Rational and Generic Cohomology

TLDR: In this article, it was shown that rational cohomology takes a stable value relative to twisting, i.e., for sufficiently large q and e (depending on I' and n), there are isomorphisms H 2(G, V(e))gH'(G(q), V'(e)), rH'((G, q), V') where the first map is restriction.

Abstract: Let G be a semisimple algebraic group defined and split over k,=GF(p). For q=p”, 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 I/ 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++xtPel of G. Theorem (6.6) states that, for sufficiently large q and e (depending on I’ and n), there are isomorphisms H”(G, V(e))gH’(G(q), V(e))rH”(G(q), V) where the first map is restriction. In particular, the cohomology groups H”(G(q), V) have a stable or “generic” value H;,,(G, V). This phenomenon had been observed empirically many times (cf. [6, 203). The computation of generic cohomology reduces essentially to the computation of rational cohomology. One (surprising) consequence is that Hi,,(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 E, we have semilinear isomorphisms H”(G, V(E)) % H”(G, V(e)) for all e 2 F. 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 H2(G, V) and Extk( K W) to the structure of Weyl modules. These results explain for example the generic values of H’ 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. A key ingredient in the proofs is an important theorem of G. Kempf [I93 on the vanishing of cohomology of certain homogeneous line bundles. This result is translated into the language of rational cohomology in (1.2), and is used in