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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 $R=S∕\left(f\right)$ 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