Homological dimensions of rigid modules

TL;DR

This paper characterizes properties of commutative Noetherian local rings using homological dimensions of modules, establishing new criteria for Gorenstein and regular rings based on injective dimensions and Ext vanishing.

Contribution

It provides novel characterizations of Gorenstein and regular rings through homological dimensions and Ext vanishing conditions, linking module properties to ring regularity.

Findings

Rings are Gorenstein if the Gorenstein injective dimension of their maximal ideal is finite.

Rings are regular if certain Ext groups vanish for some integrally closed ideals.

Local rings with maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay.

Abstract

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective dimension of the maximal ideal $\fm$ of $R$ is finite. Furthermore we prove that $R$ must be regular if a single $\Ext_{R}^{n}(I,J)$ vanishes for some integrally closed $\fm$-primary ideals $I$ and $J$ of $R$ and for some integer $n\geq \dim(R)$. Along the way we observe that local rings that admit maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay.