Homological dimensions of rigid modules

Abstract

We obtain various characterizations of commutative Noetherian local rings $(R,\mathfrak{m})$ in terms of homological dimensions of certain finitely generated modules. Our argument has a series of consequences in different directions. For example, we establish that $R$ is Gorenstein if the Gorenstein injective dimension of the maximal ideal $\mathfrak{m}$ of $R$ is finite. Moreover, we prove that $R$ must be regular if a single ${\mathsf{Ext}}_R^n(I,J)$ vanishes for some integrally closed $\mathfrak{m}$-primary ideals $I$ and $J$ of $R$ and for some positive integer $n$.