Vol. 309, No. 1, 2020

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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(f) for some local ring (S,𝔫), and a non-zero-divisor f 𝔫2 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.

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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
Authors
Olgur Celikbas
Department of Mathematics
West Virginia University
Morgantown, WV
United States