Varieties for Modules of Finite Dimensional Hopf Algebras

  • Sarah WitherspoonEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)


We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples such as finite group algebras and finite group schemes. We list open questions about tensor products of modules and projectivity, where varieties may play a role in finding answers.


  1. 1.
    L. L. Avramov and R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318.MathSciNetzbMATHGoogle Scholar
  2. 2.
    L. L. Avramov and S. B. Iyengar, Constructing modules with prescribed cohomological support, Illinois J. Math. 51 (2007), no. 1, 1–20.MathSciNetzbMATHGoogle Scholar
  3. 3.
    G. S. Avrunin and L. L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), 277–286.MathSciNetzbMATHGoogle Scholar
  4. 4.
    B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, Vol. 21, University Lecture Series, American Mathematical Society, Providence, RI, 2001.zbMATHGoogle Scholar
  5. 5.
    C. Bendel, D. K. Nakano, B. J. Parshall, and C. Pillen, Cohomology for quantum groups via the geometry of the nullcone, Mem. Amer. Math. Soc. 229 (2014), no. 1077.Google Scholar
  6. 6.
    D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge University Press, 1991.Google Scholar
  7. 7.
    D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge studies in advanced mathematics 31, Cambridge University Press, 1991.Google Scholar
  8. 8.
    D. J. Benson and E. L. Green, Non-principal blocks with one simple module, Q. J. Math. 55 (2004) (1), 1–11.MathSciNetzbMATHGoogle Scholar
  9. 9.
    D. J. Benson and S. Witherspoon, Examples of support varieties for Hopf algebras with noncommutative tensor products, Archiv der Mathematik 102 (2014), no. 6, 513–520.MathSciNetzbMATHGoogle Scholar
  10. 10.
    D. J. Benson, J. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. J. Benson, K. Erdmann, and M. Holloway, Rank varieties for a class of finite-dimensional local algebras, J. Pure Appl. Algebra 211 (2) (2007), 497–510.MathSciNetzbMATHGoogle Scholar
  12. 12.
    D. J. Benson, S. Iyengar, and H. Krause, Stratifying modular representations of finite groups, Ann. of Math. 174 (2011), 1643–1684.MathSciNetzbMATHGoogle Scholar
  13. 13.
    D. J. Benson, S. Iyengar, H. Krause, and J. Pevtsova, Stratification for module categories of finite group schemes, to appear in J. Amer. Math. Soc.Google Scholar
  14. 14.
    P. A. Bergh and K. Erdmann, The Avrunin-Scott Theorem for quantum complete intersections, J. Algebra 322 (2009), no. 2, 479–488.MathSciNetzbMATHGoogle Scholar
  15. 15.
    B. D. Boe, J. R. Kujawa, and D. K. Nakano, Cohomology and support varieties for Lie superalgebras, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6551–6590.MathSciNetzbMATHGoogle Scholar
  16. 16.
    B. D. Boe, J. R. Kujawa, and D. K. Nakano, Tensor triangular geometry for quantum groups, arxiv:1702.01289.
  17. 17.
    J. F. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104–143.MathSciNetzbMATHGoogle Scholar
  18. 18.
    J. F. Carlson and S. Iyengar, Thick subcategories of the bounded derived category of a finite group, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2703–2717.MathSciNetzbMATHGoogle Scholar
  19. 19.
    J. F. Carlson and S. Iyengar, Hopf algebra structures and tensor products for group algebras, New York J. Math. 23 (2017), 351–364.MathSciNetzbMATHGoogle Scholar
  20. 20.
    M. Duflo and V. Serganova, On associated variety for Lie superalgebras, arxiv:0507198.
  21. 21.
    K. Erdmann, M. Holloway, N. Snashall, Ø. Solberg, and R. Taillefer, Support varieties for selfinjective algebras, K-Theory 33 (2004), no. 1, 67–87.MathSciNetzbMATHGoogle Scholar
  22. 22.
    P. Etingof and V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004), no. 3, 627–654, 782–783.Google Scholar
  23. 23.
    L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239.MathSciNetzbMATHGoogle Scholar
  24. 24.
    R. Farnsteiner, Tameness and complexity of finite group schemes, Bull. London Math. Soc. 39 (2007), no. 1, 63–70.MathSciNetzbMATHGoogle Scholar
  25. 25.
    J. Feldvoss and S. Witherspoon, Support varieties and representation type of small quantum groups, Int. Math. Res. Not. (2010), no. 7, 1346–1362. Erratum, Int. Math. Res. Not. (2015), no. 1, 288–290.MathSciNetzbMATHGoogle Scholar
  26. 26.
    E. M. Friedlander, Spectrum of group cohomology and support varieties, J. K-Theory 11 (2013), 507–516.MathSciNetzbMATHGoogle Scholar
  27. 27.
    E. M. Friedlander and B. J. Parshall, On the cohomology of algebraic and related finite groups, Invent. Math. 74 (1983), no. 1, 85–117.MathSciNetzbMATHGoogle Scholar
  28. 28.
    E. M. Friedlander and B. J. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), no. 3, 553–562.MathSciNetzbMATHGoogle Scholar
  29. 29.
    E. M. Friedlander and J. Pevtsova, Representation theoretic support spaces for finite group schemes, Amer. J. Math. 127 (2005), 379–420. Correction: Amer. J. Math. 128 (2006), 1067–1068.Google Scholar
  30. 30.
    E. M. Friedlander and J. Pevtsova, \(\Pi \)-supports for modules for finite group schemes over a field, Duke Math. J. 139 (2007), no. 2, 317–368.MathSciNetzbMATHGoogle Scholar
  31. 31.
    E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270.MathSciNetzbMATHGoogle Scholar
  32. 32.
    V. Ginzburg and S. Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), 179–198.MathSciNetzbMATHGoogle Scholar
  33. 33.
    E. Golod, The cohomology ring of a finite p-group, (Russian) Dokl. Akad. Nauk SSSR 235 (1959), 703–706.MathSciNetzbMATHGoogle Scholar
  34. 34.
    I. G. Gordon, Cohomology of quantized function algebras at roots of unity, Proc. London Math. Soc. (3) 80 (2000), no. 2, 337–359.MathSciNetzbMATHGoogle Scholar
  35. 35.
    I. G. Gordon, Complexity of representations of quantised function algebras and representation type, J. Algebra 233 (2000), no. 2, 437–482.MathSciNetzbMATHGoogle Scholar
  36. 36.
    J. C. Jantzen, Kohomologie von p-Lie-Algebren und nilpotente Elemente, Abh. Math. Sem. Univ. Hamburg 56 (1986), 191–219.MathSciNetzbMATHGoogle Scholar
  37. 37.
    S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Conf. Math. Publ. 82, Amer. Math. Soc., 1993.Google Scholar
  38. 38.
    M. Mastnak, J. Pevtsova, P. Schauenburg, and S. Witherspoon, Cohomology of finite dimensional pointed Hopf algebras, Proc. London Math. Soc. (3) 100 (2010), no. 2, 377–404.MathSciNetzbMATHGoogle Scholar
  39. 39.
    T. Niwasaki, On Carlson’s conjecture for cohomology rings of modules, J. Pure Appl. Algebra 59 (1989), no. 3, 265–277.MathSciNetzbMATHGoogle Scholar
  40. 40.
    V. V. Ostrik, Support varieties for quantum groups, Functional Analysis and its Applications 32 (1998), no. 4, 237–246.MathSciNetzbMATHGoogle Scholar
  41. 41.
    I. Penkov and V. Serganova, The support of an irreducible Lie algebra representation, J. Algebra 209 (1998), no. 1, 129–142.MathSciNetzbMATHGoogle Scholar
  42. 42.
    J. Pevtsova, Representations and cohomology of finite group schemes, in Advances in Representation Theory of Algebras, EMS Series of Congress Reports (2013), 231–262.Google Scholar
  43. 43.
    J. Pevtsova and S. Witherspoon, Varieties for modules of quantum elementary abelian groups, Algebras and Rep. Th. 12 (2009), no. 6, 567–595.MathSciNetzbMATHGoogle Scholar
  44. 44.
    J. Pevtsova and S. Witherspoon, Tensor ideals and varieties for modules of quantum elementary abelian groups, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3727–3741.MathSciNetzbMATHGoogle Scholar
  45. 45.
    J. Plavnik and S. Witherspoon, Tensor products and support varieties for some noncocommutative Hopf algebras, Algebras and Rep. Th. 21 (2018), no. 2, 259–276.MathSciNetzbMATHGoogle Scholar
  46. 46.
    D. Quillen, The spectrum of an equivariant cohomology ring: I, II, Ann. Math. 94 (1971), 549–572, 573–602.MathSciNetzbMATHGoogle Scholar
  47. 47.
    S. Scherotzke and M. Towers, Rank varieties for Hopf algebras, J. Pure Appl. Algebra 215 (2011), 829–838.MathSciNetzbMATHGoogle Scholar
  48. 48.
    S. F. Siegel and S. J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. 79 (1999), 131–157.MathSciNetzbMATHGoogle Scholar
  49. 49.
    N. Snashall, Support varieties and the Hochschild cohomology ring modulo nilpotence, Proceedings of the 41st Symposium on Ring Theory and Representation Theory, 68–82, Symp. Ring Theory Represent. Theory Organ. Comm., Tsukuba, 2009.Google Scholar
  50. 50.
    N. Snashall and Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732.Google Scholar
  51. 51.
    Ø. Solberg, Support varieties for modules and complexes, Trends in Representation Theory of Algebras and Related Topics, Contemp. Math. 406, 239–270, AMS, Providence, RI, 2006.Google Scholar
  52. 52.
    D. Ştefan and C. Vay, The cohomology ring of the 12-dimensional Fomin-Kirillov algebra, Adv. Math. 291 (2016), 584–620.MathSciNetzbMATHGoogle Scholar
  53. 53.
    M. Suarez-Alvarez, The Hilton-Eckmann argument for the anti-commutativity of cup-products, Proc. Amer. Math. Soc. 132 (8) (2004), 2241–2246.MathSciNetzbMATHGoogle Scholar
  54. 54.
    A. Suslin, E. M. Friedlander, and C. P. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), no. 3, 729–759.MathSciNetzbMATHGoogle Scholar
  55. 55.
    B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSR 127 (1959), 943–944.MathSciNetzbMATHGoogle Scholar
  56. 56.
    F. Xu, Hochschild and ordinary cohomology rings of small categories, Adv. Math. 219 (2008), 1872–1893.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations