A gluing theorem for Dade P-algebras

Abstract

10.1 Let P be a finite p-group and k an algebraically closed field of characteristic p. Mimicking the situation in Proposition 7.12 and Theorem 7.14, consider a P-stable nonempty set X of subgroups of P such that any subgroup Q of P containing some R ∈ X belongs to X, and assume that, for any Q ∈ X, we have an element s(Q) in the Dade group $D_k (\bar N_P (Q))$ (cf. 1.20) in such a way that s(Q^u) coincides with the image of s(Q) in $D_k (\bar N_P (Q^u ))$ by the isomorphism induced by u ∈ P, and that, for any normal subgroup R of Q belonging to X, we have

$$ res _{\bar \bar N_{Q,R} }^{\bar N_P (Q)} (s(Q)) = \widetilde{\mathfrak{B}\mathfrak{r}}_{\bar Q}^{\bar N_{Q,R} } \left( {res_{\bar N_{Q,R} }^{\bar N_P (R)} (s(R))} \right)$$

(10.1.1,)

where we denote by $\bar N$ _Q,R and $\bar Q$ the respective images of N _P (Q) ∩ N _P (R) and Q in $\bar N$ _P (R), and we set $\bar \bar N_{Q,R} = \bar N_{Q,R} /\bar Q$.