A criterion for Gorenstein algebras to be regular
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Abstract:
In this paper we give a criterion for a left Gorenstein algebra to be AS-regular. Let $A$ be a left Gorenstein algebra such that the trivial module ${}_Ak$ admits a finitely generated minimal free resolution. Then $A$ is AS-regular if and only if its left Gorenstein index is equal to $-\inf \{i | \mathrm {Ext}_A^{\mathrm {depth}_AA}(k,k)_i\neq 0\}.$ Furthermore, $A$ is Koszul AS-regular if and only if its left Gorenstein index is $\mathrm {depth}_AA=-\inf \{i | \mathrm {Ext}_A^{\mathrm {depth}_AA}(k,k)_i\neq 0\}.$
As applications, we prove that the category of AS-regular algebras is a tensor category and that a left Noetherian $p$-Koszul, left Gorenstein algebra is AS-regular if and only if it is $p$-standard. This generalizes a result of Dong and the second author.
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Additional Information
- X.-F. Mao
- Affiliation: Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
- Address at time of publication: Department of Mathematics, Shanghai University, 200444, People’s Republic of China
- MR Author ID: 846632
- Email: 041018010@fudan.edu.cn, xuefengmao@shu.edu.cn
- Q.-S. Wu
- Affiliation: Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
- Email: qswu@fudan.edu.cn
- Received by editor(s): November 6, 2009
- Received by editor(s) in revised form: May 9, 2010
- Published electronically: October 4, 2010
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1543-1552
- MSC (2010): Primary 16E65, 16W50, 16E30, 16E10, 14A22
- DOI: https://doi.org/10.1090/S0002-9939-2010-10586-5
- MathSciNet review: 2763744