Kyoto Journal of Mathematics

Homological dimensions of rigid modules

Majid Rahro Zargar, Olgur Celikbas, Mohsen Gheibi, and Arash Sadeghi

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Abstract

We obtain various characterizations of commutative Noetherian local rings (R,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 m of R is finite. Moreover, we prove that R must be regular if a single ExtRn(I,J) vanishes for some integrally closed m-primary ideals I and J of R and for some positive integer n.

Article information

Source
Kyoto J. Math., Volume 58, Number 3 (2018), 639-669.

Dates
Received: 19 February 2016
Revised: 8 December 2016
Accepted: 9 December 2016
First available in Project Euclid: 19 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1529373739

Digital Object Identifier
doi:10.1215/21562261-2017-0033

Mathematical Reviews number (MathSciNet)
MR3843394

Zentralblatt MATH identifier
06959095

Subjects
Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D05: Homological dimension 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Keywords
Auslander’s transpose Frobenius endomorphism Gorenstein injective dimension semidualizing modules test modules Tor-rigidity vanishing of Ext and Tor

Citation

Zargar, Majid Rahro; Celikbas, Olgur; Gheibi, Mohsen; Sadeghi, Arash. Homological dimensions of rigid modules. Kyoto J. Math. 58 (2018), no. 3, 639--669. doi:10.1215/21562261-2017-0033. https://projecteuclid.org/euclid.kjm/1529373739


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