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Erschienen in:
Rings, Algebras and Modules Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

6. Derived Equivalences

verfasst von : Alexander Zimmermann

Erschienen in: Representation Theory

Verlag: Springer International Publishing

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Abstract

The derived category of an algebra was introduced in Chap. 3. In this Chap. 6 we shall study equivalences between the derived categories of two algebras. The main result is Rickard’s and Keller’s Morita theory for derived categories. We shall study many relations between this concept and stable equivalences. We also provide a long list of invariants under these equivalences and study singular categories as well as Picard groups for derived categories.

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Vorheriges Kapitel Stable Module Categories
Fußnoten
1

S. Koshitani kindly translated Okuyama’s paper [37] for me. I am very grateful to him.

 
Literatur
  1. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39(2), 436–456 (1989)View ArticleMATHMathSciNet
  2. Rickard, J.: Derived equivalences as derived functors. J. Lond. Math. Soc. 43, 37–48 (1991)
  3. Keller, B.: Bimodule complexes via strong homotopy actions. Algebras Represent. Theory 3, 357–376 (2000)
  4. Montgomery, S.: Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, vol. 82. American Mathematical Society, Providence, R.I. (1993)
  5. Brzezinski, T., Wisbauer, R.: Corings and comodules. London Mathematical Society Lecture Notes, vol. 309. Cambridge University Press, Cambridge (2003)
  6. Stasheff, J.D.: The intrinsic bracket on the deformation complex of an associative algebra. J. Pure Appl. Algebra 89, 231–235 (1993)
  7. Stasheff, J.D.: Homotopy associativity of \(H\)-spaces. I, II. Trans. Am. Math. Soc. 108, 275–292 (1963) (ibid. 108 (1963) 293–312)
  8. Zimmermann, A.: Twosided tilting complexes for Green orders and Brauer tree algebras. J. Algebra 187, 446–473 (1997)
  9. Zimmermann, A.: Twosided tilting complexes for Gorenstein orders. J. Algebra 209, 585–621 (1998)
  10. Zimmermann, A.: Fine Hochschild invariants of derived categories for symmetric algebras. J. Algebra 308, 350–367 (2007)
  11. König, S., Zimmermann, A.: Derived equivalences for group rings, with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier. Lecture Notes in Mathematics, vol. 1685. Springer, Berlin (1998)
  12. Muchtadi-Alamsyah, I.: Endomorphismes de complexes déterminés par leurs homologies, thèse de doctorat Université de Picardie (2004)
  13. Muchtadi-Alamsyah, I.: Homomorphisms of complexes via homologies. J. Algebra 294, 321–345 (2005)
  14. Rickard, J.: Infinitely many algebras derived equivalent to a block (preprint 2013), http://​arxiv.​org/​abs/​1310.​2403v1arXiv:1310.2403v1
  15. König, S., Zimmermann, A.: Tilting hereditary orders. Commun. Algebra 24(6), 1897–1913 (1996)
  16. Carlson, J., Thévenaz, J.: Torsion endo-trivial modules. Algebras Represent. Theory 3, 303–335 (2000)
  17. Carlson, J., Thévenaz, J.: The classification of endo-trivial modules. Inventiones Math. 158, 389–411 (2004)
  18. Carlson, J., Thévenaz, J.: The classification of torsion endo-trivial modules. Ann. Math. 162, 823–883 (2005)
  19. Keller, B.: Invariance and localization for cyclic homology of DG algebras. J. Pure Appl. Algebra 123, 223–273 (1998)
  20. Zimmermann, A.: Tilted symmetric orders are symmetric orders. Archiv der Mathematik 73, 15–17 (1999)
  21. Al-Nofayeeh, S.: Equivalences of derived categories for selfinjective algebras. J. Algebra 313, 897–904 (2007)
  22. Hartshorne, R.: Residues and duality, Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963/64. Lecture Notes in Mathematics, vol. 20. Springer, Berlin (1966)
  23. Broué, M.: Isométries parfaites, types de blocs, catégories dérivées. Astérisque 181–182, 61–92 (1990)
  24. Broué, M.: Isométries parfaites, types de blocs, catégories dérivées, Représentations Linéaires des Groupes Finis, Luminy 16–21 mai 1988, ed: Marc Cabanes; pages 61–92, Astérisque 181–182 (1990)
  25. Buchweitz, R.O.: Maximal Cohen-Macaulay modules and Tate cohomology, manuscript Universität Hannover (1986)
  26. Orlov, D.: Triangulated categories of singularities and \(D\)-branes in Ginzburg-Landau models. Trudy Steklov Math. Inst. 246, 240–262 (2004)
  27. Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: in honor of Yurij I. Manin. Volume II, 503–531. Progress in Mathematics, vol. 270. Birkhäuser Boston, Boston (2009)
  28. Orlov, D.: Formal completions and idempotent completions of triangulated categories of singularities. Adv. Math. 226, 206–217 (2011)
  29. Rickard, J.: Derived categories and stable equivalences. J. Pure Appl. Algebra 61, 303–317 (1989)
  30. Keller, B., Vossieck, D.: Sous les catégories dérivées. Comptes Rendus de l’Académie des Sciences Paris, Série I Mathématique 305(6), 225–228 (1987)
  31. Beligiannis, A.: The homological theory of contravariantly finite subcategories: Auslander-Reiten contexts, Gorenstein categories and (co-)stabilizations. Commun. Algebra 28(10), 4547–4596 (2000)
  32. Chen, X.-W.: The singularity category of an algebra with radical square zero. Documenta Mathematica 16, 921–936 (2011)
  33. König, S., Zimmermann, A.: Tilting selfinjective algebras and Gorenstein orders, Oxford. Q. J. Math. 48(2), 351–361 (1997)
  34. Reiner, I.: Maximal Orders. Academic Press, London (1975)
  35. Schaps, M., Zakay-Illouz, E.: Pointed Brauer trees. J. Algebra 246, 647–672 (2001)
  36. Okuyama, T.: Some examples of derived equivalent blocks of finite groups (unpublished preprint 1997)
  37. Okuyama, T.: Remarks on splendid tilting complexes, (Japanese). In: Koshitani, S. (ed.) Representations of finite groups and related topics RIMS Kokyuroku. Proceedings of the Research Institute of Mathematical Sciences, vol. 1149, pp. 53–59. Kyoto University (2000)
  38. Kunugi, N.: Derived equivalences in symmetric groups. Surikaisekikenkyusho Kokyuroku 1140, 131–135 (2000) (Research on the cohomology theory of finite groups (Japanese) (Kyoto, 1999))
  39. Kunugi, N.: Morita equivalent 3-blocks of the 3-dimensional projective special linear groups. Proc. Lond. Math. Soc. 80, 575–589 (2000)
  40. Kunugi, N., Waki, K.: Derived equivalences for the 3-dimensional special unitary groups in non-defining characteristic. J. Algebra 240, 251–267 (2001)
  41. Koshitani, S., Kunugi, N.: Broué’s conjecture holds for principal 3-blocks with elementary abelian defect group of order 9. J. Algebra 248, 575–604 (2002)
  42. Koshitani, S., Kunugi, N., Waki, K.: Broué’s conjecture for non-principal 3-blocks of finite groups. J. Pure Appl. Algebra 173, 177–211 (2002)
  43. Koshitani, S., Kunugi, N., Waki, K.: Broué’s Abelian defect group conjecture for the Held group and the sporadic Suzuki group. J. Algebra 279, 638–666 (2004)
  44. Koshitani, S., Kunugi, N., Waki, K.: Broué’s abelian defect conjecture holds for the Janko simple group \(J_4\). J. Pure Appl. Algebra 212, 1438–1456 (2008)
  45. Koshitani, S., Müller, J.: Broué’s abelian defect group conjecture holds for the Harada-Norton sporadic simple group HN. J. Algebra 324, 394–429 (2010)
  46. Rickard, J.: Equivalences of derived categories for symmeric algebras. J. Algebra 257, 460–481 (2002)
  47. König, S., Yang, D.: Silting objects, simple-minded collections, \(t\)-structures and co-\(t\)-structures for finite-dimensional algebras. Doc. Math. 19, 403–438 (2014)
  48. Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. 85, 633–668 (2012)
  49. Keller, B., Vossieck, D.: Aisles in derived categories. Bulletin de la Société mathématique de la Belgique, Série A 40, 239–253 (1988)
  50. Brenner, S., Butler, M.: Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors. Representation theory, II (Proceedings of the Second International Conference, Carleton University, Ottawa, Ontario, 1979), pp. 103–169. Lecture Notes in Mathematics, vol. 832. Springer, Berlin-New York (1980)
  51. Happel, D.: On the derived category of a finite-dimensional algebra. Commentarii Mathematici Helvetici 62, 339–389 (1987)
  52. Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)
  53. Keller, B.: Deriving DG categories. Annales Scientifiques de l’École Normale Supérieure 27(4), 63–102 (1994)MATH
  54. Broué, M.: Equivalences of blocks of group algebras. Finite dimensional algebras and related topics (Ottawa, ON, 1992), 126, NATO Advanced Science Institute Series C Mathematics Physics Science, vol. 42. Kluwer Academic Publisher, Dordrecht (1994)
  55. Külshammer, B.: Offene Probleme in der Darstellungstheorie endlicher Gruppen. Jahresbericht der Deutschen Mathematiker Vereinigung 94, 98–104 (1992)
  56. Marcus, A.: On equivalences between blocks of group algebras: reduction to the simple components. J. Algebra 184, 372–396 (1996)
  57. Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \( {\mathfrak{s}l}_2\)-categorification. Ann. Math. 167(2), 245–298 (2008)View ArticleMATHMathSciNet
  58. Holm, T.: Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type. J. Algebra 211, 159–205 (1999)
  59. Rouquier, R., Zimmermann, A.: Picard groups for derived module categories. Proc. Lond. Math. Soc. 87(3), 197–225 (2003)
  60. Fröhlich, A.: The Picard group of noncommutative rings, in particular of orders. Trans. Am. Math. Soc. 180, 1–45 (1973)
  61. Roggenkamp, K.W., Zimmermann, A.: Outer group automorphisms can become inner in the integral group ring. J. Pure Appl. Algebra 103, 91–99 (1995)
  62. Serre, J.P.: Arbres, amalgames, \(SL_2\), Société Mathématique de France, Astérisque 46, Paris (1977)
  63. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108, 37–108 (2001)
  64. Khovanov, M., Seidel, P.: Quivers, and braid group actions. J. Am. Math. Soc. 15, 203–271 (2002)
  65. Zimmermann, A.: Automorphisms of green orders and their derived categories. Algebras Represent. Theory 7(1), 19–34 (2004)
  66. Newman, M.: Integral Matrices. Academic Press, New York (1972)
  67. Keller, B.: Hochschild cohomology and derived Picard groups. J. Pure Appl. Algebra 190, 177–196 (2004)View ArticleMATHMathSciNet
Metadaten
Titel
Derived Equivalences
verfasst von
Alexander Zimmermann
Copyright-Jahr
2014
Buch
Representation Theory

Print ISBN: 978-3-319-07967-7

Electronic ISBN: 978-3-319-07968-4

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-07968-4_6

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