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.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
J. Alperin and L. Evens, Representations resolutions, and Quillen’s dimension theorem, J. Pure and Appl Alg. 22 (1981), 1–9.
J. Alperin and L. Evens, Varieties and elementary abelian groups, J. Pure and Appl Alg. 26 (1982), 221–227.
G. Avrunin, S. Scott, Quillen stratification for modules, Invent. Math 66 (1982), 277–286.
P. Bardsley, R. Richardson, Étale slices for algebraic transformation groups in characteristic p. Proc. London Math Soc 51 (1985), 295–317.
D. Benson, Representations of elementary abelian p-groups and vector bundles. CUP, 2016.
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.
V. Bondarko and V. Drozd, The representation type of finite groups, Zap. Nauncn Sem. Leningrad. Otdel Math. INst. Steklov (LOMI) 71 (1977), 24–41.
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.
J. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), 101–143.
J. Carlson, The variety of an indecomposable module is connected, Invent. Math 77 (1984), 291–299.
J. Carlson, E. Friedlander, J. Pevtsova, Modules of constant Jordan type, J. für die reine und angewandte Mathematik 614 (208), 191–234.
J. Carlson, E. Friedlander, J. Pevtsova, Vector bundles associated to Lie Algebras, J. Reine Angew. Math 716 (2016), 147–178.
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.
E. Cline, B. Parshall, L. Scott, Induced modules and affine quotients, Math. Ann 230 (1977), 1–14.
E. Cline, B. Parshall, L. Scott, Cohomology, hyperalgebras, and representations, J.Algebra 63 (1980), 98–123.
E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic cohomology, Invent. Math 39 (1977), 143–163.
S. Donkin, On projective modules for algebraic groups, J. London Math. Soc. (2) (1996), 75–88.
L. Evens, The cohomology ring of a finite group, Trans. A.M.S. 101 (1961), 224–239.
E. Friedlander, Filtrations, 1-parameter subgroups, and rational injectivity, Adv. Math. 323 (2018), 84–113. arXiv:408.2918.
E. Friedlander, Restrictions to \(G({\mathbb{F}}_p)\) and \(G_{(r)}\) of rational \(G\) -modules, Compos. Math. 147 (2011), no. 6, 1955–1978.
E. Friedlander, Support varieties for rational representations, Compositio Math 151 (2015), 765–792.
E. Friedlander, B. Parshall, On the cohomology of algebraic and related finite groups, Invent. Math 74 (1983), 85–117.
E. Friedlander, B. Parshall, Cohomology of infinitesimal and discrete groups, Math. Ann 273 (1986), 353–374.
E. Friedlander, B. Parshall, Geometry of \(p\)-Unipotent Lie Agebras, J. Algebra 109 (1987), 25–45.
E. Friedlander, B. Parshall, Modular representation theory of restricted Lie algebras, Amer. J. Math. 110 (1988), 1055–1094.
E. Friedlander, J. Pevtsova, Representation-theoretic support spaces for finite group schemes, Amer. J. of Math. 127 (2005), 379–420.
E. Friedlander, J. Pevtsova, \(\Pi \) -supports for modules for finite group schemes, Duke. Math. J. 139 (2007), 317–368.
E. Friedlander, J. Pevtsova, Generalized support varieties for finite group schemes, Documenta Math (2010), 197–222.
E. Friedlander, J. Pevtsova, Constructions for infinitesimal group schemes, Trans. A.M.S. 363 (2011), 6007–6061.
E. Friedlander, J. Pevtsova, A. Suslin, Generic and Jordan types, Invent math 168 (2007), 485–522.
E. Friedlander, A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127(1997), 209–270.
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.
R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics 52, Springer-Verlag, 1977.
N. Jacobson, Lie Algebras, Interscience tracts in Pure and Applied Mathematics 10, Wile & Sons, 1962.
J.C. Jantzen, Kohomologie von p-lie Algebren und nilpotente Elemente, Amh. Math. Sem. Univ. Hamburg 56 (1986), 191–219.
J.C. Jantzen, Representations of Algebraic groups, Second edition. Math Surveys and Monographs 107, A.M.S., 2003.
G.R. Kempf, Linear systems on homogenous spaces, Annals of Math 103 (1976), 557–591.
S. Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.
S. MacLane, Homology. Grundlehren 114, Springer, 1967.
D. Quillen, The spectrum of an equivariant cohomology ring: I, Ann. of Math 94(1971), 549–572.
D. Quillen, The spectrum of an equivariant cohomology ring: II, Ann. of Math 94(1971), 573–602.
G. Seitz, Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000), 467–502.
P. Sobaje, Support varieties for Frobenius kernels of classical groups. J. Pure and Appl. Algebra 216 (2013), 2657–2664.
P. Sobaje, Exponentiation of commuting nilpotent varieties, J. Pure and Appl. Algebra 219, (2015), 2206–2217.
R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56.
J.B. Sullivan, Representations of the hyperalgebra of an algebraic group, Amer. J. Math 100 (1978), 643–652.
A. Suslin, E. Friedlander, C. Bendel, Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), 693–728.
A. Suslin, E. Friedlander, C. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), 729–759.
R. Swan, Groups with no odd dimensionlal cohomology, J algebra 17 (1971), 401–403.
B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSSR 127 (1959), 943–944.
W. Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics 66, Springer-Verlag, 1979.
C. Weibel, An introduction to homological algebra. Cambridge Studies in Advanced Math 38, Cambridge U. Press, 1994.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
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
DOI: https://doi.org/10.1007/978-3-319-94033-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94032-8
Online ISBN: 978-3-319-94033-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)