We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.
@article{bwmeta1.element.doi-10_2478_s11533-013-0374-x,
author = {Yuriy Drozd and Volodymyr Gavran},
title = {On Cohen-Macaulay modules over non-commutative surface singularities},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {675-687},
zbl = {1327.14016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0374-x}
}
Yuriy Drozd; Volodymyr Gavran. On Cohen-Macaulay modules over non-commutative surface singularities. Open Mathematics, Tome 12 (2014) pp. 675-687. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0374-x/
[1] Artin M., Maximal orders of global dimension and Krull dimension two, Invent. Math., 1986, 84(1), 195–222 http://dx.doi.org/10.1007/BF01388739 | Zbl 0591.16002
[2] Atiyah M.F., Vector bundles over an elliptic curve, Proc. London Math. Soc., 1957, s3–7, 414–452 http://dx.doi.org/10.1112/plms/s3-7.1.414 | Zbl 0084.17305
[3] Chan D., Ingalls C., The minimal model program for orders over surfaces, Invent. Math., 2005, 161(2), 427–452 http://dx.doi.org/10.1007/s00222-005-0438-z | Zbl 1078.14005
[4] Drozd Yu.A., Greuel G.-M., Cohen-Macaulay module type, Compositio Math., 1993, 89(3), 315–338
[5] Drozd Yu.A., Greuel G.-M., Kashuba I., On Cohen-Macaulay modules on surface singularities, Mosc. Math. J., 2003, 3(2), 397–418 | Zbl 1051.13006
[6] Drozd Yu.A., Kirichenko V.V., Primary orders with a finite number of indecomposable representations, Izv. Akad. Nauk SSSR Ser. Mat., 1973, 37, 715–736 (in Russian)
[7] Hartshorne R., Residues and Duality, Lecture Notes in Math., 20, Springer, Berlin-Heidelberg-New York, 1966
[8] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, New York-Berlin-Heidelberg, 1977
[9] Grothendieck A., A General Theory of Fibre Spaces with Structure Sheaf, University of Kansas, Lawrence, 1955, available at http://www.math.jussieu.fr/~leila/grothendieckcircle/GrothKansas.pd
[10] Grothendieck A., Éléments de Géométrie Algébrique. III. Étude Cohomologique des Faisceaux Cohérents. I, Inst. Hautes Études Sci. Publ. Math., 11, Presses Universitaires de France, Paris, 1961
[11] Kahn C.P., Reflexive modules on minimally elliptic singularities, Math. Ann., 1989, 285(1), 141–160 http://dx.doi.org/10.1007/BF01442678 | Zbl 0662.14022
[12] Lipman J., Rational singularities, with application to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math., 1969, 36, 195–279 http://dx.doi.org/10.1007/BF02684604 | Zbl 0181.48903
[13] Reiner I., Maximal Orders, London Math. Soc. Monogr., 5, Academic Press, London-New York, 1975
[14] Wahl J.M., Equisingular deformations of normal surface singularities, I, Ann. Math., 1976, 104(2), 325–356 http://dx.doi.org/10.2307/1971049 | Zbl 0358.14007
[15] Yoshino Y., Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math. Soc. Lecture Note Ser., 146, Cambridge University Press, Cambridge, 1990 | Zbl 0745.13003