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Cohomology of classical algebraic groups from the functorial viewpoint. (English) Zbl 1208.20043
For the general linear group \(\text{GL}_n\) over a field of positive characteristic, it has been known that certain rational cohomology can be identified with certain extensions in the category of strict polynomial bifunctors. This relationship has been a key tool in recent finite generation results as well as for making explicit cohomological computations.
In this work, the author uses classical invariant theory to extend this identification to a finite product of general linear groups \(\text{GL}_n\), symplectic groups \(\text{Sp}_n\), and orthogonal groups \(\text{O}_{n,n}\) over arbitrary commutative rings (assuming that 2 is invertible in the ring if an orthogonal group is involved). Somewhat more precisely, for such a group \(G\), the author constructs a graded, natural map \(\text{Ext}^*_{\mathcal P}(\Gamma^*(F_G),F)\to H^*(G,F_n)\) that is compatible with cup products, where \(\mathcal P\) is an appropriate category of strict polynomial functors associated to \(G\), \(F_G\) is a certain “characteristic” functor in \(\mathcal P\) associated to \(G\), \(\Gamma^*\) denotes divided powers, \(F\) is an arbitrary functor in \(\mathcal P\), and \(F_n\) is a \(G\)-module obtained by an appropriate evaluation of \(F\) (determined by \(G\)). The map is then shown to be an isomorphism if the rank of \(G\) is sufficiently large relative to the degree of \(F\) (e.g., \(2n\geq\deg(F)\) in the case of \(\text{GL}_n\)).
A number of applications of this result are then given. First, it follows that the cohomology groups \(H^*(G,F_n)\) stabilize as \(n\) increases. Working over a field, the author constructs an external coproduct on these stable cohomology groups (for which the cup product is a section) and shows that with this coproduct the stable cohomology groups have the structure of a graded Hopf monoidal functor. Moreover, if the functor \(F\) admits a Hopf algebra structure, then the stable cohomology admits a Hopf algebra structure (without antipode). Following the work of A. Djament and C. Vespa [Ann. Sci. Éc. Norm. Supér. (4) 43, No. 3, 395-459 (2010; Zbl 1221.20036)] for finite classical groups, the author makes some explicit computations of the stable cohomology for orthogonal and symplectic groups with coefficients arising from symmetric and exterior power functors. One further consequence of the coproduct result is that the cup product \(H^*(G,(F_1)_n)\otimes H^*(G,(F_2)_n)\to H^*(G,(F_1)_n\otimes (F_2)_n)\) is necessarily an injection (again, for \(n\) sufficiently large relative to the degrees of the functors \(F_1\) and \(F_2\)).

MSC:
20G10 Cohomology theory for linear algebraic groups
18G10 Resolutions; derived functors (category-theoretic aspects)
18A25 Functor categories, comma categories
20G35 Linear algebraic groups over adèles and other rings and schemes
14L24 Geometric invariant theory
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