Cohomology of classical algebraic groups from the functorial viewpoint

Abstract

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.

Cohomology of Algebraic Groups with Coefficients in Twisted Representations

“…If r = 0 then V (0) = V and the results stated in this section can be interpreted as polynomial induction formulas. These formulas follow directly from the results of [Tou2] (improved in [Tou5]). There is no such interpretation if r ≥ 1, although the isomorphisms have similarities with induction isomorphisms.…”

“…Here k 2n # is the dual of the defining representation of Sp 2n . Similary, by [Tou2,Thm 3.24] and [Tou5,Thm 7.24,Cor 7.31] if p = 2, there are graded maps, which are isomorphisms as soon as 2n + ǫ ≥ deg F + 1:…”

“…The third isomorphism is induced by evaluating both variables of the functor on the same variable V , and then pulling back by the map Γ p (ι) where ι : X → ⊗ 2 is the canonical inclusion. The fact that is is an isomorphism is (a very particular case) of the proof [Tou2,Thm 6.6]. Finally the last isomorphism is induced by the quotient map ⊗ 2 (1) → X (1) and it is an isomorphism by [Tou2, Thm 6.6].…”

“…In [Tou2,Thm 3.17] and [Tou5,Thm 7.24] we proved the existence of a graded map, which is an isomorphism as soon as 2n ≥ deg F :…”

“…Remark 4.11. An alternative construction of the isomorphism between bifunctor cohomology and GL n -cohomology is given in [Tou2]. As shown in [Tou5], it also provides a graded isomorphism for n ≥ max{d, e}:…”