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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A criterion for Gorenstein algebras to be regular


Authors: X.-F. Mao and Q.-S. Wu
Journal: Proc. Amer. Math. Soc. 139 (2011), 1543-1552
MSC (2010): Primary 16E65, 16W50, 16E30, 16E10, 14A22
Published electronically: October 4, 2010
MathSciNet review: 2763744
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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 \vert \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 \vert \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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10586-5
PII: S 0002-9939(2010)10586-5
Keywords: Connected graded algebra, Gorenstein algebra, AS-Gorenstein algebra, AS-regular algebra
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.