Higher limits over the fusion orbit category.

The fusion orbit category F ‾ C (G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C , and whose morphisms H → K are given by the G -maps G / H → G / K modulo the action of the centralizer group C G (H). We show that the higher limits over F ‾ C (G) 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 (F). 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 (S , F , L) , the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp. [ABSTRACT FROM AUTHOR]