Finite Schur filtration dimension for modules over an algebra with Schur filtration
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Let G = GLN or SLN as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2 N: Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the Hi(G, M) are Noetherian AG-modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.
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