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.

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Mathematical Subject Classification 2010

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