Higher limits over the fusion orbit category

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