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

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

2007-06-05

arxiv.org/abs/0706.0604v3

Transformation Groups, Vol. 14, No. 3, 2009, pp. 695-711

Mathematics

Representation Theory

22 pages; final version

Scientific paper

10.1007/S00031-009-9054-0

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.

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