Higher limits over the fusion orbit category

Abstract

The fusion orbit category FCpGq of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H Ñ K are given by the G-maps G{H Ñ G{K modulo the action of the centralizer group CGpHq. We show that the higher limits over FCpGq can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G.If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category O c pFq. This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group pS, F, Lq, the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.