Homological dimensions of rigid modules

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 $n\ge \dim (R)$. Along the way we observe that local rings that admit maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay.

Major revision; title and abstract has been edited, new sections included

References in corpus (1)

• On the vanishing of Hochster's theta invariant

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• On the test properties of the Frobenius endomorphism
• On modules whose dual is of finite Gorenstein dimension
• On the vanishing of (co)homology for modules admitting certain filtrations
• On the ideal case of a conjecture of Auslander and Reiten