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The nilcentralized chain k*-functor of a block

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

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.

Keywords

Full Subcategory Grothendieck Group Group Isomorphism Primitive Idempotent Dade Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2009

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