Huneke-Wiegand conjecture and change of rings

TL;DR

This paper investigates conditions under which the tensor product of a faithful ideal and its dual is torsionfree in Cohen-Macaulay rings, establishing a bound on multiplicity that characterizes the ideal as trivial or canonical.

Contribution

It proves that in one-dimensional Cohen-Macaulay local rings with multiplicity at most 6, torsionfreeness of the tensor product characterizes the ideal as trivial or canonical, extending to monomial ideals and higher dimensions.

Findings

If $R$ has multiplicity ≤ 6, then $I$ is isomorphic to $R$ or $ m{K_R}$ when $I ensor_R I^{igvee}$ is torsionfree.

Application to monomial ideals of numerical semigroup rings.

Discussion of higher-dimensional cases.

Abstract

Let $R$ be a Cohen-Macaulay local ring of dimension one with a canonical module $\rm{K_R}$. Let $I$ be a faithful ideal of $R$. We explore the problem of when $I \otimes_RI^{\vee}$ is torsionfree, where $I^{\vee} = \operatorname{Hom_R(I, \rm{K_R})}$. We prove that if $R$ has multiplicity at most $6$, then $I$ is isomorphic to $R$ or $\rm{K_R}$ as an $R$-module, once $I\otimes_RI^{\vee}$ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher dimensional assertion is also discussed.