  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.
