We shall generalize the theory of primary decomposition and associated prime ideals of finitely g... more We shall generalize the theory of primary decomposition and associated prime ideals of finitely generated modules over a noetherian ring to general objects in an abelian R-category where R is a noetherian commutative ring. See, for example, [3], or as general references of this section. Let C be a category, where we denote by Ob(C) the object class and by C(X, Y ) the set of morphisms for objects X, Y ∈ Ob(C). By definition, the composition of morphisms in C satisfies the associative law; (f g)h = f (gh), and there is the identity morphism 1 X for any X ∈ Ob(C). Recall that C is called a preadditive category provided C(X, Y ) is an abelian group for X, Y ∈ Ob(C) and the composition of morphisms is bilinear, i.e. f (g + h) = f g + f h, (g + h)f ′ = gf ′ + hf ′ and moreover there exists the null object 0 in C. An additive category is, by definition, a preadditive category with finite coproducts. An additive category C is called an abelian category if the kernel and the cokernel exist for any morphism f and moreover the equality Cok(ker(f )) = Ker(cok(f )) holds. We recall how to construct an ideal quotient of a category. 2.1. Localization. ([4, 7.1],[8, 2.1],[3, chapter 1.3]) Let C be an additive category and let S be a collection of morphisms in C. We say that S is a
We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of... more We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type (An) are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.
Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra... more Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B = A[X 1 , . . . , Xn] is a polynomial extension of A, where X 1 , . . . , Xn are variables of positive degrees; or (b) A is a divided power DG R-algebra and B = A X 1 , . . . , Xn is a free extension of A obtained by adjunction of variables X 1 , . . . , Xn of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A → B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that Ext i B (N, N ) = 0 for all i 1, then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.
Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W wi... more Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then γ W coincides with the right derived functor RΓ W of the section functor Γ W with support in W . We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γ W with W being an arbitrary subset.
Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and n... more Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and non-invertible elements. If X (resp. Y) is a matrix factorization of / (resp. g), then we can construct the matrix factorization X §> Y of /-+• g over K [[xiyX2, -> ,x r ,yi,y2, -,ys]]i which we call the tensor product of X and y. After showing several general properties of tensor products, we will prove theorems which give bounds for the number of indecomposable components in the direct decomposition of X < §> Y.
The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of thi... more The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of this paper is to explicitly describe the obstruction to naïve liftability along extensions A → B of DG algebras, where B is projective as an underlying graded A-module. In particular, we give an explicit description of a DG B-module homomorphism which defines the obstruction to naïve liftability of a semifree DG B-module N as a certain cohomology class in Ext 1 B (N, N ⊗ B J), where J is the diagonal ideal. Our results on the obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naïve lifting property.
We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to ... more We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N if and only if there is a short exact sequence of finitely generated left R-modules 0 → Z ( φ ψ ) --→ M ⊕ Z → N → 0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory.
Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degre... more Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degree n, and let N be a semi-free DG B-module that is assumed to be bounded below as a graded module. We prove in this paper that N is liftable to A if Ext n+1 B (N, N ) = 0. Furthermore such a lifting is unique up to DG isomorphisms if Ext n B (N, N ) = 0.
Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with... more Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with cosupports in arbitrary subsets W of Spec R; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors λ W , including an explicit way to calculate λ W by the notion of Čech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat R-module is at most the Krull dimension of R. As another application, it is possible to give a functorial way to replace complexes of flat R-modules or complexes of finitely generated R-modules by complexes of pure-injective R-modules.
Proceedings of the American Mathematical Society, 1996
Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted b... more Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by δ n R (M) for each module M and for each integer n. We propose a conjecture asking if δ n R (R/m) = 0 for any positive integers n and. We prove that this is true provided the associated graded ring of R has depth not less than dim R − 1. Furthermore we show that there are only finitely many possibilities for a pair of positive integers (n,) for which δ n R (R/m) > 0.
Transactions of the American Mathematical Society, Sep 1, 1988
The fundamental module E of a normal local domain (R,m) of dimension 2 is defined by the nonsplit... more The fundamental module E of a normal local domain (R,m) of dimension 2 is defined by the nonsplit exact sequence 0-* K -* E -* xa-» 0, where K is the canonical module of R. We prove that, if R is complete with R/m ~ C, then E is decomposable if and only if R is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed. Let (R,m) be a normal local domain of dimension 2 which possesses the canonical module K. Let C(R) denote the category of finitely generated reflexive B-modules. Note that a finitely generated B-module is an object in C(R) if and only if it is a maximal Cohen-Macaulay module over R. By definition, K is a reflexive module of rank 1 and it satisfies Ext^(B/m, K) ~ R/m. (See Herzog and Kunz [8] for the details.) We denote the duality with respect to R (resp. K) by * (resp. '), that is, ( )* = HomR( ,R) and ( )' = HomR( ,K). Remark that a finitely generated B-module M lies in C(B) if and only if M** =• M, or
As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-... more As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another. R corresponding to an R-module M. Then we say that M degenerates to N if O(N) is 0 2000 Mathematics Subject Classification. Primary 13C14; Secondary 13D10.
Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. Tha... more Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.
We propose to define the notion of abstract local cohomology functors. The ordinary local cohomol... more We propose to define the notion of abstract local cohomology functors. The ordinary local cohomology functor RΓ I with support in the closed subset defined by an ideal I and the generalized local cohomology functor RΓ I,J defined in [16] are characterized as elements of the set of all the abstract local cohomology functors.
We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite va... more We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.
We shall generalize the theory of primary decomposition and associated prime ideals of finitely g... more We shall generalize the theory of primary decomposition and associated prime ideals of finitely generated modules over a noetherian ring to general objects in an abelian R-category where R is a noetherian commutative ring. See, for example, [3], or as general references of this section. Let C be a category, where we denote by Ob(C) the object class and by C(X, Y ) the set of morphisms for objects X, Y ∈ Ob(C). By definition, the composition of morphisms in C satisfies the associative law; (f g)h = f (gh), and there is the identity morphism 1 X for any X ∈ Ob(C). Recall that C is called a preadditive category provided C(X, Y ) is an abelian group for X, Y ∈ Ob(C) and the composition of morphisms is bilinear, i.e. f (g + h) = f g + f h, (g + h)f ′ = gf ′ + hf ′ and moreover there exists the null object 0 in C. An additive category is, by definition, a preadditive category with finite coproducts. An additive category C is called an abelian category if the kernel and the cokernel exist for any morphism f and moreover the equality Cok(ker(f )) = Ker(cok(f )) holds. We recall how to construct an ideal quotient of a category. 2.1. Localization. ([4, 7.1],[8, 2.1],[3, chapter 1.3]) Let C be an additive category and let S be a collection of morphisms in C. We say that S is a
We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of... more We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type (An) are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.
Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra... more Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B = A[X 1 , . . . , Xn] is a polynomial extension of A, where X 1 , . . . , Xn are variables of positive degrees; or (b) A is a divided power DG R-algebra and B = A X 1 , . . . , Xn is a free extension of A obtained by adjunction of variables X 1 , . . . , Xn of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A → B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that Ext i B (N, N ) = 0 for all i 1, then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.
Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W wi... more Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then γ W coincides with the right derived functor RΓ W of the section functor Γ W with support in W . We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γ W with W being an arbitrary subset.
Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and n... more Let K be a field and let / 6 K[[xι,x 2 , . ,x r ]] and g 6 #[[2/1,2/2, , y s ]] be non-zero and non-invertible elements. If X (resp. Y) is a matrix factorization of / (resp. g), then we can construct the matrix factorization X §> Y of /-+• g over K [[xiyX2, -> ,x r ,yi,y2, -,ys]]i which we call the tensor product of X and y. After showing several general properties of tensor products, we will prove theorems which give bounds for the number of indecomposable components in the direct decomposition of X < §> Y.
The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of thi... more The notion of naïve liftability of DG modules is introduced in [10] and . The main purpose of this paper is to explicitly describe the obstruction to naïve liftability along extensions A → B of DG algebras, where B is projective as an underlying graded A-module. In particular, we give an explicit description of a DG B-module homomorphism which defines the obstruction to naïve liftability of a semifree DG B-module N as a certain cohomology class in Ext 1 B (N, N ⊗ B J), where J is the diagonal ideal. Our results on the obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naïve lifting property.
We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to ... more We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N if and only if there is a short exact sequence of finitely generated left R-modules 0 → Z ( φ ψ ) --→ M ⊕ Z → N → 0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory.
Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degre... more Let B = A X|dX = t be an extended DG algebra by the adjunction of variable of positive even degree n, and let N be a semi-free DG B-module that is assumed to be bounded below as a graded module. We prove in this paper that N is liftable to A if Ext n+1 B (N, N ) = 0. Furthermore such a lifting is unique up to DG isomorphisms if Ext n B (N, N ) = 0.
Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with... more Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with cosupports in arbitrary subsets W of Spec R; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors λ W , including an explicit way to calculate λ W by the notion of Čech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat R-module is at most the Krull dimension of R. As another application, it is possible to give a functorial way to replace complexes of flat R-modules or complexes of finitely generated R-modules by complexes of pure-injective R-modules.
Proceedings of the American Mathematical Society, 1996
Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted b... more Let (R, m) be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by δ n R (M) for each module M and for each integer n. We propose a conjecture asking if δ n R (R/m) = 0 for any positive integers n and. We prove that this is true provided the associated graded ring of R has depth not less than dim R − 1. Furthermore we show that there are only finitely many possibilities for a pair of positive integers (n,) for which δ n R (R/m) > 0.
Transactions of the American Mathematical Society, Sep 1, 1988
The fundamental module E of a normal local domain (R,m) of dimension 2 is defined by the nonsplit... more The fundamental module E of a normal local domain (R,m) of dimension 2 is defined by the nonsplit exact sequence 0-* K -* E -* xa-» 0, where K is the canonical module of R. We prove that, if R is complete with R/m ~ C, then E is decomposable if and only if R is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed. Let (R,m) be a normal local domain of dimension 2 which possesses the canonical module K. Let C(R) denote the category of finitely generated reflexive B-modules. Note that a finitely generated B-module is an object in C(R) if and only if it is a maximal Cohen-Macaulay module over R. By definition, K is a reflexive module of rank 1 and it satisfies Ext^(B/m, K) ~ R/m. (See Herzog and Kunz [8] for the details.) We denote the duality with respect to R (resp. K) by * (resp. '), that is, ( )* = HomR( ,R) and ( )' = HomR( ,K). Remark that a finitely generated B-module M lies in C(B) if and only if M** =• M, or
As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-... more As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another. R corresponding to an R-module M. Then we say that M degenerates to N if O(N) is 0 2000 Mathematics Subject Classification. Primary 13C14; Secondary 13D10.
Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. Tha... more Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.
We propose to define the notion of abstract local cohomology functors. The ordinary local cohomol... more We propose to define the notion of abstract local cohomology functors. The ordinary local cohomology functor RΓ I with support in the closed subset defined by an ideal I and the generalized local cohomology functor RΓ I,J defined in [16] are characterized as elements of the set of all the abstract local cohomology functors.
We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite va... more We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.
Uploads
Papers by Yuji Yoshino