A perfect F-locality from a perfect F^sc -locality

Abstract

20.1 Let P be a finite p-group and F a Frobenius P-category; recall that we denote by F^sc the full subcategory of F over the set of F-selfcentralizing subgroups of P (cf. 6.1) and that in chapter 17 we have introduced the F- and F^sc-localities (cf. 17.4), introducing in 17.13 the perfect ones. In this chapter, we prove that any perfect F^sc -locality L^sc can be extended to a unique perfect F-locality L^sc. Our argument depends on a precise consequence of Theorem 17.18 (cf. 20.9); the remainder of our long proof is just routine, yet we have not found a shorter method. As Theorem 4.12, this result illustrates the fact that the full subcategory F^sc “determines” what happens in F.