Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T22:53:46.662Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectrum of group cohomology and support varieties

Published online by Cambridge University Press:  01 March 2013

Eric M. Friedlander
Eric M. Friedlander*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USAericmf@usc.edu
Get access

Abstract

We give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

Keywords

equivariant cohomologyelementary abelian subgroups20J0620C20
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 3: The Legacy of Daniel Quillen , June 2013 , pp. 507 - 516
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alperin, J.L., Evens, L., Representations, resolutions, and Quillen's dimension theorem, J. Pure & Applied Algebra 22 (1981), 1–9.Google Scholar
2.Alperin, J.L., Evens, L., Varieties and elementary abelian groups, J. Pure ʖ Applied Algebra 26 (1982), 221–227.Google Scholar
3.Atiyah, M., Segal, G., Equivariant K-theory and completion, J. Diff. Geom. 3 (1969), 1–18.Google Scholar
4.Avrunin, G. S., Annihilators of cohomology modules, J. Algebra 69 (1981), 150–154.CrossRefGoogle Scholar
5.Avrunin, G., Scott, L., Quillen stratification for modules, Inventiones Math. 66 (1982), 277–286.Google Scholar
6.Benson, D. J., Pevtsova, J., A realization theorem for modules of constant Jordan type and vector bundles, Trans. AMS. 364 (2012), 6459–6478.Google Scholar
7.Borel, A., Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–20.Google Scholar
8.Carlson, J., The varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104–143.Google Scholar
9.Carlson, J., The variety of indecomposable module is connected, Invent. Math. 77 (1984), 291–299.CrossRefGoogle Scholar
10.Carlson, J., Friedlander, E., Pevtsova, J., Modules of constant Jordan type, J. Reine Angew. Math. 614 (2008), 191–234.Google Scholar
11.Carlson, J., Friedlander, E., Suslin, A., Modules for Z/p × Z/p, Commentarrii Mathematici Helvetici 86 (2011), 609–657.Google Scholar
12.Chouinard, L., Projectivity and relative projectivity over group rings, J. Pure & Applied Algebra 7 (1976), 287–302.Google Scholar
13.Dade, E., Endo-permutation modules over p-groups, II, Annals of Math. 108 (1978), 317–346.Google Scholar
14.Duflot, J., Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), 627–637.CrossRefGoogle Scholar
15.Evens, L., A generalization of the transfer map in the cohomology of groups, Trans. AMS. 108 (1963), 45–65.Google Scholar
16.Evens, L., The cohomology of groups, Oxford University Press, (1991).Google Scholar
17.Friedlander, E. and Parshall, B., Support varieties restricted Lie algebras, Invent. Math. 86 (1986), 553–562.Google Scholar
18.Friedlander, E., Pevtsova, J., Π-supports for modules for finite group schemes, Duke. Math. J. 139 (2007), 317–368.Google Scholar
19.Friedlander, E., Pevtsova, J., Constructions for infinitesimal group schemes. Trans. Amer. Math. Soc. 363 (11) (2011), 6007–6061.Google Scholar
20.Friedlander, E., Pevtsova, J., Suslin, A., Generic and Maximal Jordan types, Invent. Math. 168 (2007), 485–522.Google Scholar
21.Friedlander, E., Suslin, A., Cohomology of finite group schemes over a field, Invent. Math 127 (1997), 235-2-53.CrossRefGoogle Scholar
22.Hopkins, M., Kuhn, N., Ravenel, D., Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553–594.Google Scholar
23.Jantzen, J., Representations of algebraic groups. 2ndedition. Math Surveys and Monographs 107, AMS 2003.Google Scholar
24.Lannes, J., Sur la cohomologie modulo p des p-groupes abèliens élémentaires, London Math Soc. Lecture Note Ser 117 (1987), 97–116.Google Scholar
25.Nakano, D., Parshall, B., Vella, D., Support varieties for algebraic groups. J. Reine Angew. Math 547 (2002), 15–49.Google Scholar
26.Quillen, D., The spectrum of an equivariant cohomology ring: I, Ann. Math. 94 (1971), 549–572.Google Scholar
27.Quillen, D., The spectrum of an equivariant cohomology ring: II, Ann. Math. 94 (1971), 573–602.CrossRefGoogle Scholar
28.Quillen, D., A cohomological criterion for p-nilpotence, J. Pure Appl. Algebra 1, 361–372.Google Scholar
29.Quillen, D. and Venkov, B.B, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 317–318.Google Scholar
30.Suslin, A., Friedlander, E., Bendel, C., Infinitesimal 1-parameter subgroups and cohomology; Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), 693–728, 729–759.Google Scholar
31.Swan, R., Groups with no odd dimensional cohomology, J. Algebra 17 (1971), 401–403.Google Scholar