Skip to main content
SpringerLink
Log in

Rational Cohomology and Supports for Linear Algebraic Groups

  • Conference paper
  • First Online:
Geometric and Topological Aspects of the Representation Theory of Finite Groups (PSSW 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 242))

Included in the following conference series:

  • 627 Accesses

Abstract

What follows are rough “notes” based upon four lectures given by the author at PIMS in Vancouver over the period June 27–June 30, 2016.

  • Lecture I. Affine groups schemes over k.

  • Lecture II. Algebraic representations.

  • Lecture III. Cohomological support varieties.

  • Lecture IV. Support varieties for linear algebraic groups.

Dedicated to David J. Benson

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Alperin and L. Evens, Representations resolutions, and Quillen’s dimension theorem, J. Pure and Appl Alg. 22 (1981), 1–9.

    Article  MathSciNet  Google Scholar 

  2. J. Alperin and L. Evens, Varieties and elementary abelian groups, J. Pure and Appl Alg. 26 (1982), 221–227.

    Article  MathSciNet  Google Scholar 

  3. G. Avrunin, S. Scott, Quillen stratification for modules, Invent. Math 66 (1982), 277–286.

    Article  MathSciNet  Google Scholar 

  4. P. Bardsley, R. Richardson, Étale slices for algebraic transformation groups in characteristic p. Proc. London Math Soc 51 (1985), 295–317.

    Article  MathSciNet  Google Scholar 

  5. D. Benson, Representations of elementary abelian p-groups and vector bundles. CUP, 2016.

    Google Scholar 

  6. D. Benson and J. Pevtsova, A realization theorem for modules of constant Jordan type and vector bundles, Trans. Amer. Math. Soc. 364 (2012), 6459–6478.

    Article  MathSciNet  Google Scholar 

  7. V. Bondarko and V. Drozd, The representation type of finite groups, Zap. Nauncn Sem. Leningrad. Otdel Math. INst. Steklov (LOMI) 71 (1977), 24–41.

    Google Scholar 

  8. A. Borel, Sur la cohomologie des espaces fibres principaux et des espaes homogenenes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–207.

    Article  MathSciNet  Google Scholar 

  9. J. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), 101–143.

    Article  MathSciNet  Google Scholar 

  10. J. Carlson, The variety of an indecomposable module is connected, Invent. Math 77 (1984), 291–299.

    Article  MathSciNet  Google Scholar 

  11. J. Carlson, E. Friedlander, J. Pevtsova, Modules of constant Jordan type, J. für die reine und angewandte Mathematik 614 (208), 191–234.

    Google Scholar 

  12. J. Carlson, E. Friedlander, J. Pevtsova, Vector bundles associated to Lie Algebras, J. Reine Angew. Math 716 (2016), 147–178.

    MathSciNet  MATH  Google Scholar 

  13. J. Carlson, Z. Lin, and D. Nakano, Support varieties for modules over Chevalley groups and classical Lie algebras, Trans. A.M.S. 360 (2008), 1870–1906.

    Google Scholar 

  14. E. Cline, B. Parshall, L. Scott, Induced modules and affine quotients, Math. Ann 230 (1977), 1–14.

    Article  MathSciNet  Google Scholar 

  15. E. Cline, B. Parshall, L. Scott, Cohomology, hyperalgebras, and representations, J.Algebra 63 (1980), 98–123.

    Article  MathSciNet  Google Scholar 

  16. E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic cohomology, Invent. Math 39 (1977), 143–163.

    Article  MathSciNet  Google Scholar 

  17. S. Donkin, On projective modules for algebraic groups, J. London Math. Soc. (2) (1996), 75–88.

    Article  MathSciNet  Google Scholar 

  18. L. Evens, The cohomology ring of a finite group, Trans. A.M.S. 101 (1961), 224–239.

    Article  MathSciNet  Google Scholar 

  19. E. Friedlander, Filtrations, 1-parameter subgroups, and rational injectivity, Adv. Math. 323 (2018), 84–113. arXiv:408.2918.

    Article  MathSciNet  Google Scholar 

  20. E. Friedlander, Restrictions to \(G({\mathbb{F}}_p)\) and \(G_{(r)}\) of rational \(G\) -modules, Compos. Math. 147 (2011), no. 6, 1955–1978.

    Google Scholar 

  21. E. Friedlander, Support varieties for rational representations, Compositio Math 151 (2015), 765–792.

    Article  MathSciNet  Google Scholar 

  22. E. Friedlander, B. Parshall, On the cohomology of algebraic and related finite groups, Invent. Math 74 (1983), 85–117.

    Article  MathSciNet  Google Scholar 

  23. E. Friedlander, B. Parshall, Cohomology of infinitesimal and discrete groups, Math. Ann 273 (1986), 353–374.

    Article  MathSciNet  Google Scholar 

  24. E. Friedlander, B. Parshall, Geometry of \(p\)-Unipotent Lie Agebras, J. Algebra 109 (1987), 25–45.

    Google Scholar 

  25. E. Friedlander, B. Parshall, Modular representation theory of restricted Lie algebras, Amer. J. Math. 110 (1988), 1055–1094.

    Article  MathSciNet  Google Scholar 

  26. E. Friedlander, J. Pevtsova, Representation-theoretic support spaces for finite group schemes, Amer. J. of Math. 127 (2005), 379–420.

    Article  MathSciNet  Google Scholar 

  27. E. Friedlander, J. Pevtsova, \(\Pi \) -supports for modules for finite group schemes, Duke. Math. J. 139 (2007), 317–368.

    Google Scholar 

  28. E. Friedlander, J. Pevtsova, Generalized support varieties for finite group schemes, Documenta Math (2010), 197–222.

    Google Scholar 

  29. E. Friedlander, J. Pevtsova, Constructions for infinitesimal group schemes, Trans. A.M.S. 363 (2011), 6007–6061.

    Article  MathSciNet  Google Scholar 

  30. E. Friedlander, J. Pevtsova, A. Suslin, Generic and Jordan types, Invent math 168 (2007), 485–522.

    Article  MathSciNet  Google Scholar 

  31. E. Friedlander, A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127(1997), 209–270.

    Article  MathSciNet  Google Scholar 

  32. W. Hardesty, D. Nakano, P. Sobaje, On the existence of mock injective modules for algebraic groups, Bull. Lond. Math. Soc. 49 (2017), 806 - 817. arXiv:1604.03840.

    Article  MathSciNet  Google Scholar 

  33. R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics 52, Springer-Verlag, 1977.

    Google Scholar 

  34. N. Jacobson, Lie Algebras, Interscience tracts in Pure and Applied Mathematics 10, Wile & Sons, 1962.

    Google Scholar 

  35. J.C. Jantzen, Kohomologie von p-lie Algebren und nilpotente Elemente, Amh. Math. Sem. Univ. Hamburg 56 (1986), 191–219.

    Article  MathSciNet  Google Scholar 

  36. J.C. Jantzen, Representations of Algebraic groups, Second edition. Math Surveys and Monographs 107, A.M.S., 2003.

    Google Scholar 

  37. G.R. Kempf, Linear systems on homogenous spaces, Annals of Math 103 (1976), 557–591.

    Article  MathSciNet  Google Scholar 

  38. S. Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.

    Article  MathSciNet  Google Scholar 

  39. S. MacLane, Homology. Grundlehren 114, Springer, 1967.

    Google Scholar 

  40. D. Quillen, The spectrum of an equivariant cohomology ring: I, Ann. of Math 94(1971), 549–572.

    Article  MathSciNet  Google Scholar 

  41. D. Quillen, The spectrum of an equivariant cohomology ring: II, Ann. of Math 94(1971), 573–602.

    Article  MathSciNet  Google Scholar 

  42. G. Seitz, Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000), 467–502.

    Article  MathSciNet  Google Scholar 

  43. P. Sobaje, Support varieties for Frobenius kernels of classical groups. J. Pure and Appl. Algebra 216 (2013), 2657–2664.

    Article  MathSciNet  Google Scholar 

  44. P. Sobaje, Exponentiation of commuting nilpotent varieties, J. Pure and Appl. Algebra 219, (2015), 2206–2217.

    Article  MathSciNet  Google Scholar 

  45. R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56.

    Article  MathSciNet  Google Scholar 

  46. J.B. Sullivan, Representations of the hyperalgebra of an algebraic group, Amer. J. Math 100 (1978), 643–652.

    Article  MathSciNet  Google Scholar 

  47. A. Suslin, E. Friedlander, C. Bendel, Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), 693–728.

    Article  MathSciNet  Google Scholar 

  48. A. Suslin, E. Friedlander, C. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), 729–759.

    Article  MathSciNet  Google Scholar 

  49. R. Swan, Groups with no odd dimensionlal cohomology, J algebra 17 (1971), 401–403.

    Article  MathSciNet  Google Scholar 

  50. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSSR 127 (1959), 943–944.

    MathSciNet  MATH  Google Scholar 

  51. W. Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics 66, Springer-Verlag, 1979.

    Google Scholar 

  52. C. Weibel, An introduction to homological algebra. Cambridge Studies in Advanced Math 38, Cambridge U. Press, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, USA

    Eric M. Friedlander

Authors
  1. Eric M. Friedlander
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric M. Friedlander .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, USA

    Jon F. Carlson

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Srikanth B. Iyengar

  3. Department of Mathematics, University of Washington, Seattle, WA, USA

    Julia Pevtsova

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Friedlander, E.M. (2018). Rational Cohomology and Supports for Linear Algebraic Groups. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation