Shifted generic cohomology

TL;DR

This paper investigates the relationship between finite group cohomology and algebraic group cohomology, showing that limits can often be eliminated, and introduces the concept of shifted m-generic modules based on cohomological properties.

Contribution

It demonstrates that, for large enough parameters, cohomology limits can be removed, and introduces shifted m-generic modules, extending understanding of cohomology in algebraic and finite groups.

Findings

Limits in generic cohomology can be eliminated in most cases.

Existence of isomorphisms between cohomology groups for shifted modules.

Introduction of the concept of shifted m-generic modules.

Abstract

The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [CPS75] and [CPSvdK77]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we show that, for sufficiently large r, depending only on the root system and m, and not on the prime p or the irreducible module L, there are isomorphisms H^m(G(p^r),L) -> H^m(G(p^r),L') -> H^m_gen(G,L') -> H^m(G,L'), where the subscript "gen" refers to generic cohomology and L' is a constructibly determined irreducible "shift" of the (arbitrary) irreducible module L for the finite Chevalley group G(p^r). By a famous theorem of Steinberg, both L and L' extend to irreducible modules for the ambient algebraic group G with p^r-restricted highest weights. This leads to the notion of a module or weight being "shifted m-generic," and thus to the title of this paper. Our approach is based on questions raised by the third author in [Stea], which we answer here in the cohomology cases. We obtain many additional results, often with formulations in the more general context of Ext^m_G(q) with irreducible coefficients.