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arXiv · 1405.5188

Homological dimensions of rigid modules

Olgur Celikbas·Mohsen Gheibi·Majid Rahro Zargar·Arash Sadeghi
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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.

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Olgur Celikbas, Mohsen Gheibi, Majid Rahro Zargar, Arash Sadeghi. 2014-05-20. Homological dimensions of rigid modules. https://doi.org/10.1215/21562261-2017-0033

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