Huneke-Wiegand conjecture and change of rings

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.