Let G be a semisimple algebraic group defined and split over k0 =GF(p). For q=pm, let G(q) be the subgroup of GF(q)-rational points. The main objective of this paper is to relate the cohomology of the finite groups G(q) to the rational cohomology of the algebraic group G. Let V be a finite dimensional rational G-module, and, for a non-negative integer e, let V(e) be the G-module obtained by "twisting" the original G-action on V by the Frobenius endomorphism x~--*x tp~ of G. Theorem (6.6) states that, for sufficiently large q and e (depending on V and n), there are isomorphisms H' (G, V(e)
)~ H"(G(q), V(e))~-H'(G(q), V)where the first map is restriction. In particular, the cohomology groups H"(G(q), V) have a stable or "generic" value Hge,, (G, V). This phenomenon had been observed empirically many times (cf. [6,20]). The computation of generic cohomology reduces essentially to the computation of rational cohomology. One (surprising) consequence is that Hgen(G, V) does not depend on the exact weight lattice for a group G of a given type cf. (6.10), though this considerably affects the structure of G(q). We also obtain that rational cohomology takes a stable value relative to twisting-i.e., for sufficiently large ~,, we have semilinear isomorphisms H' (G, V(e)) ~H" (G, V(e)) for all e>~. This paper contains many new results on rational cohomology beyond those required for the proof of the main theorem. We mention in particular the vanishing theorems (2.4) and (3.3), and especially the results (3.9) through (3.11) which relate HZ(G, V) and Ext,(V, W) to the structure of Weyl modules. These results explain for example the generic values of H 1 determined in [6], cf. (7.6). Also, it is shown in Theorem (3.12) that every finite dimensional rational G-module has a finite resolution by finite dimensional acyclic G-modules.
