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Cohomology of Algebraic Groups with Coefficients in Twisted Representations

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

Abstract

This article is a survey on the cohomology of a reductive algebraic group with coefficients in twisted representations. A large part of the paper is devoted to the advances obtained by the theory of strict polynomial functors initiated by Friedlander and Suslin in the late 90s. The last section explains that the existence of certain ‘universal classes’ used to prove cohomological finite generation is equivalent to some recent ‘untwisting theorems’ in the theory of strict polynomial functors. We actually provide thereby a new proof of these theorems.

2010 Mathematics Subject Classification

Primary 20G10 Secondary 18G15 

Notes

Acknowledgements

The author thanks the anonymous referee for very carefully reading a first version of the article and detecting several mistakes.

References

  1. 1.
    K. Akin, D. A. Buchsbaum, J. Weyman, Schur functors and Schur complexes. Adv. in Math. 44 (1982), no. 3, 207–278.MathSciNetCrossRefGoogle Scholar
  2. 2.
    H.H. Andersen, J.C. Jantzen, Cohomology of induced representations for algebraic groups. Math. Ann. 269 (1984), no. 4, 487–525.MathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Bendel, D. Nakano and C. Pillen, On the vanishing ranges for the cohomology of finite groups of Lie type, Int. Math. Res. Not. (2011).Google Scholar
  4. 4.
    M. Chałupnik, Extensions of strict polynomial functors. Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 773–792.MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Chałupnik, Derived Kan Extension for strict polynomial functors, Int. Math. Res. Not. IMRN 2015, no. 20, 10017–10040.MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. C. Chevalley 1956–1958, Classification des groupes de Lie algébriques, Paris 1958 (Secr. Math.).Google Scholar
  7. 7.
    E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic cohomology. Invent. Math. 39 (1977), no. 2, 143–163.MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Cline, B. Parshall, L. Scott, Detecting rational cohomology of algebraic groups, J. London Math. Soc. (2) 28 (1983), no. 2, 293–300.MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Eilenberg, S. MacLane, On the groups \(H(\Pi ,n)\). II. Methods of computation. Ann. of Math. (2) 60, (1954). 49–139.Google Scholar
  10. 10.
    L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Z. Fiedorowicz and S. Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Mathematics, vol. 674, Springer, Berlin, 1978.CrossRefGoogle Scholar
  12. 12.
    V. Franjou, E. Friedlander, Cohomology of bifunctors. Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 514–544.MathSciNetCrossRefGoogle Scholar
  13. 13.
    V. Franjou, E. Friedlander, A. Scorichenko, A. Suslin, General linear and functor cohomology over finite fields, Ann. of Math. (2) 150 (1999), no. 2, 663–728.MathSciNetCrossRefGoogle Scholar
  14. 14.
    V. Franjou, E. Friedlander, T. Pirashvili, L. Schwartz, Rational representations, the Steenrod algebra and functor homology, Panor. Synthèses, 16, Soc. Math. France, Paris, 2003.Google Scholar
  15. 15.
    E. Friedlander, Lectures on the cohomology of finite group schemes, in Rational representations, the Steenrod algebra and functor homology, 27–53, Panor. Synthèses, 16, Soc. Math. France, Paris, 2003.Google Scholar
  16. 16.
    E. Friedlander, B. Parshall, On the cohomology of algebraic and related finite groups. Invent. Math. 74 (1983), no. 1, 85–117.MathSciNetCrossRefGoogle Scholar
  17. 17.
    E. Friedlander, B. Parshall, Cohomology of Lie algebras and algebraic groups. Amer. J. Math. 108 (1986), no. 1, 235–253 (1986).MathSciNetCrossRefGoogle Scholar
  18. 18.
    E. Friedlander, A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997) 209–270.MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. A. Green, Polynomial representations of \({\rm GL}_{n}\). Lecture Notes in Mathematics, 830. Springer-Verlag, Berlin-New York, 1980.Google Scholar
  20. 20.
    J. C. Jantzen, Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. American Mathematical Society, Providence, RI, 2003.Google Scholar
  21. 21.
    H. Krause, Koszul, Ringel and Serre Duality for strict polynomial functors, Compos. Math. 149 (2013), no. 6, 996–1018.MathSciNetCrossRefGoogle Scholar
  22. 22.
    I. G. Macdonald, Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1979.Google Scholar
  23. 23.
    E. Noether, Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik \(p\). Nachr. Ges. Wiss.Göttingen (1926) 28–35, Collected papers, Springer Verlag, Berlin 1983, pp. 495–492.Google Scholar
  24. 24.
    B. Parshall, L. Scott, D. Stewart, Shifted generic cohomology, Compositio Math. 149 (2013), 1765–1788.MathSciNetCrossRefGoogle Scholar
  25. 25.
    V. T. Pham, Applications des foncteurs strictement polynomiaux, Ph.D. Thesis, Paris 13 University, 2015 (under the direction of L. Schwartz and A. Touzé).Google Scholar
  26. 26.
    D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field ,Ann. of Math.(2) 96 (1972), 552–586.Google Scholar
  27. 27.
    I. Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe, 1927, in Gesammelte Abhandlungen. Vol III (eds. Alfred Brauer und Hans Rohrbach) Springer-Verlag, Berlin-New York, 1973, pp. 68–85.Google Scholar
  28. 28.
    R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963) 33–56.MathSciNetCrossRefGoogle Scholar
  29. 29.
    R. Steinberg, Lectures on Chevalley groups. Notes prepared by John Faulkner and Robert Wilson. Revised and corrected edition of the 1968 original. With a foreword by Robert R. Snapp. University Lecture Series, 66. American Mathematical Society, Providence, RI, 2016.Google Scholar
  30. 30.
    A. Touzé, Universal classes for algebraic groups, Duke Math. J. 151 (2010), no. 2, 219–249.MathSciNetCrossRefGoogle Scholar
  31. 31.
    A. Touzé, Cohomology of classical algebraic groups from the functorial viewpoint, Adv. Math. 225 (2010) 33–68.MathSciNetCrossRefGoogle Scholar
  32. 32.
    A. Touzé, Troesch complexes and extensions of strict polynomial functors, Ann. Sci. E.N.S. 45 (2012), no. 1, 53–99.MathSciNetCrossRefGoogle Scholar
  33. 33.
    A. Touzé, A construction of the universal classes for algebraic groups with the twisting spectral sequence. Transform. Groups 18 (2013), no. 2, 539–556.MathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Touzé, Foncteurs strictement polynomiaux et applications, Habilitation Thesis, Paris 13 University, 2014.Google Scholar
  35. 35.
    A. Touzé, W. van der Kallen, Bifunctor cohomology and Cohomological finite generation for reductive groups, Duke Math. J. 151 (2010), no. 2, 251–278.MathSciNetCrossRefGoogle Scholar
  36. 36.
    W. van der Kallen, Cohomology with Grosshans graded coefficients, In: Invariant Theory in All Characteristics, Edited by: H. E. A. Eddy Campbell and David L. Wehlau, CRM Proceedings and Lecture Notes, Volume 35 (2004) 127-138, Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  37. 37.
    W. C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, 66. Springer-Verlag, New York-Berlin, 1979.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Laboratoire Painlevé, Cité Scientifique - Bâtiment M2Université Lille 1 - Sciences et TechnologiesVilleneuve d’Ascq CedexFrance

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