Rational cohomology and cohomological stability in generic representation theory

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_GLm(Fq) (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.