Rational cohomology and cohomological stability in generic representation theory

Abstract

With F q a finite field of characteristic p , let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]( q ) be the category whose objects are functors from finite dimensional F q -vector spaces to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-vector spaces. Friedlander and Suslin have introduced a category [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] of "strict polynomial functors" which has the same relationship to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]( q ) that the category of rational GL m -modules has to the category of GL m ( F q )-modules. Our main theorem says that, for all finite objects F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /], and all s , the natural restriction map from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] ( F ( k ) , G ( k ) ) to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] ( F, G ) is an isomorphism for all large enough k and q . Here F ( k ) denotes F twisted by the Frobenius k times. This combines with an analogous theorem of Cline, Parshall, Scott, and van der Kallen to show that, for all finite F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /], and all s , evaluation on an m dimensional vector space V m induces an isomorphism from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /] ( F, G ) to Ext s GL m ( F q ) ( F ( V m ), F ( V m )) for all large enough m and q . Thus group cohomology of the finite general linear groups has often been identified with MacLane (or Topological Hochschild) cohomology.