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We prove two theorems on the vanishing of Ext over commutative Noetherian local rings. Our first theorem shows that there are no Burch ideals which are rigid over non-regular local domains. Our second theorem reformulates a conjecture of Huneke-Wiegand in terms of the vanishing of Ext, and highlights its relation with the celebrated Auslander-Reiten conjecture. We also discuss several consequences of our results, for example, about the rigidity of the Frobenius endomorphism in prime characteristic p and a generalization of a result of Araya., Comment: 12 pages. Souvik Dey is a new coauthor. One of our main theorems (Theorem 1.2) is improved. Outcomes about the rigidity of the Frobenius endomorphism in prime characteristic are added

In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show that the conjecture is true for two periodic modules. We also make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension. Along the way we investigate the interaction between the vanishing of Tate (co)homology and torsion in tensor products of modules, and obtain new results that are of independent interest., Comment: 18 pages

In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersection rings. Along the way, we also give a splitting criteria for vector bundles on smooth complete intersections., Comment: Revised version, some typos corrected

In this paper we revisit an example of Celikbas and Takahashi concerning the reflexivity of tensor products of modules. We study Tor-rigidity and the Hochster--Huneke graph with vertices consisting of minimal prime ideals, and determine a condition with which the aforementioned example cannot occur. Our result, in particular, corroborates the Second Rigidity Theorem of Huneke and Wiegand., Comment: 8 pages

In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory, and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules. In the second part of the paper we consider a depth equality, called the depth formula, which has been initially introduced by Auslander and developed further by Huneke and Wiegand. As an application of our main result, we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension., Comment: Final version; to appear in Acta Math. Sin. (Engl. Ser.)

In this paper, we introduce and study the notion of linkage of modules by reflexive homomorphisms. This notion unifies and generalizes several known concepts of linkage of modules and enables us to study the theory of linkage of modules over Cohen-Macaulay rings rather than the more restrictive Gorenstein rings. It is shown that several known results for Gorenstein linkage are still true in the more general case of module linkage over Cohen-Macaulay rings. We also introduce the notion of colinkage of modules and establish an adjoint equivalence between the linked and colinked modules., Comment: 43 pages,Journal of the Mathematical Society of Japan, to appear

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 pages

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.

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.

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/s10231-020-01019-9

Let R be a commutative Noetherian local ring with residue field k. Let X be a resolving subcategory of finitely generated R-modules. This paper mainly studies when X contains k or consists of totally reflexive modules. It is proved that X does so if X is closed under cosyzygies. A conjecture of Dao and Takahashi is also shown to hold in several cases.

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.

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:1407.6544

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. In particular, our main result recovers theorems of Araya, and Ono and Yoshino simultaneously., Comment: 6 pages

The depth of tensor product of modules over a Gorenstein local ring is studied. For finitely generated modules M and N over a Gorenstein local ring R, under some assumptions on the vanishing of finite number of Tate and relative homology modules, the depth($M\otimes N$) is determined in terms of the depth(M) and depth(N)., Comment: The main theorem has been improved. Some new results have been added

A celebrated conjecture of Auslander and Reiten claims that a finitely generated module $M$ that has no extensions with $M\oplus \Lambda$ over an Artin algebra $\Lambda$ must be projective. This conjecture is widely open in general, even for modules over commutative Noetherian local rings. Over such rings, we prove that a large class of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Along the way we obtain a new characterization of regularity in terms of the injective dimensions of certain ideals., Comment: 10 pages

For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depth M + depth N= depth(M\otimes N) +depth R$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology $\hat{Tor}_{i}^{R}(M,N)$ vanishes for all $i\in\mathbb{Z}$. We establish the same conclusion under weaker hypotheses: if $M$ and $N$ are $\mathcal{G}$-relative Tor-independent, then the vanishing of $\hat{Tor}_{i}^{R}(M,N)$ for all $i\le 0$ is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of $\mathcal{G}$-relative homology., Comment: To appear in Journal of Algebra

We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$., Comment: Revised version, Lemma 3.3 has been added

Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring $R$, if a Cohen-Macaulay $R$-module $M$ of grade $g$ is linked to an $R$-module $N$ by a Gorenstein ideal $c$, such that $Ass_R(M)\cap Ass_R(N)=\emptyset$, then $M\otimes_RN$ is isomorphic to direct sum of copies of $R/a$, where $a$ is a Gorenstein ideal of $R$ of grade $g+1$. We give a criterion for the depth of a local ring $(R,m,k)$ in terms of the homological dimensions of the modules linked to the syzygies of the residue field $k$. As a result we characterize a local ring $(R,m,k)$ in terms of the homological dimensions of the modules linked to the syzygies of $k$., Comment: To appear in Journal of Commutative Algebra

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:1407.6544

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 Algebra

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective dimension of the maximal ideal $\fm$ of $R$ is finite. Furthermore we prove that $R$ must be regular if a single $\Ext_{R}^{n}(I,J)$ vanishes for some integrally closed $\fm$-primary ideals $I$ and $J$ of $R$ and for some integer $n\geq \dim(R)$. Along the way we observe that local rings that admit maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay., Comment: Major revision; title and abstract has been edited, new sections included

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 Algebra

Let R be a complete intersection ring and let M and N be R-modules. It is shown that the vanishing of Ext^i_R(M,N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n-1. We also estimate the complete intersection dimension of the dual of M, in terms of vanishing of the cohomology modules, Ext^i_R(M,N)., Comment: 11 pages

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 pages

For a horizontally linked module, over a commutative semiperfect Noetherian ring $R$, the connections of its invariants reduced grade, Gorenstein dimension and depth are studied. It is shown that under certain conditions the depth of a horizontally linked module is equal to the reduced grade of its linked module. The connection of the Serre condition $(S_n)$ on an $R$--module of finite Gorenstein dimension with the vanishing of the local cohomology groups of its linked module is discussed., Comment: 21 pages, Theorem 2.7 is corrected

Inspired by the works in linkage theory of ideals, the concept of sliding depth of extension modules is defined to prove the Cohen-Macaulyness of linked module if the base ring is merely Cohen-Macaulay. Some relations between this new condition and other module-theory conditions such as G-dimension and sequentially Cohen-Macaulay are established. By the way several already known theorems in linkage theory are improved or recovered by new approaches., Comment: 12 Pages