We show that the stable module $\infty$-category of a finite group $G$ decomposes in three different ways as a limit of the stable module $\infty$-categories of certain subgroups of $G$. Analogously to Dwyer's terminology for homology decompositions, we call these the centraliser, normaliser, and subgroup decompositions. We construct centraliser and normaliser decompositions and extend the subgroup decomposition (constructed by Mathew) to more collections of subgroups. The key step in the proof is extending the stable module $\infty$-category to be defined for any $G$-space, then showing that this extension only depends on the $S$-equivariant homotopy type of a $G$-space. The methods used are not specific to the stable module $\infty$-category, so may also be applicable in other settings where an $\infty$-category depends functorially on $G$.

中文翻译：

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稳定模块 ∞$\infty$-category 的分解

我们证明了稳定的模块$\infty$- 有限群的范畴$G$以三种不同的方式分解作为稳定模块的限制$\infty$- 某些亚组的类别$G$. 类似于 Dwyer 的同调分解术语，我们称它们为集中器、归一器和子群分解。我们构造集中器和归一器分解，并将子组分解（由 Mathew 构造）扩展到更多子组集合。证明的关键步骤是扩展 stable 模块$\infty$- 为任何定义的类别$G$-space，然后表明这个扩展只依赖于$\mathrm{小号}$- a 的等变同伦类型$G$-空间。使用的方法不是特定于 stable 模块$\infty$-类别，因此也可能适用于其他设置$\infty$-类别在功能上取决于$G$.