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Universal classes for algebraic groups. (English) Zbl 1196.20052
Let \(G\) be a reductive linear algebraic group over a field \(k\) of positive characteristic \(p\). The group \(G\) is said to have the cohomological finite generation (CFG) property if for every finitely generated commutative \(k\)-algebra \(A\) on which \(G\) acts rationally by \(k\)-algebra automorphisms, the cohomology ring \(H^*(G,A)\) is finitely generated as a \(k\)-algebra. W. van der Kallen observed that the CFG property would hold for all such \(G\) if certain universal cohomology classes (in the cohomology of the general linear group \(\text{GL}_n\)) could be constructed. The main result of this paper is the construction of such classes. Proofs of the CFG property are presented by the author and W. van der Kallen [in Duke Math. J. 151, No. 2, 251-278 (2010; Zbl 1196.20053)].
More precisely, the result here is the construction of classes \(c[d]\) in \(H^{2d}(\text{GL}_n,\Gamma^d(\mathfrak{gl}_n^{(1)}))\) where \(\Gamma^d\) denotes the divided power functor and \(\mathfrak{gl}_n^{(1)}\) denotes the adjoint representation (the Lie algebra of \(\text{GL}_n\)) once twisted by Frobenius. Further \(c[1]\) is shown to be non-zero, and there is a cup product relationship between \(c[d]\) and \(c[1]\) for higher \(d\). These classes generalize those constructed by E. M. Friedlander and A. Suslin [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] in their proof of the finite generation of \(H^*(G,k)\) for a finite group scheme \(G\).
The construction of the desired classes is first reduced to large values of \(n\). From there, the problem can be translated to bifunctor cohomology for strict polynomial bifunctors. Note that the work of Friedlander and Suslin made use of strict polynomial functors. Here functors are replaced with bifunctors. Working with bifunctors, explicit coresolutions of \(\Gamma^d(\mathfrak{gl}^{(1)})\) are constructed from which the desired cocycles are obtained. The construction of the resolutions requires investigation of \(p\)-complexes and a tensor product property of complexes obtained from \(p\)-complexes.

20G10 Cohomology theory for linear algebraic groups
18G10 Resolutions; derived functors (category-theoretic aspects)
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[1] K. Akin, D. Buchsbaum, and J. Weyman, Schur functors and Schur complexes , Adv. in Math. 44 (1982), 207–278. · Zbl 0497.15020
[2] H. Cartan, “Constructions multiplicatives” in Séminaire Henri Cartan 7 , no. 1, Secrétariat mathématique, Paris, 1954/55, exp. no. 4. · Zbl 0055.41801
[3] M. ChałUpnik, Extensions of strict polynomial functors , Ann. Sci. École Norm. Sup. (4) 38 (2005), 773–792. · Zbl 1089.20029
[4] V. Franjou and E. Friedlander, Cohomology of bifunctors , Proc. London Math. Soc. 97 (2008), 514–544. · Zbl 1153.20042
[5] V. Franjou, E. Friedlander, A. Scorichenko, and A. Suslin, General linear and functor cohomology over finite fields , Ann. of Math. (2) 150 (1999), 663–728. JSTOR: · Zbl 0952.20035
[6] V. Franjou, J. Lannes, and L. Schwartz, Autour de la cohomologie de Mac Lane des corps finis , Invent. Math. 115 (1994), 513–538. · Zbl 0798.18009
[7] E. Friedlander and A. Suslin, Cohomology of finite group schemes over a field , Invent. Math. 127 (1997), 209–270. · Zbl 0945.14028
[8] A. Grothendieck, Sur quelques points d’algèbre homologique , Tôhoku Math. J. (2) 9 (1957), 119–221. · Zbl 0118.26104
[9] W. Haboush, Reductive groups are geometrically reductive , Ann. of Math. (2) 102 (1975), 67–83. JSTOR: · Zbl 0316.14016
[10] M. M. Kapranov, On the q-analog of homological algebra , · Zbl 1433.17025
[11] C. Kassel and M. Wambst, Algèbre homologique des \(N\)-complexes et homologie de Hochschild aux racines de l’unité , Publ. Res. Inst. Math. Sci. 34 (1998), 91–114. · Zbl 0992.18010
[12] S. Mac Lane, Homology , Classics Math., Springer, Berlin, 1995.
[13] M. Nagata, Invariants of a group in an affine ring , J. Math. Kyoto Univ. 3 (1963/1964), 369–377. · Zbl 0146.04501
[14] P. Real, Homological perturbation theory and associativity . Homology Homotopy Appl. 2 (2000), 51–88. · Zbl 0949.18005
[15] V. Srinivas and W. Van Der Kallen, Finite Schur filtration dimension for modules over an algebra with Schur filtration , Transform. Groups 14 (2009), 695–711. · Zbl 1192.20032
[16] B. Totaro, Projective resolutions of representations of \(\mathrm GL(n)\) . J. Reine Angew. Math. 482 (1997), 1–13. · Zbl 0859.20034
[17] A. Touzé, Cohomologie rationnelle du groupe linéaire et extensions de bifoncteurs , Ph.D. dissertation, University of Nantes, Nantes, France, 2008.
[18] A. Touzé and W. Van Der Kallen, Bifunctor cohomology and cohomological finite generation for reductive groups , Duke Math. J. 151 (2010), 251–278. · Zbl 1196.20053
[19] A. Troesch, Une résolution injective des puissances symétriques tordues , Ann. Inst. Fourier (Grenoble) 55 (2005), 1587–1634. · Zbl 1077.18009
[20] W. Van Der Kallen, “Cohomology with Grosshans graded coefficients” in Invariant Theory in All Characteristics , CRM Proc. Lecture Notes 35 , Amer. Math. Soc., Providence, 2004, 127–138. · Zbl 1080.20039
[21] -, “A reductive group with finitely generated cohomology algebras” in Algebraic Groups and Homogeneous Spaces , Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, 301–314. · Zbl 1140.20037
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