Huneke-Wiegand conjecture and change of rings

Abstract

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

16 pages, and the title of this article was changed on 24, Oct, 2013

References in corpus (2)

• Auslander-Reiten conjecture and Auslander-Reiten duality
• Necessary conditions for the depth formula over Cohen-Macaulay local rings

Cited by in corpus (1)

• Factorization Properties of Leamer Monoids