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.
Let R be a Gorenstein local domain of dimension one. We show that a nonfree maximal Cohen--Macaulay R-module M possessing more than one nonfree indecomposable summand in the middle term of the almost split sequence ending in M has a nonvanishing self extension. In other words, we show that the Huneke--Wiegand conjecture is affirmative for such R-modules M.
We let R be a one-dimensional graded complete intersection, satisfying certain degree conditions which are satisfied whenever R is a numerical semigroup ring of embedding dimension at least three. We show that a graded maximal Cohen-Macaulay R-module M satisfies the Huneke-Wiegand Conjecture provided there exists an Auslander-Reiten sequence ending in M whose middle term has at least two nonfree direct summands.
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.
In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove thateach 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.
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 analog of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question …
In this paper we explore consequences of the vanishing of ${\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline x)$ is a non-trivial fiber product ring, for some regular sequence $\underline x$ of $R$. Equivalently, the maximal ideal of $R/(\underline x)$ decomposes as a direct sum of two nonzero ideals. Gorenstein quasi-fiber product rings are AB-rings and are Ext-bounded. We show in Theorem 3.31 that quasi-fiber product rings satisfy a sharpened form of the Auslander-Reiten Conjecture. We also make some observati…
We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup rings.
Let $R$ be a Cohen-Macaulay local ring of dimension one with a canonical module $\rm{K_R}$. Let $I$ be a faithful ideal of $R$. We explore the problem of when $I \otimes_RI^{\vee}$ is torsionfree, where $I^{\vee} = \operatorname{Hom_R(I, \rm{K_R})}$. We prove that if $R$ has multiplicity at most $6$, then $I$ is isomorphic to $R$ or $\rm{K_R}$ as an $R$-module, once $I\otimes_RI^{\vee}$ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher dimensional assertion is also discussed.
Huneke and Wiegand conjectured that, if $M$ is a finitely generated, non-free, torsion-free module with rank over a one-dimensional Cohen-Macaulay local ring $R$, then the tensor product of $M$ with its algebraic dual has torsion. This conjecture, if $R$ is Gorenstein, is a special case of a celebrated conjecture of Auslander and Reiten on the vanishing of self extensions that stems from the representation theory of finite-dimensional algebras. If $R$ is a one-dimensional Cohen-Macaulay ring such that $R=S/(f)$ for some local ring $(S, \mathfrak{n})$, and a non zero-divisor $f \in \mathfrak{n}…
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.
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Under various hypotheses, it is proved that the center of $\mbox{End}_R(M)$ coincides with the endomorphism ring of the trace ideal of $M$. These results are exploited to establish results for balanced and rigid modules, and to settle certain cases of a conjecture of Huneke and Wiegand.
In this paper we are concerned with the vanishing of $\textnormal{Tor}$ over complete intersection rings. Building on results of C. Huneke, D. Jorgensen and R. Wiegand, and, more recently, H. Dao, we obtain new results showing that good depth properties on the $R$-modules $M$, $N$ and $M\otimes_RN$ force the vanishing of $\textnormal{Tor}^{R}_{i}(M,N)$ for all $i\geq 1$.
Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when M tensor N has torsion. We also give constructive formulas for producing a module in the isomorphism class of the torsion submodule of M tensor N. In some cases we determine bounds on the length and minimal number of generators of this module. We focus on the case where R is a numerical semigroup ring with the goal of making progress on the Huneke-Wiegand Conjecture.
We establish new results on (co)homology vanishing and Ext-Tor dualities, and derive a number of freeness criteria for finite modules over Cohen-Macaulay local rings. In the main application, we settle the long-standing Auslander-Reiten conjecture for the class of Cohen-Macaulay Burch rings, among other results toward this and related problems, e.g., the Tachikawa and Huneke-Wiegand conjectures. We also derive results on further topics of interest such as Cohen-Macaulayness of tensor products and Tor-independence, and inspired by a paper of Huneke and Leuschke we obtain characterizations of wh…
In this paper we exploit properties of Dao's eta-pairing as well as techniques of Huneke, Jorgensen, and Wiegand to study the vanishing of Tor_i(M,N) of finitely generated modules M, N over complete intersections. We prove vanishing of Tor_i(M, N) for all positive integers i under depth conditions on M, N, and their tensor product. Our arguments improve a result of Dao and establish a new connection between the vanishing of Tor and the depth of tensor products.
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 intersecti…
Tensor products usually have nonzero torsion. This is a central theme of Auslander's paper "Modules over unramified regular local rings"; the theme continues in the work of Huneke and Wiegand. The main focus in this note is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that its tensor product with any module M has torsion, unless M is very special. Important examples of such modules N are the Frobenius power of a ring, viewed as a module over itself, that is a complete intersection domain of positive characteristic.
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 condi…
Let $R$ be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules $(M,N)$ over $R$ we study applications of rigidity of $\Tor^R(M,N)$, based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connections and applications to projective hypersurfaces such as intersection dimension of subvarieties and cohomological criterion for splitting of vector bundles are discussed.
We introduce the notion of Burch submodules and weakly $\mathfrak m$-full submodules of modules over local rings and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test property. We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or $M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-full in $X$, is 1-Tor rigid and a test module provided that $X$ is faithful (and $X/M$ has finite length when $M$ is weakly $\mathfrak m$-full). As an application, we …
The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, García-Sánchez and Leamer construct a monoid S_Γ^s whose elements correspond to arithmetic sequences in a numerical monoid Γof step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and inves…
We show that under suitable hypothesis (which are sharp in certain sense) that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power, generalizing a result of Huneke and Swanson in dimension two. We also prove a version of this in the singular setting, which we show to be intimately related to the problem of finding global sections of ample line bundles on projective varieties. In particular, we show that a graded analog of our formula for core would imply a remarkable conjecture of Kawamata predicting that every a…
