Noncommutative graded algebras of finite Cohen-Macaulay representation type

Ueyama, Kenta

doi:10.1090/proc/12527

Noncommutative graded algebras of finite Cohen-Macaulay representation type
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Proc. Amer. Math. Soc. 143 (2015), 3703-3715 Request permission

Abstract:

Let $A$ be an AS-Cohen-Macaulay algebra. We show that if $A$ is of finite Cohen-Macaulay representation type, then $A$ is a noncommutative graded isolated singularity. This is a noncommutative analogue of a well-known theorem of Auslander and is a generalization of Jørgensen’s theorem. Besides, we give an example of a noncommutative quadric hypersurface of finite Cohen-Macaulay representation type in a quantum ${\mathbb P}^3$ which is not a domain. We also give all indecomposable graded maximal Cohen-Macaulay modules over it explicitly.

References

M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
Maurice Auslander, Isolated singularities and existence of almost split sequences, Representation theory, II (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986, pp. 194–242. MR 842486, DOI 10.1007/BFb0075297
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
David Eisenbud and Jürgen Herzog, The classification of homogeneous Cohen-Macaulay rings of finite representation type, Math. Ann. 280 (1988), no. 2, 347–352. MR 929541, DOI 10.1007/BF01456058
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, 2nd ed., London Mathematical Society Student Texts, vol. 61, Cambridge University Press, Cambridge, 2004. MR 2080008, DOI 10.1017/CBO9780511841699
Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
Craig Huneke and Graham J. Leuschke, Two theorems about maximal Cohen-Macaulay modules, Math. Ann. 324 (2002), no. 2, 391–404. MR 1933863, DOI 10.1007/s00208-002-0343-3
Peter Jørgensen, Local cohomology for non-commutative graded algebras, Comm. Algebra 25 (1997), no. 2, 575–591. MR 1428799, DOI 10.1080/00927879708825875
Peter Jørgensen, Properties of AS-Cohen-Macaulay algebras, J. Pure Appl. Algebra 138 (1999), no. 3, 239–249. MR 1691464, DOI 10.1016/S0022-4049(98)00086-3
Peter Jørgensen, Finite Cohen-Macaulay type and smooth non-commutative schemes, Canad. J. Math. 60 (2008), no. 2, 379–390. MR 2398754, DOI 10.4153/CJM-2008-018-0
Peter Jørgensen and James J. Zhang, Gourmet’s guide to Gorensteinness, Adv. Math. 151 (2000), no. 2, 313–345. MR 1758250, DOI 10.1006/aima.1999.1897
Graham J. Leuschke, Endomorphism rings of finite global dimension, Canad. J. Math. 59 (2007), no. 2, 332–342. MR 2310620, DOI 10.4153/CJM-2007-014-1
Izuru Mori, Homological properties of balanced Cohen-Macaulay algebras, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1025–1042. MR 1938744, DOI 10.1090/S0002-9947-02-03166-5
Izuru Mori, Co-point modules over Koszul algebras, J. London Math. Soc. (2) 74 (2006), no. 3, 639–656. MR 2286437, DOI 10.1112/S002461070602326X
I. Mori and K. Ueyama, Ample group actions on AS-regular algebras and noncommutative graded isolated singularities, to appear in Trans. Amer. Math. Soc., arXiv:1404.5045.
I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366. MR 1887637, DOI 10.1090/S0894-0347-02-00387-9
S. Paul Smith and Michel Van den Bergh, Noncommutative quadric surfaces, J. Noncommut. Geom. 7 (2013), no. 3, 817–856. MR 3108697, DOI 10.4171/JNCG/136
J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian $\textrm {PI}$ rings, J. Algebra 168 (1994), no. 3, 988–1026. MR 1293638, DOI 10.1006/jabr.1994.1267
Kenta Ueyama, Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities, J. Algebra 383 (2013), 85–103. MR 3037969, DOI 10.1016/j.jalgebra.2013.02.022
M. Vancliff, K. Van Rompay, and L. Willaert, Some quantum $\textbf {P}^3$s with finitely many points, Comm. Algebra 26 (1998), no. 4, 1193–1208. MR 1612220, DOI 10.1080/00927879808826193
Michel van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), no. 2, 662–679. MR 1469646, DOI 10.1006/jabr.1997.7052
Amnon Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), no. 1, 41–84. MR 1195406, DOI 10.1016/0021-8693(92)90148-F