On the vanishing of theta invariant and a conjecture of Huneke and Wiegand

TL;DR

This paper proves a special case of Huneke and Wiegand's conjecture on torsion in tensor products of modules over certain Cohen-Macaulay rings, using Hochster's theta invariant.

Contribution

It establishes the conjecture for modules with finite projective dimension over a specific class of Cohen-Macaulay rings using Hochster's theta invariant.

Findings

Modules with finite projective dimension satisfy the torsion condition.

Applications to torsion properties of tensor products of modules.

Connections to the Auslander-Reiten conjecture.

Abstract

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 $f \in \mathfrak{n}^2$ on $S$, we make use of Hochster's theta invariant and prove that such $R$-modules $M$ which have finite projective dimension over $S$ satisfy the proposed torsion condition of the conjecture. Along the way we give several applications of our argument pertaining to torsion properties of tensor products of modules.