Rational Cohomology and Cohomological Stability in Generic Representation Theory

With Fq a finite field of characteristic p, let F(q) be the category whose objects are functors from finite dimensional Fq–vector spaces to F̄p– vector spaces. Extension groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology with twisted coefficients. Furthermore, evaluation on an m dimensional vector space Vm induces a homorphism from Ext F(q) (F, G) to finite group cohomology Ext GLm(Fq) (F (Vm), G(Vm)). E. Friedlander and A. Suslin have introduced a category P of “strict polynomial functors” which has the same relationship to the category of rational GLm–modules that F(q) has to the category of GLm(Fq)–modules. Our main theorem says that, for all finite objects F,G ∈ P , and all s, the natural restriction map Ext P (F(k),G(k)) −→ Ext s F(q)(F,G) is an isomophism for all large enough k and q. Here F(k) denotes F twisted by the Frobenious k times. This theorem is an analogue of an old stability theorem of E. Cline, B. Parshall, L. Scott, and W. van der Kallen relating rational GLm–modules to GLm(Fq)–modules. These two theorems then combine with an observation of Friedlander and Suslin to show that, for all finite F,G ∈ P , and all s, the natural map Ext F(q)(F, G) −→Ext s GLm(Fq) (F (Vm), G(Vm)) is an isomorphism for all large enough m and q. Thus group cohomology of the finite general linear groups has often been identified with (the more computable) MacLane cohomology.