The nilcentralized chain k^*-functor of a block

Abstract

11.1 Let us come back to our setting in chapter 7. Let G be a finite group, b a block of G, (P, e) a maximal Brauer (b, G)-pair and F_(b,G) the associated Frobenius P-category, and let us denote by (F_(b,G))^nc the full subcategory of F_(b,G) over the set of nilcentralized objects. In chapter 7 we have seen that, for any F_(b,G)-nilcentralized subgroup Q of P, the group F_(b,G)(Q) can be canonically lifted to a k^*-group $\hat F_{(b,G)} (Q)$ . In this chapter, we discuss the possibility of lifting whole subcategories of (F_(b,G))^nc to “k^*-categories”†. These results allow us to construct a suitable functor $\widehat{\mathfrak{a}\mathfrak{u}\mathfrak{t}}_{(F_{(b,G)} )^{nc} }$ from the proper category of chains of (F_(b,G))^nc (cf. A2.8) to the category of finite k^*-groups k^*-Gr, which lifts the automorphism functor $\mathfrak{a}\mathfrak{u}\mathfrak{t}_{(F_{(b,G)} )^{nc} }$ introduced in Proposition A2.10 below. This is the main tool for the definition of the Grothendieck group of the pair $(F_{(b,G)} , \widehat{\mathfrak{a}\mathfrak{u}\mathfrak{t}}_{(F_{(b,G)} )^{nc} } )$ in chapter 14.

For the full subcategory over the selfcentralizing objects, the question on the existence of such an extension was raised by Markus Linckelmann at Durham in 2002.