Fusion-stable endosplit $p$-permutation resolutions

Abstract

Let $k$ be a field of characteristic $p>0$, $\mathcal{F}$ a saturated fusion system over a finite $p$-group $P$, and $V$ an indecomposable capped endopermutation $kP$-module. Let $D_k^Ω(P)$ be the subgroup of the Dade group $D_k(P)$ generated by all the relative syzygies. It is known that $V$ has an endosplit $p$-permutation resolution if and only if the Dade class $[V]$ belongs to $D_k^Ω(P)$. We show that the resolution can be chosen to be $\mathcal{F}$-stable if and only if $V$ is $\mathcal{F}$-stable. As an application, we prove the following folklore result: if two blocks of finite groups are Morita equivalent via a bimodule with an endopermutation $kP$-source $V$ such that $[V]\in D_k^Ω(P)$, then these two blocks are splendidly Rickard equivalent.