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On two examples by Iyama and Yoshino

Part of: Theory of modules and ideals Homological methods

Published online by Cambridge University Press:  09 February 2011

Bernhard Keller ,
Daniel Murfet  and
Michel Van den Bergh
Bernhard Keller
Affiliation:
UFR de Mathématiques, Université Denis Diderot – Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France (email: keller@math.jussieu.fr)
Daniel Murfet
Affiliation:
Hausdorff Center for Mathematics, Endenicher Allee 62, D-53115 Bonn, Germany (email: murfet@math.uni-bonn.de)
Michel Van den Bergh
Affiliation:
Departement WNI, Universiteit Hasselt, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium (email: michel.vandbenbergh@uhasselt.be)
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Abstract

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In a recent paper, Iyama and Yoshino considered two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen–Macaulay modules in terms of linear algebra data. In this paper, we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov’s result on the graded singularity category.

Keywords

singularity categorymaximal Cohen–Macaulay modules

MSC classification

Secondary: 13C14: Cohen-Macaulay modules 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 591 - 612
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Artin, M. and Zhang, J. J., Noncommutative projective schemes, Adv. Math. 109 (1994), 228–287.CrossRefGoogle Scholar
[2]Auslander, M., Platzeck, M. and Reiten, I., Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46.CrossRefGoogle Scholar
[3]Auslander, M. and Reiten, I., Graded modules and their completions, in Topics in algebra, Part 1 (Warsaw, 1988), Banach Center Publications, vol. 26, Part 1 (PWN, Warsaw, 1990), 181–192.CrossRefGoogle Scholar
[4]Beligiannis, A., The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and (co-)stabilization, Comm. Algebra 28 (2000), 4547–4596.CrossRefGoogle Scholar
[5]Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209–234.Google Scholar
[6]Bondal, A. and Polishchuk, A., Homological properties of associative algebras: the method of helices, Russian Acad. Sci. Izv. Math. 42 (1994), 219–260.Google Scholar
[7]Buchweitz, R.-O., Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings (1987), 155, unpublished manuscript, available at https://tspace.library.utoronto.ca/handle/1807/16682.Google Scholar
[8]Buchweitz, R.-O., Eisenbud, D. and Herzog, J., Cohen–Macaulay modules on quadrics, in Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) (Springer, Berlin, 1987), 58–116.CrossRefGoogle Scholar
[9]Dyckerhoff, T., Compact generators in categories of matrix factorizations, Preprint (2009), arXiv:0904.4713v3.Google Scholar
[10]Elkik, R., Solution d’équations au-dessus d’anneaux henséliens, in Quelques problèmes de modules (Sém. Géom. Anal., École Norm. Sup., Paris, 1971–1972) (Soc. Math. France, Paris, 1974) Astérisque 16, 116–132.Google Scholar
[11]Gabriel, P., Auslander–Reiten sequences and representation-finite algebras, in Representation theory, I (Proc. Workshop, Carleton University, Ottawa, Ontario, 1979), Lecture Notes in Mathematics, vol. 831 (Springer, Berlin, 1980), 1–71.CrossRefGoogle Scholar
[12]Iyama, O. and Yoshino, Y., Mutation in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (2008), 117–168.CrossRefGoogle Scholar
[13]Keller, B., Deriving DG-categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), 63–102.CrossRefGoogle Scholar
[14]Keller, B., On the construction of triangle equivalences, in Derived equivalences for group rings, Lecture Notes in Mathematics, vol. 1685 (Springer, Berlin, 1998), 155–176.CrossRefGoogle Scholar
[15]Keller, B., On triangulated orbit categories, Doc. Math. 10 (2005), 551–581 (electronic).CrossRefGoogle Scholar
[16]Keller, B. and Reiten, I., Acyclic Calabi–Yau categories (with an appendix by Michel Van den Bergh), Compositio Math. 144 (2008), 1332–1348.CrossRefGoogle Scholar
[17]Krause, H., The stable derived category of a Noetherian scheme, Compositio Math. 141 (2005), 1128–1162.CrossRefGoogle Scholar
[18]Levy, L. and Odenthal, C., Krull–Schmidt theorems in dimension 1, Trans. Amer. Math. Soc. 348 (1996), 3391–3455.CrossRefGoogle Scholar
[19]Matsumura, H., Commutative ring theory (Cambridge University Press, Cambridge, 1989).Google Scholar
[20]Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.CrossRefGoogle Scholar
[21]Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).CrossRefGoogle Scholar
[22]Orlov, D., Projective bundles, monoidal transformations and derived functors of coherent sheaves, Russian Acad. Sci. Izv. Math. 41 (1993), 133–141.Google Scholar
[23]Orlov, D., Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), 240–262, Algebr. Geom. Metody, Svyazi i Prilozh.Google Scholar
[24]Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Vol. II, Progress in Mathematics, vol. 270 (Birkhäuser, Boston, MA, 2009), 503–531 (English summary).CrossRefGoogle Scholar
[25]Orlov, D., Formal completions and idempotent completions of triangulated categories of singularities, Preprint (2009), arXiv:0901.1859v1.Google Scholar
[26]Raynaud, M. and Gruson, L., Critères de platitude et de projectivité. Techniques de ‘platification’ d’un module, Invent. Math. 13 (1971), 1–89.CrossRefGoogle Scholar
[27]Rickard, J., Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436–456.CrossRefGoogle Scholar
[28]Rouquier, R., Dimensions of triangulated categories, J. K-Theory 1 (2008), 193–256.CrossRefGoogle Scholar
[29]Schoutens, H., Projective dimension and the singular locus, Comm. Algebra 31 (2003), 217–239.CrossRefGoogle Scholar
[30]Serre, J. P., Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278.CrossRefGoogle Scholar
[31]Wiegand, R., Local rings of finite Cohen–Macaulay type, J. Algebra 203 (1998), 156–168.CrossRefGoogle Scholar
[32]Yoshino, Y., Cohen–Macaulay modules over Cohen–Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar