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

Olgur Celikbas

Vol. 309 (2020), No. 1, 103–144

DOI: 10.2140/pjm.2020.309.103

Abstract

Huneke and Wiegand conjectured that, if $M$ is a finitely generated, nonfree, 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 $R=S∕\left(f\right)$ for some local ring $\left(S,\mathfrak{𝔫}\right)$, 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 conclusion of the conjecture. Along the way we give several applications of our argument pertaining to torsion properties of tensor products of modules.

Keywords

complete intersection dimension, complexity, theta invariant, torsion in tensor products of modules, vanishing of $\mathrm{Ext}$ and $\mathrm{Tor}$

Mathematical Subject Classification 2010

Primary: 13D07

Secondary: 13C13, 13C14, 13H10

Milestones

Received: 5 May 2019

Revised: 4 February 2020

Accepted: 18 June 2020

Published: 26 December 2020