Rational cohomology and cohomological stability in generic representation theory, by Nicholas J. Kuhn

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 vector spaces over the algebraic closure of Fp. 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 V_m induces an exact functor from F(q) to the category of modules over GL(m,q).

E.Friedlander and A.Suslin have introduced a category P of ``strict polynomial functors'' which has the same relationship to the category of rational GL_m modules that F(q) has to the category of GL(m,q) modules. Our main theorem says that, for all finite objects F,G in P, and all s, the natural restriction

             Ext^s_{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 GL_m modules to GL(m,q) modules. These two theorems then combine with an observation of Friedlander and Suslin to show that, for all finite F,G in P, and all s, the natural map

          Ext^s_{F(q)}(F,G) ---> Ext^s_{GL(m,q)}(F(V_m),G(V_m))
is an isomorphism for all large enough m and q. Thus group cohomology of the finite general linear groups in the stable range (a.k.a. stable K-theory of Fq with twisted coefficients) has often been identified with (the more computable) MacLane cohomology.

Nicholas J. Kuhn <njk4x@virginia.edu>