On the ideal case of a conjecture of Huneke and Wiegand

TL;DR

This paper investigates a conjecture by Huneke and Wiegand regarding torsion in tensor products over one-dimensional local domains, proving it for a broad class of ideals and exploring higher-dimensional analogs.

Contribution

It extends the conjecture's validity to a large class of ideals and examines higher-dimensional cases and related questions involving Cohen-Macaulay modules.

Findings

Conjecture holds for many ideals in one-dimensional local domains.

Higher-dimensional analogs are established for integrally closed ideals.

Affirmative results for first syzygies of maximal Cohen-Macaulay modules.

Abstract

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 on the conjecture and give an affirmative answer for first syzygies of maximal Cohen-Macaulay modules.