The localizers in a Frobenius P-category

Abstract

18.1 Let P be a finite p-subgroup and F a Frobenius P-category. As a matter of fact, independently of the possible existence of a perfect F-locality L (cf. 17.13), the groups L(Q) where Q runs over the set of subgroups of P have a direct characterization in terms of F — we call them the F-localizers — and their existence and uniqueness admits a direct proof. In this chapter we give this characterization and such a direct proof; this allows us, in the next chapter, to determine the (p-)solvable Frobenius P-categories. Moreover, we show the functorial nature of the F-localizers — provided we replace F by the proper category of F-chains $\mathfrak{c}\mathfrak{h}*(F)$ of F (cf. A2.8) — by constructing with them a suitable functor from $\mathfrak{c}\mathfrak{h}*(F)$ — the localizing functor of F — which actually holds some kind of “universal property” (cf. 18.20 below).