  We prove an effective stabilization result for the sheaf cohomology groups of
line bundles on flag varieties parametrizing complete flags in k^n, as well as
for the sheaf cohomology groups of polynomial functors applied to the cotangent
sheaf Omega on projective space. In characteristic zero, these are natural
consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are
non-trivial. Unlike many important contexts in modular representation theory,
where the prime characteristic p is assumed to be large relative to n, in our
study we fix p and let n go to infinity. We illustrate the general theory by
providing explicit stable cohomology calculations in a number of cases of
interest. Our examples yield cohomology groups where the number of
indecomposable summands has super-polynomial growth, and also show that the
cohomological degrees where non-vanishing occurs do not form a connected
interval. In the case of polynomial functors of Omega, we prove a Kunneth
formula for stable cohomology, and show the invariance of stable cohomology
under Frobenius, which combined with the Steinberg tensor product theorem
yields calculations of stable cohomology for an interesting class of simple
polynomial functors arising in the work of Doty. The results in the special
case of symmetric powers of Omega provide a nice application to commutative
algebra, yielding a sharp vanishing result for Koszul modules of finite length
in all characteristics.
