Linkage of finite G_C-dimension modules

Let R be a semiperfect commutative Noetherian ring and C a semidualizing R-module. We study the theory of linkage for modules of finite G_C-dimension. For a horizontally linked R-module M of finite G_C-dimension, the connection of the Serre condition (S_n) with the vanishing of certain relative cohomology modules of its linked module is discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1407.654

A note on the depth formula and vanishing of cohomology

It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of cohomology of a module of finite complete intersection dimension.Comment: 12 page

Maximal Cohen-Macaulay tensor products

In this paper we are concerned with the following question: if the tensor product of finitely generated modules $M$ and $N$ over a local complete intersection domain is maximal Cohen-Macaulay, then must $M$ or $N$ be a maximal Cohen-Macaulay? Celebrated results of Auslander, Lichtenbaum, and Huneke and Wiegand, yield affirmative answers to the question when the ring considered has codimension zero or one, but the question is very much open for complete intersection domains that have codimension at least two, even open for those that are one-dimensional, or isolated singularities. Our argument exploits Tor-rigidity and proves the following, which seems to give a new perspective to the aforementioned question: if $R$ is a complete intersection ring which is an isolated singularity such that dim($R$) > codim($R$), and the tensor product $M\otimes_R N$ is maximal Cohen-Macaulay, then $M$ is maximal Cohen-Macaulay if and only if $N$ is maximal Cohen-Macaulay.Comment: This is a pre-print of an article published in Annali di Matematica. The final authenticated version is available online at: https://doi.org/10.1007/s10231-020-01019-

Two generalizations of Auslander-Reiten duality and applications

This paper extends Auslander-Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander-Reiten conjecture

Linkage of modules with respect to a semidualizing module

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1507.00036, arXiv:1407.654

Linkage of modules and the Serre conditions

Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing $R$--module. The connection of the Serre condition $(S_n)$ on a horizontally linked $R$-module of finite \gc-dimension with the vanishing of certain cohomology modules of its linked module is discussed. As a consequence, it is shown that under some conditions Cohen-Macaulayness is preserved under horizontally linkage.Comment: 21 pages, final version will appear in Journal of Pure and Applied Algebr

On modules with reducible complexity

In this paper we generalize a result, concerning a depth equality over local rings, proved independently by Araya and Yoshino, and Iyengar. Our result exploits complexity, a concept which was initially defined by Alperin for finitely generated modules over group algebras, introduced and studied in local algebra by Avramov, and subsequently further developed by Bergh.Comment: 8 page

Bounds on depth of tensor products of modules

Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. We employ Auslander's transpose in the study of the vanishing of Tor and obtain useful bounds for the depth of the tensor product $M\otimes_{R}N$. An application of our main argument shows that, if $M$ is locally free on the the punctured spectrum of $R$, then either \depth(M\otimes_{R}N)\geq \depth(M)+\depth(N)-\depth(R), or \depth(M\otimes_{R}N)\leq \cod(R). Along the way we generalize an important theorem of D. A. Jorgensen and determine the number of consecutive vanishing of \Tor_i^R(M,N) required to ensure the vanishing of all higher \Tor_i^R(M,N).Comment: Grant information included. To appear in Journal of Pure and Applied Algebr

Tensoring with the Frobenius endomorphism

Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has positive dimension and prime characteristic. Li proved that the tensor product of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism ${}^{\varphi^n}\!R$ is not maximal Cohen-Macaulay provided that $M$ has rank and $n\gg 0$. We replace the rank hypothesis with the weaker assumption that $M$ is locally free on the minimal prime ideals of $R$. As a consequence, we obtain, if $R$ is a one-dimensional non-regular complete reduced local ring that has a perfect residue field and prime characteristic, then ${}^{\varphi^n}\!R \otimes_{R}{}^{\varphi^n}\!R$ has torsion for all $n\gg0$. This property of the Frobenius endomorphism came as a surprise to us since, over such rings $R$, there exist non-free modules $M$ such that $M\otimes_{R}M$ is torsion-free

Notes on linkage of modules

Let R be a Cohen-Macaulay local ring. It is shown that under some mild conditions, the Cohen-Macaulayness property is preserved under linkage. We also study the connection of (S_n) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module