Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T16:56:18.008Z Has data issue: false hasContentIssue false
  • English
  • Français

GRADED MORITA EQUIVALENCES FOR GEOMETRIC AS-REGULAR ALGEBRAS

Published online by Cambridge University Press:  02 August 2012

IZURU MORI  and
KENTA UEYAMA
IZURU MORI
Affiliation:
Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka 422-8529, Japan e-mail: [email protected]
KENTA UEYAMA
Affiliation:
Department of Information Science and Technology, Graduate School of Science and Technology, Shizuoka University, Shizuoka 422-8529, Japan e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Classification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

Keywords

16W5016D9016S3816S3716E65
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 2 , May 2013 , pp. 241 - 257
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Artin, M. and Schelter, W., Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171–216.Google Scholar
2.Artin, M., Tate, J. and Van den Bergh, M., Some algebras associated to automorphisms of elliptic curves in The Grothendieck Festschrift volume 1, Progress in Mathematics, vol. 86 (Birkhauser, Boston, 1990), 33–85.Google Scholar
3.Artin, M., Tate, J. and Van den Bergh, M., Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), 335–388.CrossRefGoogle Scholar
4.Artin, M. and Zhang, J. J., Noncommutative projective schemes, Adv. Math. 109 (1994), 228–287.Google Scholar
5.Jørgensen, P., Local cohomology for non-commutative graded algebras, Commun. Algebra 25 (1997), 575–591.Google Scholar
6.Minamoto, H., Ampleness of two-sided tilting complexes, Int. Math. Res. Not. 1 (2012), 67–101.CrossRefGoogle Scholar
7.Minamoto, H. and Mori, I., The structure of AS-Gorenstein algebras, Adv. Math. 226 (2011), 4061–4095.Google Scholar
8.Mori, I., Noncommutative projective schemes and point schemes in Algebras, rings and their representations (World Scientific, Hackensack, NJ, 2006), 215–239.CrossRefGoogle Scholar
9.Mori, I., Co-point modules over Koszul algebras, J. London Math. Soc. 74 (2006), 639–656.CrossRefGoogle Scholar
10.Mori, I., Co-point modules over Frobenius Koszul algebras, Commun. Algebra 36 (2008), 4659–4677.Google Scholar
11.Mori, I., Classification problems in noncommutative algebraic geometry and representation theory in Representations of algebras and related topics, EMS Series of Congress Reports (Eur. Math. Soc., Zurich, 2011), 355–406.CrossRefGoogle Scholar
12.Mori, I., B-cosntrcution and C-construction, Commun. Algebra, to appear.Google Scholar
13.de Naeghel, K. and Van den Bergh, M., Ideal classes of three-dimensional Sklyanin algebras, J. Algebra 276 (2004), 515–551.Google Scholar
14.Nafari, M., Vancliff, M. and Zhang, J., Classifying quadratic quantum ℙ2s by using graded skew Clifford algebras, J. Algebra 346 (2011), 152–164.Google Scholar
15.Smith, S. P., Some finite dimensional algebras related to elliptic curves in Representation theory of algebras and related topics (Mexico City, 1994), CMS Conference Proceedings, vol. 19 (Amer. Math. Soc., Providence, RI, 1996), 315–348.Google Scholar
16.Ueyama, K., Graded Morita equivalences for generic Artin-Schelter regular algebras, Kyoto J. Math. 51 (2011), 485–501.Google Scholar
17.Van den Bergh, M., Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), 662–679.CrossRefGoogle Scholar
18.Van Rompay, K., Segre product of Artin-Schelter regular algebras of dimension 2 and embeddings in quantum ℙ3's, J. Algebra 180 (1996), 483–512.CrossRefGoogle Scholar
19.Vitoria, J., Equivalences for noncommutative projective spaces, (2011), arXiv: 1001.4400v3.Google Scholar
20.Zhang, J. J., Twisted graded algebras and equivalences of graded categories, Proc. London Math. Soc. 72 (1996), 281–311.CrossRefGoogle Scholar