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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
DOI: https://doi.org/10.1090/S0002-9939-2010-10586-5
Published electronically: October 4, 2010
MathSciNet review: 2763744
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Abstract | References | Similar Articles | Additional Information

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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  • [AS] M. Artin and W.F. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171-216. MR 917738 (88k:16003)
  • [Be] R. Berger, Koszulity for nonquadratic algebras, J. Algebra 239 (2001), 705-734. MR 1832913 (2002d:16034)
  • [BGS] A.A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. MR 1322847 (96k:17010)
  • [DW] Z.-C. Dong and Q.-S. Wu, Non-commutative Castelnuovo-Mumford regularity and AS-regular algebras, J. Algebra 322 (2009), 122-136. MR 2526379 (2010g:16019)
  • [FM] Y. Félix and A. Murillo, Gorenstein graded algebras and the evaluation map, Canad. Math. Bull. 41 (1998), 28-32. MR 1618931 (99c:57069)
  • [HL] J.-W. He and D.-M. Lu, Higher Koszul algebras and A-infinity algebras, J. Algebra 293 (2005), 335-362. MR 2172343 (2006m:16030)
  • [LWZ] D.-M. Lu, Q.-S. Wu and J.J. Zhang, Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007), 4945-4975. MR 2320655 (2008f:16083)
  • [Men] C. Menini, Cohen-Macaulay and Gorenstein finitely graded rings, Rend. Sem. Mat. Univ. Padova 79 (1988), 123-152. MR 964026 (89i:13030)
  • [LPWZ] D.-M. Lu, J.-H. Palmieri, Q.-S. Wu and J.-J. Zhang, Koszul equivalences in $ A_{\infty}$-algebras, New York J. Math. 14 (2008), 325-378. MR 2430869 (2010b:16017)
  • [Sm] S.P. Smith, Some finite-dimensional algebras related to elliptic curves, CMS Conf. Proc., Vol. 19, 315-348, Amer. Math. Soc., 1996. MR 1388568 (97e:16053)
  • [SZ] D.R. Stephenson and J.J. Zhang, Growth of graded Noetherian rings, Proc. Amer. Math. Soc. 125 (1997), 1593-1605. MR 1371143 (97g:16033)
  • [YZ] Y. Ye and P. Zhang, Higher Koszul complexes, Sci. China Ser. A 46 (2003), 118-128. MR 1977972 (2004f:16016)

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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: https://doi.org/10.1090/S0002-9939-2010-10586-5
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.

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