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Inventiones mathematicae

, Volume 172, Issue 1, pp 117–168 | Cite as

Mutation in triangulated categories and rigid Cohen–Macaulay modules

  • Osamu IyamaEmail author
  • Yuji Yoshino
Article

Abstract

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Keywords

Exact Sequence Direct Summand Cluster Algebra Abelian Category Triangulate Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Artin, M., Verdier, J.-L.: Reflexive modules over rational double points. Math. Ann. 270(1), 79–82 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Auslander, M.: Coherent functors. In: Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pp. 189–231. Springer, New York (1966)Google Scholar
  3. 3.
    Auslander, M.: Rational singularities and almost split sequences. Trans. Am. Math. Soc. 293(2), 511–531 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Auslander, M.: Functors and morphisms determined by objects. In: Representation Theory of Algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), pp. 1–244. Lect. Notes Pure Appl. Math., vol. 37. Dekker, New York (1978)Google Scholar
  5. 5.
    Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen–Macaulay approximations. In: Colloque en l’h onneur de Pierre Samuel (Orsay, 1987). Mem. Soc. Math. France (N.S.), vol. 38, pp. 5–37 (1989)Google Scholar
  6. 6.
    Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Auslander, M., Reiten, I.: Stable equivalence of dualizing R-varieties. Adv. Math. 12, 306–366 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Auslander, M., Reiten, I.: The Cohen–Macaulay type of Cohen–Macaulay rings. Adv. Math. 73(1), 1–23 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Auslander, M., Reiten, I.: DTr-periodic modules and functors. Representation theory of algebras (Cocoyoc, 1994). CMS Conf. Proc., vol. 18, pp. 39–50. Am. Math. Soc., Providence, RI (1996)Google Scholar
  11. 11.
    Auslander, M., Reiten, I., Smalo, S.O.: Representation theory of Artin algebras. Camb. Stud. Adv. Math., vol. 36. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  12. 12.
    Auslander, M., Smalo, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Baur, K., Marsh, R.: A geometric description of m-cluster categories. arXiv:math.RT/0607151Google Scholar
  14. 14.
    Beligiannis, A., Reiten, I.: Homological Aspects of Torsion Theories. Mem. Am. Math. Soc., vol. 188. Am. Math. Soc. (2007)Google Scholar
  15. 15.
    Benson, D.J.: Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, 2nd edn. Camb. Stud. Adv. Math., vol. 30. Cambridge University Press, Cambridge (1998)Google Scholar
  16. 16.
    Bernšteĭ n, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem. Uspehi Mat. Nauk 28(2), 19–33 (1973)MathSciNetGoogle Scholar
  17. 17.
    Bezrukavnikov, R., Kaledin, D.: McKay equivalence for symplectic resolutions of quotient singularities. Tr. Mat. Inst. Steklova 246, 20–42 (2004) (translation in Proc. Steklov Inst. Math. 246(3), 13–33 (2004))MathSciNetGoogle Scholar
  18. 18.
    Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors. Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), pp. 103–169, Lect. Notes Math., vol. 832. Springer, Berlin New York (1980)Google Scholar
  19. 19.
    Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Buan, A., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi–Yau categories and unipotent groups. arXiv:math/0701557Google Scholar
  21. 21.
    Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Buan, A., Marsh, R., Reiten, I.: Cluster mutation via quiver representations. Comm. Math. Helv. 83(1), 143–177 (2008) (arXiv:math.RT/0412077)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Math. USSR-Izv. 35(3), 519–541 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358(3), 1347–1364 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Caldero, P., Keller, B.: From triangulated categories to cluster algebras II. Ann. Sci. Éc. Norm. Supér., IV. Sér. 39(6), 983–1009 (2006)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Crawley-Boevey, W.W.: On tame algebras and bocses. Proc. Lond. Math. Soc. (3) 56(3), 451–483 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Derksen, H., Weyman, J.: On the canonical decomposition of quiver representations. Compos. Math. 133(3), 245–265 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Drozd, Y.A.: Tame and wild matrix problems. Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979). Lect. Notes Math., vol. 832, pp. 242–258. Springer, Berlin New York (1980)Google Scholar
  30. 30.
    Drozd, Y.A., Greuel, G.-M.: Tame-wild dichotomy for Cohen–Macaulay modules. Math. Ann. 294(3), 387–394 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Erdmann, K., Holm, T.: Maximal n-orthogonal modules for selfinjective algebras. Proc. Am. Math. Soc. (to appear), arXiv:math.RT/0603672Google Scholar
  32. 32.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Gabriel, P., Roĭ ter, A.V.: Representations of finite-dimensional algebras. Encyclopaedia Math. Sci. 73, Algebra, VIII, 1–177. Springer, Berlin (1992) (With a chapter by Keller, B.)Google Scholar
  35. 35.
    Geiss, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. 165(3), 589–632 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Gonzalez-Sprinberg, G., Verdier, J.-L.: Structure multiplicative des modules reflexifs sur les points doubles rationnels. Geometrie algebrique et applications, I (La Rabida, 1984). Travaux en Cours, vol. 22, pp. 79–110. Hermann, Paris (1987)Google Scholar
  37. 37.
    Gorodentsev, A.L., Rudakov, A.N.: Exceptional vector bundles on projective spaces. Duke Math. J. 54(1), 115–130 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. Lond. Math. Soc. Lect. Note Ser., vol. 119. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  39. 39.
    Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology 39(6), 1155–1191 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Iyama, O.: Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Iyama, O.: Maximal orthogonal subcategories of triangulated categories satisfying Serre duality. Oberwolfach, Rep. 2(1), 353–355 (2005)Google Scholar
  43. 43.
    Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Am. J. Math. (to appear), arXiv:math.RT/0605136Google Scholar
  44. 44.
    Kac, V.G.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3), 565–576 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Keller, B.: Deriving DG categories. Ann. Sci. Éc. Norm. Supér., IV. Sér. 27(1), 63–102 (1994)zbMATHGoogle Scholar
  47. 47.
    Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)zbMATHMathSciNetGoogle Scholar
  48. 48.
    Keller, B., Vossieck, D.: Aisles in derived categories. Deuxieme Contact Franco–Belge en Algebre (Faulx-les-Tombes, 1987). Bull. Soc. Math. Belg. Ser. A 40(2), 239–253 (1988)zbMATHMathSciNetGoogle Scholar
  49. 49.
    Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Keller, B., Reiten, I.: Acyclic Calabi–Yau categories are cluster categories. arXiv:math/0610594Google Scholar
  51. 51.
    Koenig, S., Zhu, B.: From triangulated categories to abelian categories–cluster tilting in a general framework. Math. Z. (to appear), arXiv:math.RT/0605100Google Scholar
  52. 52.
    Krause, H.: A Brown representability theorem via coherent functors. Topology 41(4), 853–861 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Krause, H.: Cohomological quotients and smashing localizations. Am. J. Math. 127(6), 1191–1246 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Kurano, K.: Private communicationGoogle Scholar
  55. 55.
    Miyashita, Y.: Tilting modules of finite projective dimension. Math. Z. 193(1), 113–146 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Am. Math. Soc. 15(2), 295–366 (2002)zbMATHCrossRefGoogle Scholar
  57. 57.
    Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. (2) 39(3), 436–456 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Riedtmann, C., Schofield, A.: On a simplicial complex associated with tilting modules. Comment. Math. Helv. 66(1), 70–78 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Rudakov, A.N.: Helices and vector bundles. Seminaire Rudakov. Lond. Math. Soc. Lect. Note Ser., vol. 148. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  60. 60.
    Schofield, A.: Semi-invariants of quivers. J. Lond. Math. Soc. (2) 43(3), 385–395 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Tabuada, G.: On the structure of Calabi–Yau categories with a cluster tilting subcategory. Doc. Math. 12, 193–213 (2007)zbMATHMathSciNetGoogle Scholar
  63. 63.
    Thomas, H.: Defining an m-cluster category. J. Algebra 318(1), 37–46 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Van den Bergh, M.: Non-commutative crepant resolutions. The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)Google Scholar
  66. 66.
    Watanabe, K.: Certain invariant subrings are Gorenstein. I, II. Osaka J. Math. 11, 1–8 (1974); ibid. 11, 379–388 (1974)Google Scholar
  67. 67.
    Yoshino, Y.: Cohen–Macaulay modules over Cohen–Macaulay rings. Lond. Math. Soc. Lect. Note Ser., vol. 146. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  68. 68.
    Yoshino, Y.: Rigid Cohen–Macaulay modules over a three dimensional Gorenstein ring. Oberwolfach, Rep. 2(1), 345–347 (2005)Google Scholar
  69. 69.
    Zhu, B.: Generalized cluster complexes via quiver representations. J. Algebr. Comb. (to appear), arXiv:math.RT/0607155Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Nagoya UniversityNagoyaJapan
  2. 2.Okayama UniversityOkayamaJapan

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