Finite Schur filtration dimension for modules over an algebra with Schur filtration

TL;DR

This paper proves that modules with Schur filtrations over certain algebraic groups have finite filtration dimension, ensuring finitely many nonzero cohomology groups and noetherian properties of these cohomologies.

Contribution

It establishes finite Schur filtration dimension for modules over reductive groups in positive characteristic, extending previous results to broader cases.

Findings

Modules with Schur filtrations have finite filtration dimension.

Finitely many nonzero cohomology groups H^i(G,M).

Cohomology modules H^i(G,M) are noetherian over A^G.

Abstract

Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a resolution involving Schur functors of the ideal of the diagonal in a product of Grassmannians.