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Shifted generic cohomology

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 lea