The nilcentralized chain k*-functor of a block

Part of the Progress in Mathematics book series (PM, volume 274)


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.


Full Subcategory Grothendieck Group Group Isomorphism Primitive Idempotent Dade Group 
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© Birkhäuser Verlag AG 2009

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