Skip to main content
SpringerLink
Log in

Mutation in triangulated categories and rigid Cohen–Macaulay modules

  • Published:
Inventiones mathematicae Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artin, M., Verdier, J.-L.: Reflexive modules over rational double points. Math. Ann. 270(1), 79–82 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auslander, M.: Coherent functors. In: Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pp. 189–231. Springer, New York (1966)

    Google Scholar 

  3. Auslander, M.: Rational singularities and almost split sequences. Trans. Am. Math. Soc. 293(2), 511–531 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

  6. Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  7. Auslander, M., Reiten, I.: Stable equivalence of dualizing R-varieties. Adv. Math. 12, 306–366 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  8. Auslander, M., Reiten, I.: The Cohen–Macaulay type of Cohen–Macaulay rings. Adv. Math. 73(1), 1–23 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  11. Auslander, M., Reiten, I., Smalo, S.O.: Representation theory of Artin algebras. Camb. Stud. Adv. Math., vol. 36. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  12. Auslander, M., Smalo, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Baur, K., Marsh, R.: A geometric description of m-cluster categories. arXiv:math.RT/0607151

  14. Beligiannis, A., Reiten, I.: Homological Aspects of Torsion Theories. Mem. Am. Math. Soc., vol. 188. Am. Math. Soc. (2007)

  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. Bernšteĭ n, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem. Uspehi Mat. Nauk 28(2), 19–33 (1973)

    MathSciNet  Google Scholar 

  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))

    MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Buan, A., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi–Yau categories and unipotent groups. arXiv:math/0701557

  21. Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Buan, A., Marsh, R., Reiten, I.: Cluster mutation via quiver representations. Comm. Math. Helv. 83(1), 143–177 (2008) (arXiv:math.RT/0412077)

    Article  MathSciNet  MATH  Google Scholar 

  23. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)

    MATH  MathSciNet  Google Scholar 

  24. Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Math. USSR-Izv. 35(3), 519–541 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  25. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358(3), 1347–1364 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Caldero, P., Keller, B.: From triangulated categories to cluster algebras II. Ann. Sci. Éc. Norm. Supér., IV. Sér. 39(6), 983–1009 (2006)

    MATH  MathSciNet  Google Scholar 

  27. Crawley-Boevey, W.W.: On tame algebras and bocses. Proc. Lond. Math. Soc. (3) 56(3), 451–483 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  28. Derksen, H., Weyman, J.: On the canonical decomposition of quiver representations. Compos. Math. 133(3), 245–265 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  30. Drozd, Y.A., Greuel, G.-M.: Tame-wild dichotomy for Cohen–Macaulay modules. Math. Ann. 294(3), 387–394 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  31. Erdmann, K., Holm, T.: Maximal n-orthogonal modules for selfinjective algebras. Proc. Am. Math. Soc. (to appear), arXiv:math.RT/0603672

  32. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)

    MATH  MathSciNet  Google Scholar 

  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.)

  35. Geiss, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. 165(3), 589–632 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  37. Gorodentsev, A.L., Rudakov, A.N.: Exceptional vector bundles on projective spaces. Duke Math. J. 54(1), 115–130 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  39. Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology 39(6), 1155–1191 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  40. Iyama, O.: Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  42. Iyama, O.: Maximal orthogonal subcategories of triangulated categories satisfying Serre duality. Oberwolfach, Rep. 2(1), 353–355 (2005)

    Google Scholar 

  43. Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Am. J. Math. (to appear), arXiv:math.RT/0605136

  44. Kac, V.G.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3), 565–576 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  46. Keller, B.: Deriving DG categories. Ann. Sci. Éc. Norm. Supér., IV. Sér. 27(1), 63–102 (1994)

    MATH  Google Scholar 

  47. Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  49. Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  50. Keller, B., Reiten, I.: Acyclic Calabi–Yau categories are cluster categories. arXiv:math/0610594

  51. Koenig, S., Zhu, B.: From triangulated categories to abelian categories–cluster tilting in a general framework. Math. Z. (to appear), arXiv:math.RT/0605100

  52. Krause, H.: A Brown representability theorem via coherent functors. Topology 41(4), 853–861 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  53. Krause, H.: Cohomological quotients and smashing localizations. Am. J. Math. 127(6), 1191–1246 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  54. Kurano, K.: Private communication

  55. Miyashita, Y.: Tilting modules of finite projective dimension. Math. Z. 193(1), 113–146 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  56. Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Am. Math. Soc. 15(2), 295–366 (2002)

    Article  MATH  Google Scholar 

  57. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. (2) 39(3), 436–456 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  58. Riedtmann, C., Schofield, A.: On a simplicial complex associated with tilting modules. Comment. Math. Helv. 66(1), 70–78 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  59. Rudakov, A.N.: Helices and vector bundles. Seminaire Rudakov. Lond. Math. Soc. Lect. Note Ser., vol. 148. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  60. Schofield, A.: Semi-invariants of quivers. J. Lond. Math. Soc. (2) 43(3), 385–395 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  61. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  62. Tabuada, G.: On the structure of Calabi–Yau categories with a cluster tilting subcategory. Doc. Math. 12, 193–213 (2007)

    MATH  MathSciNet  Google Scholar 

  63. Thomas, H.: Defining an m-cluster category. J. Algebra 318(1), 37–46 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  64. Van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  65. Van den Bergh, M.: Non-commutative crepant resolutions. The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)

    Google Scholar 

  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. Yoshino, Y.: Cohen–Macaulay modules over Cohen–Macaulay rings. Lond. Math. Soc. Lect. Note Ser., vol. 146. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  68. Yoshino, Y.: Rigid Cohen–Macaulay modules over a three dimensional Gorenstein ring. Oberwolfach, Rep. 2(1), 345–347 (2005)

    Google Scholar 

  69. Zhu, B.: Generalized cluster complexes via quiver representations. J. Algebr. Comb. (to appear), arXiv:math.RT/0607155

Download references

Author information

Authors and Affiliations

  1. Nagoya University, Nagoya, 464-8602, Japan

    Osamu Iyama

  2. Okayama University, Okayama, 700-8530, Japan

    Yuji Yoshino

Authors
  1. Osamu Iyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuji Yoshino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Osamu Iyama.

Additional information

Dedicated to Professor Idun Reiten on the occasion of her 65th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iyama, O., Yoshino, Y. Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. math. 172, 117–168 (2008). https://doi.org/10.1007/s00222-007-0096-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-007-0096-4

Keywords

Search

Navigation