Stable sheaf cohomology on flag varieties

TL;DR

This paper establishes stabilization results for sheaf cohomology on flag varieties and projective space, providing explicit calculations and revealing complex behaviors in modular settings, with applications to algebra and representation theory.

Contribution

It proves new stabilization theorems for sheaf cohomology in positive characteristic and develops explicit stable cohomology calculations, including a Kunneth formula and Frobenius invariance.

Findings

Cohomology groups exhibit super-polynomial growth in indecomposable summands.

Non-vanishing degrees of cohomology do not form a connected interval.

Stable cohomology of polynomial functors is invariant under Frobenius and satisfies a Kunneth formula.

Abstract

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.