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Bifunctor cohomology and cohomological finite generation for reductive groups. (English) Zbl 1196.20053
Let $$G$$ be a reductive linear algebraic group over a field $$k$$. 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.
The main result of this paper is the impressive fact that any such $$G$$ has the CFG property.
Over a field of characteristic zero, the fact is well-known from invariant theory, so the case of prime characteristic is the focus here. This problem (or special cases thereof) has seen significant study by many people, particularly in full generality by the second author. A nice discussion is given of some of this history along with equivalent formulations of the theorem. Also, some consequences are given for the cohomology module $$H^*(G,M)$$ for a Noetherian $$A$$-module $$M$$ on which $$G$$ acts compatibly.
The CFG property for an arbitrary such $$G$$ is first reduced to the case of the general linear group $$\text{GL}_n$$ over an algebraically closed field (of prime characteristic). The second author [in CRM Proceedings & Lecture Notes 35, 127-138 (2004; Zbl 1080.20039)] had previously shown the result for some small $$n$$ via the construction of certain universal cohomology classes in $$H^*(\text{GL}_n,\Gamma^*(\mathfrak{gl}_n^{(1)}))$$ satisfying certain divided power relations, where $$\Gamma^*$$ 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. The existence of such classes (without the divided power relations) was shown in general by the first author [in Duke Math. J. 151, No. 2, 219-249 (2010; Zbl 1196.20052)]. 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 proof of the CFG property in part parallels the argument of Friedlander and Suslin.
The authors present two proofs of how the CFG property follows from the existence of the universal cohomology classes constructed by Touzé. One proof makes further investigation of bifunctor cohomology beyond the aforementioned work of Touzé, obtaining further classes and relations, and then follows the argument in the aforementioned work of van der Kallen. The second proof simply uses the universal classes as constructed by Touzé and an inductive argument with Frobenius kernels.

##### MSC:
 20G10 Cohomology theory for linear algebraic groups 14L24 Geometric invariant theory 18G10 Resolutions; derived functors (category-theoretic aspects)
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##### References:
 [1] K. Akin, D. A. Buchsbaum, and J. Weyman, Schur functors and Schur complexes , Adv. in Math. 44 (1982), 207–278. · Zbl 0497.15020 [2] D. J. Benson, Representations and Cohomology, II: Cohomology of Groups and Modules , 2nd ed., Cambridge Stud. Adv. Math. 31 , Cambridge Univ. Press, Cambridge, 1998. · Zbl 0908.20002 [3] H. Borsari and W. Ferrer Santos, Geometrically reductive Hopf algebras , J. Algebra 152 (1992), 65–77. · Zbl 0803.16038 [4] H. Cartan and S. Eilenberg, Homological Algebra , Princeton Univ. Press, Princeton, 1956. · Zbl 0075.24305 [5] E. Cline, B. Parshall, L. Scott, and W. Van Der Kallen, Rational and generic cohomology , Invent. Math. 39 (1977), 143–163. · Zbl 0336.20036 [6] L. Evens, The cohomology ring of a finite group , Trans. Amer. Math. Soc. 101 (1961), 224–239. JSTOR: · Zbl 0104.25101 [7] Y. FéLix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory , Grad. Texts in Math. 205 , Springer, New York, 2001. [8] V. Franjou and E. M. Friedlander, Cohomology of bifunctors , Proc. Lond. Math. Soc. (3) 97 (2008), 514–544. · Zbl 1153.20042 [9] E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field , Invent. Math. 127 (1997), 209–270. · Zbl 0945.14028 [10] F. D. Grosshans, Contractions of the actions of reductive algebraic groups in arbitrary characteristic , Invent. Math. 107 (1992), 127–133. · Zbl 0778.20018 [11] -, Algebraic Homogeneous Spaces and Invariant Theory , Lecture Notes in Math. 1673 , Springer, Berlin, 1997. · Zbl 0886.14020 [12] J. C. Jantzen, Representations of Algebraic Groups , Math. Surveys Monogr. 107 , Amer. Math. Soc., Providence, 2003. · Zbl 1034.20041 [13] S. Mac Lane, Homology , reprint of 1975 edition, Classics Math., Springer, Berlin, 1995. [14] W. S. Massey, Products in exact couples , Ann. of Math. (2) 59 (1954), 558–569. · Zbl 0057.15204 [15] O. Mathieu, Filtrations of $$G$$-modules , Ann. Sci. École Norm. Sup. (4) 23 (1990), 625–644. · Zbl 0748.20026 [16] M. Nagata, Invariants of a group in an affine ring , J. Math. Kyoto Univ. 3 (1963/1964), 369–377. · Zbl 0146.04501 [17] E. Noether, Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik $$p$$, Nachr. Ges. Wiss. Göttingen (1926), 28–35. · JFM 52.0106.01 [18] V. L. Popov, On Hilbert’s theorem on invariants (in Russian), Dokl. Akad. Nauk SSSR 249 (1979), 551–555.; English translation in Soviet Math. Dokl. 20 (1979), 1318–1322. · Zbl 0446.14004 [19] T. A. Springer, Invariant Theory , Lecture Notes in Math. 585 , Springer, Berlin, 1977. · Zbl 0346.20020 [20] 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 [21] A. Suslin, E. M. Friedlander, and C. P. Bendel, Infinitesimal $$1$$-parameter subgroups and cohomology , J. Amer. Math. Soc. 10 (1997), 693–728. JSTOR: · Zbl 0960.14023 [22] A. Touzé, Universal classes for algebraic groups , Duke Math. J. 151 (2010), 219–250. · Zbl 1196.20052 [23] W. Van Der Kallen, “Cohomology with Grosshans graded coefficients” in Invariant Theory in All Characteristics (Kingston, Ontario, Canada) , CRM Proc. Lecture Notes 35 (2004), Amer. Math. Soc., Providence, 2004, 127–138. · Zbl 1080.20039 [24] -, “A reductive group with finitely generated cohomology algebras” in Algebraic Groups and Homogeneous Spaces (Mumbai, 2004) , Tata Inst. Fund. Res. Studies in Math., Narosa, New Delhi, 2007. [25] W. C. Waterhouse, Geometrically reductive affine group schemes , Arch. Math. (Basel) 62 (1994), 306–307. · Zbl 0804.14022
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