Universal classes for algebraic groups

Abstract

Abstract. We exhibit cocycles representing certain classes in the cohomology of the algebraic group GLn with coefficients in the representation Γ * (gl (1) n ). These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras [18].

A construction of the universal classes for algebraic groups with the twisting spectral sequence

“…We first transpose the problem in the framework of strict polynomial functors as in [T1,Section 1.2]. Since the representation gl n is defined over F p , we have an isomorphism gl…”

Cohomology of Algebraic Groups with Coefficients in Twisted Representations

“…As already mentioned, a direct proof of theorem 5.1 when (F , X) = P k (1, 1) is given in [Tou1,TVdK]. The construction is combinatorial, that is, we construct an explicit resolution of the representation Γ d (gl n ) (1) .…”

Lectures on Bifunctors and Finite Generation of Rational Cohomology Algebras

“…Let us say that G satisfies the cohomological finite generation property (CFG) if, whenever G acts on a commutative algebra A of finite type over k, the cohomology algebra H * (G, A) is also finitely generated over k. So my conjecture was that if the base ring k is a field and an affine algebraic group (or group scheme) G over k satisfies property (FG) then it actually satisfies the stronger property (CFG). This was proved by Touzé [30], by constructing classes c [m] in Ext groups in the category of strict polynomial bifunctors of Franjou and Friedlander [10]. If the base field has characteristic zero then there is little to do, because then (FG) implies that H >0 (G, A) vanishes.…”

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This text is an updated version of material used for a course at Université de Nantes, part of 'Functor homology and applications', April 23-27, 2012. The proof [30], [31] by Touzé of my conjecture on cohomological finite generation (CFG) has been one of the successes of functor homology. We will not treat this proof in any detail.

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