The Grothendieck groups of a Frobenius P-category

Abstract

14.1 Let P be a finite p-group and F a Frobenius P-category. As in chapter 11, let us denote by F^nc the full subcategory of F over the F-nilcentralized subgroups of P (cf. 4.3) and let us consider the proper category of F^nc -chains $\mathfrak{c}\mathfrak{h}*(F^{nc} )$ (cf. A2.8); recall that we have the automorphism functor from $\mathfrak{c}\mathfrak{h}*(F^{nc} )$ to the category of groups (cf. Proposition A2.10)

$$ \mathfrak{a}\mathfrak{u}\mathfrak{t}_{F^{nc} } :\mathfrak{c}\mathfrak{h}*(F^{nc} ) \to \mathfrak{G}\mathfrak{r}$$

(14.1.1)

which maps any $\mathfrak{c}\mathfrak{h}*(F^{nc} )$ -object ($\mathfrak{q}$ , Δ _n ) on its $\mathfrak{c}\mathfrak{h}*(F^{nc} )$ -automorphism group — noted $F(\mathfrak{q})$ ; explicitly, we identify $F(\mathfrak{q})$ with the subgroup of $F(\mathfrak{q}(n))$ formed by the elements stabilizing the images $\mathfrak{q}(i)_n = \mathfrak{q}(in)(\mathfrak{q}(\mathfrak{i}))$ in $\mathfrak{q}(n)$ of all the groups $\mathfrak{q}(i)$ determined by the functor $\mathfrak{q}:\Delta _n \to F^{nc}$ for 0 ≤ i ≤ n.