AI 中文总结
本文证明饱和融合系下满足特定 Dade 类条件的稳定内置换模具有内分裂 $p$-置换分解，并由此推出 Morita 等价的两个块是华丽 Rickard 等价的。
AI 中文摘要
设 $k$ 为特征 $p$ 的域。我们证明：若 $\mathcal{F}$ 是有限 $p$-群 $P$ 上的饱和融合系，且 $V$ 是 $\mathcal{F}$-稳定的不可分解带顶内置换 $kP$-模，其 Dade 类 $[V]$ 属于由所有相对合冲生成的 Dade 群 $D_k(P)$ 的子群 $D_k^\Omega(P)$，则 $V$ 具有 $\mathcal{F}$-稳定的内分裂 $p$-置换分解。作为应用，我们证明以下民间结果：若有限群的两个块通过具有内置换 $kP$-源 $V$ 的双模 Morita 等价，且 $[V]\in D_k^\Omega(P)$，则这两个块是华丽 Rickard 等价的。
英文摘要
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.
CommentsRewrote the abstract and introduction; corrected some typos; added some details
