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Mosc. Math. J., 2003, Volume 3, Number 2, Pages 397–418 (Mi mmj92)  

This article is cited in 14 scientific papers (total in 14 papers)

On Cohen–Macaulay modules on surface singularities

Yu. A. Drozdab, G.-M. Greuelc, I. Kashubac

a National Taras Shevchenko University of Kyiv
b Max Planck Institute for Mathematics
c Technical University of Kaiserslautern

Abstract: We study Cohen–Macaulay modules over normal surface singularities. Using the method of Kahn and extending it to families of modules, we classify Cohen–Macaulay modules over cusp singularities and prove that a minimally elliptic singularity is Cohen–Macaulay tame if and only if it is either simple elliptic or cusp. As a corollary, we obtain a classification of Cohen–Macaulay modules over log-canonical surface singularities and hypersurface singularities of type $T_{pqr}$ especially they are Cohen–Macaulay tame. We also calculate the Auslander–Reiten quiver of the category of Cohen–Macaulay modules in the considered cases.

Key words and phrases: Cohen–Macaulay modules, Cohen–Macaulay tame and wild rings, normal surface singularities, minimally elliptic singularities, cusp singularities, log-canonical singularities, hypersurface singularities, Auslander–Reiten quiver.


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MSC: Primary 13C14, 13C05; Secondary 16G50, 14J17
Received: February 18, 2002

Citation: Yu. A. Drozd, G.-M. Greuel, I. Kashuba, “On Cohen–Macaulay modules on surface singularities”, Mosc. Math. J., 3:2 (2003), 397–418

Citation in format AMSBIB
\by Yu.~A.~Drozd, G.-M.~Greuel, I.~Kashuba
\paper On Cohen--Macaulay modules on surface singularities
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 2
\pages 397--418

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    This publication is cited in the following articles:
    1. Burban I., Kreussler B., “Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstraß cubics”, J. Reine Angew. Math., 584 (2005), 45–82  crossref  mathscinet  zmath  isi  elib
    2. Ile R., “Deforming syzygies of liftable modules and generalised Knörrer functors”, Collect. Math., 58:3 (2007), 255–277  mathscinet  zmath  isi  elib
    3. Burban I., Iyama O., Keller B., Reiten I., “Cluster tilting for one-dimensional hypersurface singularities”, Adv. Math., 217:6 (2008), 2443–2484  crossref  mathscinet  zmath  isi
    4. Gustavsen T.S., Ile R., “Representation Theory for Log-Canonical Surface Singularities”, Ann Inst Fourier (Grenoble), 60:2 (2010), 389–416  crossref  mathscinet  zmath  isi
    5. Bhosle U.N., “Coherent systems on a nodal curve of genus one”, Math Nachr, 284:14–15 (2011), 1829–1845  crossref  mathscinet  zmath  isi  elib
    6. Monsky P., “Hilbert-Kunz theory for nodal cubics, via sheaves”, J Algebra, 346:1 (2011), 180–188  crossref  mathscinet  zmath  isi
    7. Crabbe A., Leuschke G.J., “Wild hypersurfaces”, J Pure Appl Algebra, 215:12 (2011), 2884–2891  crossref  mathscinet  zmath  isi
    8. Voloshyn D.E., Drozd Yu.A., “Derived Categories of Nodal Curves”, Ukr. Math. J., 64:8 (2013), 1177–1184  crossref  mathscinet  zmath  isi  elib
    9. E. A. Makedonskii, “On Wild and Tame Finite-Dimensional Lie Algebras”, Funct. Anal. Appl., 47:4 (2013), 271–283  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. Drozd Yu.A., Gavran V.S., “On Cohen-Macaulay Modules Over Non-Commutative Surface Singularities”, Cent. Eur. J. Math., 12:5 (2014), 675–687  crossref  mathscinet  zmath  isi  elib
    11. Drozd Yu.A., Tovpyha O., “Graded Cohen-Macaulay Rings of Wild Cohen-Macaulay Type”, J. Pure Appl. Algebr., 218:9 (2014), 1628–1634  crossref  mathscinet  zmath  isi  elib
    12. Burban I., Gnedin W., “Cohen-Macaulay Modules Over Some Non-Reduced Curve Singularities”, J. Pure Appl. Algebr., 220:12 (2016), 3777–3815  crossref  mathscinet  zmath  isi
    13. Bondarenko V.M., Bortosh M.Yu., “Indecomposable and Isomorphic Objects in the Category of Monomial Matrices Over a Local Ring”, Ukr. Math. J., 69:7 (2017), 1034–1050  crossref  isi  scopus
    14. Burban I., Drozd Yu., “Generalities on Maximal Cohen-Macaulay Modules”, Mem. Am. Math. Soc., 248:1178 (2017), 1+  crossref  isi  scopus
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