We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for a subgroup $H$, often the normalizer of a $p$-subgroup of $G$. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories.

中文翻译：

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复合体的相对射影性和格林对应

我们研究了复合物类别的格林对应版本，包括同伦类别和派生类别。对应关系是在有限群 $G$ 上定义的类别与子群 $H$ 的相同类别之间的等价，通常是 $G$ 的 $p$-子群的归一化器。我们提出了一个基本公式，用于确定模块或复合体的类别何时具有格林对应关系，并将其应用于许多示例。在某些情况下，等价是三角化类别的等价，在特殊情况下，它是张量三角化类别的等价。