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Mathematics > Commutative Algebra

arXiv:2207.06170 (math)
[Submitted on 13 Jul 2022 (v1), last revised 7 Jun 2023 (this version, v3)]

Title:Quasi-injective dimension

Authors:Mohsen Gheibi
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Abstract:Following our previous work about quasi-projective dimension, in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module $M$ over a local ring $R$ is finite, then it is equal to the depth of $R$. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then $R$ is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then $R$ is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.
Comments: The final version, to appear in J. Pure and Applied Algebra
Subjects: Commutative Algebra (math.AC)
MSC classes: 13D05, 13D07, 13H10
Cite as: arXiv:2207.06170 [math.AC]
  (or arXiv:2207.06170v3 [math.AC] for this version)
  https://doi.org/10.48550/arXiv.2207.06170

Submission history

From: Mohsen Gheibi [view email]
[v1] Wed, 13 Jul 2022 13:07:49 UTC (12 KB)
[v2] Fri, 5 Aug 2022 19:41:58 UTC (13 KB)
[v3] Wed, 7 Jun 2023 13:16:28 UTC (13 KB)
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