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

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.

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
Received: 5 May 2019
Revised: 4 February 2020
Accepted: 18 June 2020
Published: 26 December 2020
Olgur Celikbas
Department of Mathematics
West Virginia University
Morgantown, WV
United States