Advertisement

Noncommutative Geometry in the Framework of Differential Graded Categories

  • Snigdhayan Mahanta
Part of the Progress in Mathematics book series (PM, volume 279)

Summary

In this survey we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.

Keywords

Abelian Variety Noncommutative Geometry Coherent Sheave Triangulate Category Noncommutative Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Artin and J. J. Zhang. Noncommutative projective schemes. Adv. Math., 109(2):228–287, 1994.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P. Balmer. Presheaves of triangulated categories and reconstruction of schemes. Math. Ann., 324(3):557–580, 2002.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Beilinson and V. Vologodsky. A DG guide to Voevodsky’s motives. Geom. Funct. Anal., 17(6):1709–1787, 2008.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. A. Beĭlinson. The derived category of coherent sheaves on P n. Selecta Math. Soviet., 3(3):233–237, 1983/84. Selected translations.Google Scholar
  5. [5]
    R. Bezrukavnikov. Perverse coherent sheaves (after Deligne). math/0005152.Google Scholar
  6. [6]
    R. Bezrukavnikov, I. Mirković, and D. Rumynin. Singular localization and intertwining functors for reductive Lie algebras in prime characteristic. Nagoya Math. J., 184:1–55, 2006.MATHMathSciNetGoogle Scholar
  7. [7]
    J. Block. Duality and equivalence of module categories in noncommutative geometry I. math.QA/0509284.Google Scholar
  8. [8]
    J. Block. Duality and equivalence of module categories in noncommutative geometry II: Mukai duality for holomorphic noncommutative tori. math.QA/0604296.Google Scholar
  9. [9]
    A. Bondal and D. Orlov. Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math., 125(3):327–344, 2001.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A. Bondal and M. van den Bergh. Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J., 3(1):1–36, 258, 2003.Google Scholar
  11. [11]
    A. I. Bondal and M. M. Kapranov. Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat., 53(6):1183–1205, 1337, 1989.Google Scholar
  12. [12]
    A. I. Bondal, M. Larsen, and V. A. Lunts. Grothendieck ring of pretriangulated categories. Int. Math. Res. Not., (29):1461–1495, 2004.Google Scholar
  13. [13]
    A. I. Bondal and D. Orlov. Semiorthogonal decomposition for algebraic varieties. alg-geom/9506012.Google Scholar
  14. [14]
    P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan. Chern character for twisted complexes. In Geometry and dynamics of groups and spaces, volume 265 of Progr. Math., pages 309–324. Birkhäuser, Basel, 2008.Google Scholar
  15. [15]
    A. Connes. Noncommutative geometry. Academic Press Inc., San Diego, CA, 1994.MATHGoogle Scholar
  16. [16]
    A. Connes. Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Math. (N.S.), 5(1):29–106, 1999.Google Scholar
  17. [17]
    A. Connes. Noncommutative geometry and the Riemann zeta function. In Mathematics: frontiers and perspectives, pages 35–54. Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  18. [18]
    A. Connes, M. Marcolli, and N. Ramachandran. KMS states and complex multiplication. Selecta Math. (N.S.), 11(3-4):325–347, 2005.Google Scholar
  19. [19]
    K. J. Costello. Topological conformal field theories and Calabi-Yau categories. Adv. Math., 210(1):165–214, 2007.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    J. Denef and F. Loeser. Motivic integration and the Grothendieck group of pseudo-finite fields. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 13–23, Beijing, 2002. Higher Ed. Press.Google Scholar
  21. [21]
    C. Deninger. Number theory and dynamical systems on foliated spaces. Jahresber. Deutsch. Math.-Verein., 103(3):79–100, 2001.Google Scholar
  22. [22]
    C. Deninger. Two-variable zeta functions and regularized products. Doc. Math., (Extra Vol.):227–259 (electronic), 2003. Kazuya Kato’s fiftieth birthday.Google Scholar
  23. [23]
    V. Drinfeld. DG quotients of DG categories. J. Algebra, 272(2):643–691, 2004.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    D. Dugger and B. Shipley. K-theory and derived equivalences. Duke Math. J., 124(3):587–617, 2004.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    E. Frenkel. Lectures on the Langlands program and conformal field theory. In Frontiers in number theory, physics, and geometry. II, pages 387–533. Springer, Berlin, 2007.Google Scholar
  26. [26]
    E. M. Friedlander, A. Suslin, and V. Voevodsky. Introduction. In Cycles, transfers, and motivic homology theories, volume 143 of Ann. of Math. Stud., pages 188–238. Princeton Univ. Press, Princeton, NJ, 2000.Google Scholar
  27. [27]
    P. Gabriel. Des catégories abéliennes. Bull. Soc. Math. France 90, 323-448, MR 38:1411, 1962.Google Scholar
  28. [28]
    V. Ginzburg. Lectures on Noncommutative Geometry. math.AG/0506603.Google Scholar
  29. [29]
    E. Ha and F. Paugam. Bost-Connes-Marcolli systems for Shimura varieties.I. Definitions and formal analytic properties. IMRP Int. Math. Res. Pap., (5):237–286, 2005.Google Scholar
  30. [30]
    U. Jannsen. Motives, numerical equivalence, and semi-simplicity. Invent. Math., 107(3):447–452, 1992.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    M. Kapranov. The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups. math.AG/0001005.Google Scholar
  32. [32]
    M. Kapranov. Noncommutative geometry based on commutator expansions. J. Reine Angew. Math., 505:73–118, 1998.MATHMathSciNetGoogle Scholar
  33. [33]
    M. Kashiwara. The Riemann-Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci., 20(2):319–365, 1984.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    B. Keller. Deriving DG categories. Ann. Sci. École Norm. Sup. (4), 27(1):63–102, 1994.Google Scholar
  35. [35]
    B. Keller. Invariance and localization for cyclic homology of DG algebras. J. Pure Appl. Algebra, 123(1-3):223–273, 1998.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    B. Keller. A-infinity algebras, modules and functor categories. In Trends in representation theory of algebras and related topics, volume 406 of Contemp. Math., pages 67–93. Amer. Math. Soc., Providence, RI, 2006.Google Scholar
  37. [37]
    B. Keller. On differential graded categories. In International Congress of Mathematicians. Vol. II, pages 151–190. Eur. Math. Soc., Zürich, 2006.Google Scholar
  38. [38]
    B. Keller and I. Reiten. Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. math/0512471.Google Scholar
  39. [39]
    M. Kontsevich. Notes on motives in finite characteristic. math/0702206.Google Scholar
  40. [40]
    M. Kontsevich and Y. Soibelman. Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I. math.RA/0606241.Google Scholar
  41. [41]
    M. Larsen and V. A. Lunts. Motivic measures and stable birational geometry. Mosc. Math. J., 3(1):85–95, 259, 2003.Google Scholar
  42. [42]
    M. Larsen and V. A. Lunts. Rationality criteria for motivic zeta functions. Compos. Math., 140(6):1537–1560, 2004.MATHMathSciNetGoogle Scholar
  43. [43]
    O. A. Laudal. Noncommutative algebraic geometry. In Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), volume 19, pages 509–580, 2003.Google Scholar
  44. [44]
    E. Looijenga. Motivic measures. Astérisque, (276):267–297, 2002. Séminaire Bourbaki, Vol. 1999/2000.Google Scholar
  45. [45]
    J. Lurie. Derived algebraic geometry II: Noncommutative algebra. math/0702299.Google Scholar
  46. [46]
    J. Lurie. Derived algebraic geometry III: Commutative algebra. math/0703204.Google Scholar
  47. [47]
    V. Lyubashenko and O. Manzyuk. A-infinity-bimodules and Serre A-infinity-functors. math/0701165.Google Scholar
  48. [48]
    Y. I. Manin. Correspondences, motifs and monoidal transformations. Mat. Sb. (N.S.), 77 (119):475–507, 1968.Google Scholar
  49. [49]
    M. Marcolli. Arithmetic noncommutative geometry, volume 36 of University Lecture Series. American Mathematical Society, Providence, RI, 2005. With a foreword by Yu. I. Manin.Google Scholar
  50. [50]
    S. Mukai. Duality between D(X) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J., 81:153–175, 1981.MATHMathSciNetGoogle Scholar
  51. [51]
    D. Orlov. Derived categories of coherent sheaves on abelian varieties and equivalences between them. Izv. Ross. Akad. Nauk Ser. Mat., 66(3):131–158, 2002.MathSciNetGoogle Scholar
  52. [52]
    J. Plazas. Arithmetic structures on noncommutative tori with real multiplication. math.QA/0610127.Google Scholar
  53. [53]
    A. Polishchuk and A. Schwarz. Categories of holomorphic vector bundles on noncommutative two-tori. Comm. Math. Phys., 236(1):135–159, 2003.MATHCrossRefMathSciNetGoogle Scholar
  54. [54]
    B. Poonen. The Grothendieck ring of varieties is not a domain. Math. Res. Lett., 9(4):493–497, 2002.MATHMathSciNetGoogle Scholar
  55. [55]
    A. L. Rosenberg. The spectrum of abelian categories and reconstruction of schemes. In Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), volume 197 of Lecture Notes in Pure and Appl. Math., pages 257–274. Dekker, New York, 1998.Google Scholar
  56. [56]
    S. Schwede and B. Shipley. Stable model categories are categories of modules. Topology, 42(1):103–153, 2003.MATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    G. Tabuada. Théorie homotopique des dg-categories. arXiv:0710.4303.Google Scholar
  58. [58]
    G. Tabuada. Invariants additifs de DG-catégories. Int. Math. Res. Not., (53):3309–3339, 2005.Google Scholar
  59. [59]
    G. Tabuada. Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories. C. R. Math. Acad. Sci. Paris, 340(1):15–19, 2005.MATHMathSciNetGoogle Scholar
  60. [60]
    G. Tabuada. On the structure of Calabi-Yau categories with a cluster tilting subcategory. Doc. Math., 12:193–213 (electronic), 2007.Google Scholar
  61. [61]
    G. Tabuada. Higher K-theory via universal invariants. Duke Math. J., 145(1):121–206, 2008.MATHCrossRefMathSciNetGoogle Scholar
  62. [62]
    R. W. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. In The Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 247–435. Birkhäuser Boston, Boston, MA, 1990.Google Scholar
  63. [63]
    B. Toën. Lectures on dg-categories. For the Sedano winter school on K-theory Sedano, January 2007.Google Scholar
  64. [64]
    B. Toën. The homotopy theory of dg-categories and derived Morita theory. Invent. Math., 167(3):615–667, 2007.MATHCrossRefMathSciNetGoogle Scholar
  65. [65]
    B. Toën and G. Vezzosi. From HAG to DAG: derived moduli stacks. In Axiomatic, enriched and motivic homotopy theory, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 173–216. Kluwer Acad. Publ., Dordrecht, 2004.Google Scholar
  66. [66]
    B. Toën and G. Vezzosi. Homotopical algebraic geometry. I. Topos theory. Adv. Math., 193(2):257–372, 2005.Google Scholar
  67. [67]
    M. van den Bergh. Blowing up of non-commutative smooth surfaces. Mem. Amer. Math. Soc., 154(734):x+140, 2001.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. of Math.University of TorontoTorontoCanada

Personalised recommendations