }} for a ∈ N. The maps defined by g −b 7→ f −1 g −b 2 (R) f ) {{f −a }} 7→ {{f −a g −1 }} provide RhF i-linear isomorphisms2 Hg1 (R/f ) → (0 :Hf,g p−1 2 (R) g), respectively, using Frobenius actions g and Hf1 (R/g) → (0 :Hf,g Fnat and 2 (R) f ) and V = (0 :H 2 (R) g), f p−1 Fnat on Hg1 (R/f ) and Hf1 (R/g). Letting U = (0 :Hf,g f,g the RhF i-linear exact sequence 0 → U ∩ V → U ⊕ V → U + V → 0 may therefore be expressed as follows. (7.3) 2 Both 2 (R) f g) → 0 0 → R/(f, g) → Hf1 (R/g) ⊕ Hg1 (R/f ) → (0 :Hf,g of these embeddings are in fact special cases of the embedding described in Proposition VI.6. 84 The sequences and together yield a four-term exact sequence of RhF i-modules. (7.4) fg 2 2 0 → R/(f, g) → Hf1 (R/g) ⊕ Hg1 (R/f ) → Hf,g (R)fed −→ Hf,g (R)nat → 0 We regard the exact sequences 7.1 and 7.4 as the augmentations of a pair of (cohomologically indexed) complexes of RhF i-modules ∆ ∆•f (R) : R/f → Hf1 (R)fed and 2 ∆ ∆•f,g (R) : R/(f, g) → Hf1 (R/g) ⊕ Hg1 (R/f ) → Hf,g (R)fed Both of these complexes, in codimensions c = 1 and c = 2, satisfy H i (∆ ∆•f (R)) = 0 for i < c and H c (∆ ∆•f (R)) ≃ Hfc (R)nat , where f denotes either f or f, g, respectively. The main result of this chapter, which represents original work of the author and Eric Canton appearing in [CL20], is that a complex of RhF i-modules completely analogous to the c = 1 and c = 2 complexes above can be constructed for permutable regular sequences of arbitrary length. The higher codimension ∆ ∆•f (R) complexes are the main technical tool required for the applications presented in Chapter VIII. Before we begin the construction, we set some terminology and notation that will be used throughout both this chapter and the next. Definition VII.1. Let R be a ring and f = f1 , . . . , fc be a regular sequence in R. Call f permutable if fσ(1) , . . . , fσ(c) is a regular sequence for all permutations σ on the set {1, . . . , c}. Permutability is automatic for regular sequences in a local ring [BH98, Proposition 1.1.6], or for regular sequences of homogeneous elements in a standard graded ring [BH98, Exercise 1.5.23]. Let [c] := {1, . . . , c}, and for a subset T ⊆ [c], let f T denote the subsequence of f indexed by the elements of T . Let Te = [c] \ T , with f Te the 85 complementary subsequence to f T in f . The permutability of f is equivalent to the hypothesis that f T is a regular sequence for all subsets T ⊆ [c] [BH98, Exercise 1.2.21]. 7.1 Construction of the ∆ ∆ Complex Let R be a Noetherian ring, and fix a permutable regular sequence f = f1 , . . . , fc ∈ Q R. For T ⊆ [c], let fT = i∈T fi , and write f = f[c] for convenience. For a ≥ 1, recall that we denote f [a] := f1a , . . . , fca regardless of whether a is a power of the [a] characteristic, and the notation f T denotes the subsequence of f [a] indexed by T ⊆ [c]. Since f is permutable, we can use Proposition VI.6 to obtain identifications (7.5) Hfc−i (R/f Te ) = (0 :Hfc (R) f Te ) T for each subset T ⊆ [c]. Let M = Hfc (R), and observe that there are inclusions ιT,S : (0 :M f Se) ֒→ (0 :M f Te ) whenever S ⊆ T . Definition VII.2. Let R be a Noetherian ring and let f be a permutable regular sequence of codimension c ≥ 1. Let M = Hfc (R). The ∆ ∆-complex of f , denoted ∆ ∆•f (R), is the chain complex (M • , ∂ • ) defined as follows. • Mi = • ∂i = L |S|=i (0 :M Pc j=1 (−1) f Se) for 0 ≤ i ≤ c j i dj where dij |(0:M f Se ) is the direct sum of the inclusion maps ιS,T : (0 :M f Se) ֒→ (0 :M f Te ) ranging over the sets T ⊇ S of size |T | = i + 1 such that T \ S is the jth element of T , enumerated so that ta < tb (as elements of [c]) when a < b. The choice of differentials gives {M i }ci=0 the structure of a semi-cosimplicial Ri+1 i i module [Wei94, Def. 8.1.9, Ex. 8.1.6], which is to say, di+1 k dj = dj dk−1 for j < k. Checking that ∂ i+1 ∂ i = 0 is similar to checking that the chain maps in a Čech 86 complex square to zero, and depends only on the semi-cosimplicial structure [Wei94, Def. 8.2.1]. For each 1 ≤ n ≤ c, we may define a quotient complex ∆ ∆•f (R)n of ∆ ∆•f (R) in the following manner. Note that this definition depends on the specific order of elements in the sequence f1 , . . . , fc . If σ : [c] → [c] is a non-trivial bijection, then setting gi = fσ(i) , 1 ≤ i ≤ c, ∆ ∆ complex of the regular sequence g1 , . . . , gc would have a distinct collection of quotients. Definition VII.3. Let R be a Noetherian ring and let f be a permutable regular sequence of codimension c ≥ 1. For fixed n such that 1 ≤ n ≤ c, define the complex ∆ ∆•f (R)n by ∆ ∆if (R)n := M S⊂[n], |S|=i with differentials ∂0i = Pn j=1 (−1) j i dj,n (0 :M f Se) ⊆ ∆ ∆if (R) defined so that the map dij,n |(0:M f Se ) for S ⊆ [n] is the direct sum of inclusions ιS,T : (0 :M f Se) ֒→ (0 :M f Te ) ranging only over the sets T of size |T | = i + 1 such that S ⊆ T ⊆ [n] and such that T \ S is the jth element of T , enumerated so that ti < tj (as elements of [c]) when i < j. ∆•f (A)c = ∆ ∆•f (R). For each n, there For example, ∆ ∆•f (R)n = (0 → R/f → 0) and ∆ is a surjection of complexes πn : ∆ ∆•f (R)n → ∆ ∆•f (R)n−1 defined term-by-term in the obvious way, and we let Kn• denote the kernel of πn 0 Kn• ∆ ∆•f (R)n ∆ ∆•f (R)n−1 0 For our calculation of the cohomology of ∆ ∆•f (R), the key observations are as follows. Proposition VII.4. Let R be a Noetherian ring and let f be a permutable regular sequence of codimension c ≥ 1. For fixed n such that 1 ≤ n ≤ c, where all set f are taken within [c], we have the following. complements (e.g. [n]) 87 1. ∆ ∆•f (R)n = ∆ ∆•f [n] (R/f [n] f ). 2. Kn• = Hf1n (∆ ∆•f [n−1] (R/f [n] f ))[−1]. where [−1] denotes the right-shift operator on cohomologically indexed complexes. ∆if (R)n is the direct sum of annihilators Proof. Let M = Hfc (R). The module ∆ (0 :M f Se) ranging over all subsets S ⊆ [n] of size |S| = i. In particular, we have [c] − [n] ⊆ [c] − S for all such S, so that (0 :M f [c]−S ) = (0 :(0:M f [c]−[n] ) f [n]−S ) = (0 :Mn f [n]−S ) where Mn = Hfn[n] (R/f [c]−[n] ), and thus, ∆ ∆if (R)n = ∆ ∆if [n] (R/f [c]−[n] ). The agreement ∆•f [n] (R/f [c]−[n] ) is a straightforward of the differentials in the complexes ∆ ∆•f (R)n and ∆ consequence of their definitions. ∆•f (R)n−1 is the direct sum of the Concerning Kni , the kernel of ∆ ∆•f (R)n ։ ∆ annihilators (0 :M f Se) ranging over subsets S ⊆ [n] of size |S| = i such that n ∈ S. We therefore have an isomorphism HfiS (R/f [c]−S ) = Hf1n  Hfi−1 (R/f [c]−S ) S−{n}  = Hf1n 0 :Mn f [n−1]−(S−{n})

where, once again, Mn = Hfn[n] (R/f [c]−[n] ). The sets S − {n} for S ⊆ [n] of size |S| = i such that n ∈ S correspond precisely to the subsets S ′ ⊆ [n − 1] of size |S ′ | = i − 1. Thus, Kni = ∆ ∆i−1 f [n−1] (R/f [c]−[n] ). Confirming agreement of the corresponding differentials is straightforward. Example VII.5. Suppose c = 4. When n = 0, ∆ ∆if (R)n = (0 :M f ) ∼ = R/f for i = 0 ∆n for 1 ≤ n ≤ 4, identifying (0 :M f Se) with and ∆ ∆if (R)n = 0 for i > 0. We show ∆ HfiS (R/f Se). Components of ∂ i corresponding to di1 , di2 , di3 , and di4 are indicated in red, blue, dashed red, and dashed blue, with (dashed or solid) red indicating a sign change. 88 n ∆ ∆•f (R)n Hf11 (R/f {2,3,4} ) 1 R/f Hf11 (R/f {2,3,4} ) 2 ⊕ R/f Hf12 (R/f {1,3,4} ) Hf11 (R/f {2,3,4} ) 3 Hf21 ,f2 (R/f {3,4} ) R/f R/f Hf21 ,f2 (R/f {3,4} ) ⊕ ⊕ Hf12 (R/f {1,3,4} ) Hf21 ,f3 (R/f {2,4} ) ⊕ ⊕ Hf13 (R/f {1,2,4} ) Hf22 ,f3 (R/f {1,4} ) Hf11 (R/f {2,3,4} ) Hf21 ,f2 (R/f {3,4} ) ⊕ ⊕ Hf12 (R/f {1,3,4} ) Hf21 ,f 3 (R/f {2,4} ) ⊕ ⊕ Hf13 (R/f {1,2,4} ) Hf22 ,f3 (R/f {1,4} ) ⊕ Hf21 ,f4 (R/f {2,3} ) Hf14 (R/f {1,2,3} ) Hf22 ,f4 (R/f {1,3} ) 4 Hf31 ,f2 ,f3 (R/f4 ) Hf31 ,f2 ,f3 (R/f4 ) ⊕ ⊕ Hf31 ,f2 ,f4 (R/f3 ) ⊕ Hf31 ,f3 ,f4 (R/f2 ) Hf4 (R) ⊕ Hf23 ,f4 (R/f {1,2} ) Hf32 ,f3 ,f4 (R/f1 ) The subcomplexes K2• , K3• , and K4• are displayed with terms generally to the lower right. For example, K4• consists of the terms in ∆ ∆•f (R)4 that involve the local cohomology of quotients R/f S for subsets S ⊆ {1, 2, 3}. 7.2 Computing the Cohomology of the ∆ ∆ Complex We continue with the notation of the last section. Let R be a Noetherian ring, Q let f be a permutable regular sequence of codimension c ≥ 1, and let f = ci=1 fi . We denote by ∆ ∆•f (R)+ the augmented chain complex equal to ∆ ∆•f (R) in degrees ≤ c, 89 with augmentation ∆ ∆c+1 (R)+ := Hfc (R) and differential ∂ c : ∆ ∆cf (R)+ → ∆ ∆c+1 (R)+ f f given by the multiplication by f map Hfc (R) → Hfc (R). Lemma VII.6. Let R be a Noetherian ring, let f be a permutable regular sequence of codimension c ≥ 1. Suppose H i (∆ ∆•f (R)+ ) = 0 for all 1 ≤ i ≤ c + 1, and let h ∈ R extend f to a permutable regular sequence f1 , . . . , fc , h of codimension c + 1. Then H i (Hh1 (∆ ∆f (R)+ )) = 0 for all 1 ≤ i ≤ c + 1. Proof. For the sake of notational convenience, write ∆ ∆• = ∆ ∆•f (R)+ . We compute Hh1 via the double complex Č(h; R) ⊗R ∆ ∆• , i.e. 0 → ∆ ∆• → (∆ ∆• )h → 0, as shown below. 0 0 0 0 0 ∆ ∆0 ∆ ∆1 ··· ∆ ∆c f ∆ ∆c+1 0 0 (∆ ∆ 0 )h (∆ ∆ 1 )h ··· (∆ ∆ c )h f (∆ ∆c+1 )h 0 0 0 0 0. By hypothesis, H i (∆ ∆• ) = 0 for all i, so H i ((∆ ∆• )h ) = 0 for all i as well. If we first take cohomology horizontally, we therefore obtain E1 = E∞ = 0. The cohomology of the total complex of the double complex is therefore zero. If, on the other hand, we take vertical cohomology first, then we arrive at a double complex with one nonzero row, 0 Hh1 (∆ ∆0 ) ··· Hh1 (∆ ∆c+1 ) 0. 1 The modules H i (H(h) (∆ ∆• )) appear now as the horizontal cohomology, with E2 = E∞ , and since only a single row is nonzero, they yield the cohomology of the total complex, which vanishes. We are now ready to prove the main theorem of this chapter. 90 Theorem VII.7. Let R be a Noetherian ring, and let f be a permutable regular sequence of codimension c ≥ 1. Then H i (∆ ∆•f (R)) = 0 for 0 ≤ i < c, and H c (∆ ∆•f (R)) ∼ = Q Hfc (R), with augmentation map isomorphic to multiplication by f := c1 fi . If R has prime characteristic p > 0, then ∆ ∆•f (R) is a complex of RhF i-modules considered with their Fedder actions, and the induced Frobenius action on the aug∆•f (R)) is the natural action. mentation Hfc (R) ∼ = H c (∆ Proof. The differentials of ∆ ∆•f (R) are direct sums of inclusions of submodules of M = Hfc (R), and by Proposition VI.6 these inclusions are Fedder-action linear in characteristic p > 0. Our statement about the induced Frobenius action on H c (∆ ∆•f (R)) is proven at the end of this argument; the bulk of this proof is calculation of the cohomology. We proceed by by induction on c, with base case c = 1. With f = f1 , the complex ∆ ∆•f (R) is 0 R/f (r+f R)7→{{r/f }} Hf1 (R) 0. The map R/f → Hf1 (R) shown above is clearly injective, so H 0 (∆ ∆•f (R)) = 0. Moreover, the image of R/f → Hf1 (R) is precisely the kernel of multiplication by f . Multiplication by f on Hf1 (R) is surjective, and the exactness of 0 → R/f → f Hf1 (R) → − Hf1 (R) → 0 implies that H 1 (∆ ∆•f (R)) = Hf1 (R). Now assume the theorem has been proven for any permutable regular sequence of codimension c ≥ 1 in a Noetherian ring. Let f , h = f1 , . . . , fc , h ∈ R be a permutable regular sequence of codimension c + 1. From Proposition VII.4, ∆ ∆•f (R/h) is the • of this quotient ∆•f ,h (R) = ∆ ∆•f (R)c+1 . The kernel Kc+1 quotient complex ∆ ∆•f ,h (R)c of ∆ is isomorphic to Hh1 (∆ ∆•f (R))[−1], giving us the short exact sequence of complexes 91 shown below. (7.6) Hh1 (∆ ∆•f (R))[−1] 0 ∆ ∆•f (R/h) ∆ ∆•f ,h (R) 0,
By Lemma VII.6 and the induction hypothesis, H i Hh1 (∆ ∆•f (R))[−1] = 0 for
∆f ,h (R)) = Hfc (R). Likewise, we i ≤ c, and that H c+1 Hh1 (∆ ∆•f (R))[−1] = H c+1 (∆ may apply the induction hypothesis to the ∆ ∆• complex of the regular sequence f of ∆•f (R/h)) = codimension c in R/h to obtain H i (∆ ∆•f (R/h)) = 0 for i < c and H c (∆ Hfc (R/h). We now study the long exact sequence in cohomology from the short exact se∆•1 := ∆ ∆•f (R/h), and quence (7.6). To simplify notation, let ∆ ∆• := ∆ ∆•f ,h (R), ∆ K • := Hh1 (∆ ∆•f (R))[−1]. We obtain ··· H i (K • ) H i (∆ ∆• ) δ H i (∆ ∆•1 ) H i+1 (K • ) ··· We immediately see that H i (∆ ∆• ) = 0 for i < c. Using that H c (K • ) = 0, we have an exact sequence (7.7) 0 H c (∆ ∆• ) H c (∆ ∆•1 ) δ H c+1 (K • ) H c+1 (∆ ∆• ) 0. We claim that δ is injective. To see this, we start with recalling the construction of δ. We begin with the map from row c to row c + 1 in the short exact sequence of complexes. 0 0 Kc K ∆ ∆c ∆ ∆c1 c ∂K c ∂∆ ∆ c+1 c+1 ∆ ∆ c ∂∆ ∆ 0 1 0 ∆c1 = (0 :M h) and K c = Let M = Hfc+1 ,h (R). We identify ∆ ι:∆ ∆c1 ֒→ ∆ ∆c denote the obvious splitting. Lc Lc (0 : f ) (0 : 0 h) ⊕ ( M i M i=1 i=1 (0 :M fi )) c ∂K 0 M c ∂∆ ∆ M ι 0 Lc i=1 (0 :M fi ). Let (0 :M h) c ∂∆ ∆ 0 0 1 0 92 A class {{η}}∆∆1 ∈ H c (∆ ∆•1 ) is represented by η ∈ (0 :M h), where we write a subscript {{· · ·}}C to indicate the complex C • with respect to which we’re taking cohomology. By definition, c c+1 δ({{η}}∆∆1 ) = {{∂∆ (K). ∆ (ι(η))}}K ∈ H c c If δ({{η}}∆∆1 ) = 0, then ∂∆ ∆ (ι(η)) is in the image of ∂K , which is Pc j=1 (0 :M fj ). Note c that ∂∆ ∆ ◦ ι is just the inclusion map (0 :M h) ֒→ M , possibly up to a sign change, so Pc c to say that ∂∆ ∆ (ι(η)) ∈ j=1 (0 :M fj ) means precisely that ! c X c−1 η ∈ (0 :M h) ∩ (0 :M fj ) = Im(∂∆ ∆1 ), j=1 Thus, {{η}}∆∆1 = 0, and δ is injective. We conclude that H c (∆ ∆• ) = 0 from the sequence (7.7). Following the reasoning of Pc c the last paragraph, the image of ∂∆ ∆ is (0 :M h)+ j=1 (0 :M fj ), which is the kernel of Q multiplication by hf = h ( c1 fj ). Thus, the augmentation map ∆ ∆c+1 ։ H c+1 (∆ ∆• ) is (by definition) the quotient M ։ M/(0 :M f h). Since multiplication by f h is surjective on M , this provides an isomorphism ϕ : H c+1 (∆ ∆• ) → M , 0 c Im(∂∆ ∆) ∆ ∆c+1 aug. H c+1 (∆ ∆• ) 0 ϕ 0 (0 :M f h) M fh M 0 So, under the isomorphism ϕ that identifies H c+1 (∆ ∆• ) with M = Hfc+1 ,h (R), the aug. augmentation map ∆ ∆c+1 −−→ H c+1 (∆ ∆• ) is isomorphic to the multiplication map fh M −→ M , as claimed. Except for the final statement about the induced Frobenius action in characteristic p > 0, we have proven the theorem. The final statement comes down to a direct calculation. For a given representative c+1 η∈∆ ∆c+1 = Hfc+1 (∆ ∆• ) sends {{η}}∆∆ to ,h (R)fed , the induced Frobenius action F on H ∼ {{Ffed (η)}}∆∆ . Under the identification ϕ : H c+1 (∆ ∆• ) − → Hfc+1 ,h (R), the augmentation 93 map η 7→ {{η}}∆∆ is the multiplication η 7→ (f h)η. Thus, F ((f h)η) = (f h)Ffed (η) = (f h)p Fnat (η) = Fnat ((f h)η) so F = Fnat , as desired. CHAPTER VIII Application to Closed Support Due to the isomorphism in Theorem IV.4, for a Noetherian ring S and an Smodule M , every local cohomology module of M is isomorphic to one of the form HIi (M ) for I an ideal satisfying i ≤ ht(I) + 1. When S is Cohen-Macaulay and M = S (cf. [Hel01, Theorem 3]) our attention is therefore restricted to the cases ht(I) HI ht(I)+1 (S) and HI ht(I) (S). The module HI (R) has a finite set of associated primes (see Theorem II.18), so questions about the support or associated primes of the local cohomology HIi (S) for S a Cohen-Macaulay ring can, without loss of generality, be ht(I)+1 posed entirely for modules of the form HI (S). Given a regular ring R and a regular sequence f = f1 , . . . , fc , the closed support problem for R/f is, by the preceding discussion, determined by the behavior of modht(I/f )+1 ules of the form HI/f (R/f ) where I is an ideal of R containing f . Indeed, the hypersurface support theorems of Hochster and Núñez-Betancourt [HNB17, Corollary 4.13] or Katzman and Zhang [KZ17, Theorem 7.1] may be interpreted as the ht(I/f )+1 statement that in prime characteristic p > 0, all modules of the form HI/f (R/f ) have closed support. Our main application in this chapter, representing original work of the author and Eric Canton appearing in [CL20], states that if f = f1 , . . . , fc is a permutable regular sequence in a regular ring R of prime characteristic p > 0, then the module 94 95 ht(I/f R)+c HI/f R (R/f R) has closed support for any ideal I satisfying the vanishing hypoth- esis HIi (R) = 0 for ht(I) < i < ht(I) + c. Note that the hypothesis on I is vacuous if c = 1, and is satisfied automatically if R/I is Cohen-Macaulay, due to a result of Peskine and Szpiro [PS73]. To establish notation, assume from this point onward that the regular sequence Q f = f1 , . . . , fc is permutable, and let f = ci=1 fi . Given a subset T ⊆ [c], recall that Q we use f T to denote the subsequence of f indexed by T , and let fT = i∈T fi . For any such subset T of size |T | = b, let NfaT denote the kernel of the ith differential ∂ a : b ∆a+1 ∆ ∆af T (R) → ∆ f T (R) when a < b, and let Nf T denote the kernel of the augmentation map ∆ ∆bf T (R) → H b (∆ ∆bf T (R)). By Theorem VII.7, we have the following statements. • Nf1T = R/f T . • NfbT fits into an exact sequence f T 0 → NfbT → HfbT (R)fed −→ HfbT (R)nat → 0 • For all values 1 ≤ a < b, there is an exact sequence of the form 0 → NfaT → M S⊆T, |S|=b−a → 0. HfaT −S (R/f S ) → Nfa+1 T The compatibility of the differentials ∂ i with the Fedder actions of each term in the ∆ ∆•f T (R) complex implies that each module of the form NfaT carries an induced Frobenius action. The short exact sequences displayed above may therefore be understood over RhF i, and by Proposition III.13, the long exact sequences that result from applying a functor ΓI (−) consist entirely of RhF i-linear maps. Note that the vanishing hypotheses of the following theorem are automatically satisfied if R/I is Cohen-Macaulay or if c = 1. 96 Theorem VIII.1. Let R be a regular ring of prime characteristic p > 0, let f = f1 , . . . , fc be a permutable regular sequence of codimension c ≥ 1, and let I ⊇ f be an ht(I/f R)+c ideal such that HIi (R) = 0 for ht(I) < i < ht(I) + c. The module HI/f R (R/f R) has Zariski closed support in Spec(R/f R). Proof. For convenience, write t = ht(I/f R) = ht(I) − c. We will make heavy use of the NfaT notation introduced in the preceding discussion. Our first aim is to show that the following three statements hold for all b such that 1 ≤ b ≤ c, (i) For any subset T ⊆ [c] of size |T | = b, and for all a satisfying max(1, 3 − c + b) ≤ a ≤ b (an empty range of values if c ≤ 2), it holds that HIj (NfaT ) = 0 whenever t + c + 2 − a ≤ j ≤ t + 2c − 1 − b. (ii) For any subset T ⊆ [c] of size |T | = b, and for all a satisfying max(1, 2 − c + b) ≤ a ≤ b (an empty range if c = 1) the module HIt+c+1−a (NfaT ) is finitely generated over RhF i. (iii) For any subset T ⊆ [c] of size |T | = b, and for all max(1, 2 − c + b) ≤ a ≤ b (an empty range of values if c = 1), the module HIt+2c−b (NfaT ) has a finite set of associated primes. The proof is by induction on b, beginning with the case b = 1. We will actually start by showing that the statements hold whenever b = a, i.e., for the modules NfbT when |T | = b. This immediately implies the b = 1 case, since Nf1j = R/fj . So, fix 1 ≤ b ≤ c and let T ⊆ [c] be a subset of size |T | = b. Concerning the module NfbT , we have an exact sequence f T 0 → NfbT → HfbT (R)fed −→ HfbT (R)nat → 0 97 From the long exact sequence induced by ΓI (−) along with our vanishing hypothesis HIi (R) = 0 for t + c + 1 ≤ i ≤ t + 2c − 1, it is readily verified that (i) so long as c ≥ 3, HIj (NfbT ) = 0 for t + c + 2 − b ≤ j ≤ t + 2c − 1 − b. (ii) So long as c ≥ 2, HIt+c+1−b (NfbT ) is an RhF i homomorphic image of HIt (Hfc (R)nat ), and is therefore finitely generated over RhF i (see Theorem III.15, and recall that Hfc (R)nat is unit and finitely generated). Finally, (iii) so long as c ≥ 2, HIt+2c−b (R/f T ) is isomorphic to a submodule of HIt+c (Hfc (R)), and hence has a finite set of associated primes (see Theorem III.16). Now take b in the range 2 ≤ b ≤ c, since the case b = 1 is proven. Suppose that the statements (i)–(iii) have been shown for subsets S ⊆ [c] of size |S| < b, and fix T ⊆ [c] any subset of size b. For this set T , we will demonstrate the claims (i)–(iii) about the modules NfaT by a decreasing induction on a. The case a = b has already been shown. Fix a < b and suppose we’ve proven (i)–(iii) for the modules NfrT whenever a < r ≤ b. We will show that the statements hold for the module NfaT using the short exact sequence 0 → NfaT → M S⊆T, |S|=b−a HfaT −S (R/f S ) → Nfa+1 → 0, T To show claim (i), fix j in the range t + c + 2 − a ≤ j ≤ t + 2c − 1 − b and consider the exact sequence ) → HIj (NfaT ) → · · · → HIj−1 (Nfa+1 T M S⊆T, |S|=b−a HIj (HfaT −S (R/f S )) → · · · note that for each subset S ⊆ T of size |S| = b − a, we have HIj (HfaT −S (R/f S )) = HIj+a (R/f S ) where R/f S = Nf1S . The inequality t + c + 1 ≤ j + a ≤ t + 2c − 1 − (b − a) gives us vanishing HIj+a (Nf1S ) = 0 for each subset S ⊆ T of size b − a by induction, since 98 |S| < |T |. Since t+c+2−(a+1) ≤ j −1 ≤ t+2c−1−b, we also have HIj−1 (Nfa+1 )=0 T by induction, since a + 1 > a. The vanishing of HIj (NfaT ) follows at once. For claim (ii), the relevant exact sequence is · · · → HIt+c−a (Nfa+1 ) → HIt+c+1−a (NfaT ) → T M S⊆T, |S|=b−a HIt+c+1 (Nf1S ) → · · · Since a ≥ max(1, 2 − c + b), we have that 1 ≥ 3 − c + (b − a), so claim (i) for the module Nf1S implies that HIt+c+1 (Nf1S ) = 0. Thus, HIt+c+1−a (NfaT ) is a ) by some (RhF i-stable) submodule. As RhF i-modules, quotient of HIt+c−a (Nfa+1 T t+c+1−(a+1) HI (Nfa+1 ) is finitely generated by induction on a, so the same is true of its T image HIt+c+1−a (NfaT ). To show claim (iii), consider the exact sequence ) → HIt+2c−b (NfaT ) → · · · → HIt+2c−b−1 (Nfa+1 T M S⊆T, |S|=b−a HIt+2c−b+a (NS1 ) → · · · The condition a ≥ 2 − c + b implies that a + 1 ≥ 3 − c + b, so claim (i) for the shows that HIt+2c−1−b (Nfa+1 ) = 0. Thus HIt+2c−b (NfaT ) is isomorphic to module Nfa+1 T T t+2c−(b−a) a submodule of a direct sum of modules of the form HI t+2c−(b−a) subset of size b − a. Since 2 − c + (b − a) ≤ 1, each HI (NS1 ), for S ⊆ T a (NS1 ) has a finite set of associated primes. The induction is complete and the claims (i)–(iii) have been demonstrated. We are now ready to show that HIt+c (R/f R) = HIt+c (Nf1R ) has closed support. This is known in the case c = 1 (see Theorem V.2). For c ≥ 2, there is an exact sequence · · · → HIt+c−1 (Nf2R ) → HIt+c (Nf1R ) → M S⊆T, |S|=1 HIt+c+1 (Nf1S ) → · · · Since 2 = max(1, 2 + c − c), the module HIt+c+1−2 (Nf2R ) is finitely generated over RhF i, and thus, any RhF i homomorphic image will have closed support by Theorem 99 t+2c−(c−1) III.1. Additionally, 1 ≥ 2 − c + (c − 1), so HI (Nf1S ) (for each singleton set S ⊆ T ) has a finite set of associated primes. The claim about the support of HIt+c (Nf1R ) follows at once. 8.0.1 Nesting of Supports In this subsection, we remark that the support of the local cohomology of a complete intersection cut out by a regular sequence f1 , . . . , fc has a curious nesting property in relation to the supports of the local cohomologies of the complete intersections defined by subsequences of f1 , . . . , fc . Theorem VIII.2. Let R be a Cohen-Macaulay ring of prime characteristic p > 0, let f = f1 , . . . , fc be a permutable regular sequence, and let I be an ideal containing f R. For T ⊆ [c], let f T be the ideal generated by the subsequence of f1 , . . . , fc indexed by T . For any δ ≥ 0, ht(I/f T )+δ Supp HI/f T ht(I/f R)+δ (R/f T ) ⊆ Supp HI/f R (R/f R) In particular, if h = f1 , . . . , fc , g1 , . . . , gt is a maximal length regular sequence in I and if h is permutable, then ht(I)+δ Supp HI ht(I/f R)+δ (R) ⊆ Supp HI/f R δ (R/h) (R/f R) ⊆ Supp HI/h Proof. Let h = f1 , . . . , fc , g1 , . . . , gt be a maximal length regular sequence contained in I. Via the obvious inclusions T ⊆ [c] ⊆ [c + t], we may write f R = h[c] and f T = hT . Let b = |T |. Observe that ht(I/f T )+δ HI/f T   (R/h (R/f T ) = HIδ Hht+c−b ) , T [c+t]−T and that ht(I/f R)+δ HI/f R   (R/f R) = HIδ Hht [c+t]−[c] (R/h[c] ) . 100 Let A = R/hT , and consider the ring R/h[c] as being cut out from A by a regular sequence of length c − b (indexed by the set [c] − T ). The result follows by a straightforward induction using the following lemma. Lemma VIII.3. Let A be a Noetherian ring, let f1 , · · · , ft , h ∈ A be a permutable regular sequence, and let f = f1 , . . . , ft . Let I be an ideal containing f , h. Then for any δ ≥ 0, δ t Supp HIδ (Hft+1 ,h (R)) ⊆ Supp HI (Hf (R/h)) Proof. Suppose that, after replacing R by RP for some P ∈ Spec(R), we obtain vanishing HIδ (Hft (R/h)) = 0. We would like to show that HIδ (Hft+1 ,h (R)) = 0, where we recall that 1 t H t (R/hn ) Hft+1 ,h (R) = Hf (Hh (R)) = lim −→ f n It would therefore suffice to show that HIδ (Hft (R/hn )) = 0 for all n ≥ 1. By hypoth- esis, this is true when n = 1, so fix n > 1 and suppose for the sake of induction that HIδ (Hft (R/hj )) = 0 for all j < n. Note that (hn :R h) = hn−1 R, i.e., the annihilator of h in R/hn R is hn−1 R/hn R, isomorphic as an R-module to R/hR. Mapping R/hR onto the image of hn−1 , we get hn−1 0 → R/h −−−→ R/hn → R/hn−1 → 0 inducing the exact sequence · · · → Hft−1 (R/hn−1 ) → Hft (R/h) → Hft (R/hn ) → Hft (R/hn−1 ) → Hft+1 (R/h) → · · · The arithmetic rank of f is t, so Hft+1 (R/h) = 0. Since h is permutable, the sequence hn−1 , f1 , · · · , ft is a regular, and consequently, f1 , . . . , ft is an R/hn−1 -regular sequence. Since depthf (R/hn−1 ) = t, we get Hht−1 (R/hn−1 ) = 0. 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