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If M is a nonzero finitely generated module over a commutative Noetherian local ring R such that M has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that M has finite projective dimension, and hence a result of Foxby implies that R is Gorenstein. We prove that the same conclusion holds for certain nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general. Along the way, we also prove new results, independent of the reducing dimensions, concerning modules of finite Gorenstein dimension. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove that each ideal from such class satisfies the conclusion of the conjecture in question. We also study the relation between the class of Burch ideals and that of the ideals we define, and construct several examples that corroborate our results. [ABSTRACT FROM AUTHOR]

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 ⊗ R N is maximal Cohen–Macaulay, then M is maximal Cohen–Macaulay if and only if N is maximal Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand. [ABSTRACT FROM AUTHOR]

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and Şega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of the vanishing of Tor. This has led to the notion of Tor-persistent rings. Our main result shows that the class of Tor-persistent local rings is closed under a number of standard procedures in ring theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules. [ABSTRACT FROM AUTHOR]

In 2007 Huneke and Wiegand announced in an erratum that one of the conclusions of their depth formula theorem is flawed due to an incorrect convention for the depth of the zero module. Since then, the deleted claim has remained unresolved. In this paper we give examples to prove that the deleted claim is false, in general. Moreover, we point out several places in the literature which relied upon this deleted claim or the initial argument from 2007. [ABSTRACT FROM AUTHOR]

In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Froberg concerning Arf numerical semigroup rings. [ABSTRACT FROM AUTHOR]

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the longstanding 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. [ABSTRACT FROM AUTHOR]

We obtain various characterizations of commutative Noetherian local rings (R,m) in terms of homological dimensions of certain finitely generated modules. Our argument has a series of consequences in different directions. For example, we establish that R is Gorenstein if the Gorenstein injective dimension of the maximal ideal m of R is finite. Moreover, we prove that R must be regular if a single ExtnR(I,J) vanishes for some integrally closed m-primary ideals I and J of R and for some positive integer n. [ABSTRACT FROM AUTHOR]

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 φnR is not maximal Cohen–Macaulay provided that M has rank and n≫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 φnR⊗RφnR has torsion for all n≫0. 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⊗RM is torsion-free. [ABSTRACT FROM AUTHOR]

We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings--among them Gorenstein rings, the original domain of Tate homology. Closely related to Gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone. Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced. [ABSTRACT FROM AUTHOR]

We compare two generalizations of Tate homology to the realm of associative rings: stable homology and the J-completion of Tor, also known as complete homology. For finitely generated modules, we show that the two theories agree over Artin algebras and over commutative noetherian rings that are Gorenstein, or local and complete. [ABSTRACT FROM AUTHOR]

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $$(R,\mathfrak m)$$ is Gorenstein if and only if it admits an integrally closed $$\mathfrak m$$ -primary ideal of finite Gorenstein dimension. This is accomplished through a detailed study of certain test complexes. Along the way we construct such a test complex that detect finiteness of Gorenstein dimension, but not that of projective dimension. [ABSTRACT FROM AUTHOR]

We study the homological behavior ofmodules over local rings modulo exact zero-divisors.We obtain new results which are in some sense "opposite" to those known for modules over local rings modulo regular elements. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian local ring. Assume that R has a pair { x, y} of exact zerodivisors such that dim R/( x, y) ≥ 2 and all totally reflexive R/( x)-modules are free. We show that the first and second Brauer-Thrall type theorems hold for the category of totally reflexive R-modules. More precisely, we prove that, for infinitely many integers n, there exists an indecomposable totally reflexive R-module of multiplicity n. Moreover, if the residue field of R is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive R-modules of multiplicity n. [ABSTRACT FROM AUTHOR]

We study homological properties of testmodules that are, in principle, modules that detect finite homological dimensions. Themain outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that if a commutative Noetherian complete local ring R admits a test module of finite Gorenstein dimension, then R is Gorenstein. [ABSTRACT FROM AUTHOR]

We give an application of the New Intersection Theorem and prove the following: let $$R$$ be a local complete intersection ring of codimension $$c$$ and let $$M$$ and $$N$$ be nonzero finitely generated $$R$$ -modules. Assume $$n$$ is a nonnegative integer and that the tensor product $$M\otimes _{R}N$$ is an $$(n+c)$$ th syzygy of some finitely generated $$R$$ -module. If $${{\mathrm{Tor}}}^{R}_{>0}(M,N)=0$$ , then both $$M$$ and $$N$$ are $$n$$ th syzygies of some finitely generated $$R$$ -modules. [ABSTRACT FROM AUTHOR]

We define a variant of Hochster's θ pairing and prove that it is constant in flat families of modules over hypersurfaces with isolated singularities. As a consequence, we show that the θ pairing factors through the Grothendieck group modulo algebraic equivalence. Moreover, our result allows us, in certain situations, to translate the properties of the θ pairing in characteristic zero [established in Moore et al. (Adv Math, 226(2):1692-1714, ) and Polishchuk and Vaintrob (Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations. Preprint, arXiv:1002, )] to the characteristic p setting. We also give an application of our result to the rigidity of Tor over hypersurfaces. [ABSTRACT FROM AUTHOR]

Let R be a local ring, and let M and N be finitely generated R-modules such that M has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules $${{\rm {Ext}}_R^i(M,N)}$$ which generalizes Buchweitz's notion of the Herbrand difference. We exploit this pairing to examine the number of consecutive vanishing of $${{\rm {Ext}}_R^i(M,N)}$$ needed to ensure that $${{\rm {Ext}}_R^i(M,N)=0}$$ for all $${i\gg 0}$$ . Our results recover and improve on most of the known bounds in the literature, especially when R has dimension at most two. [ABSTRACT FROM AUTHOR]