Abstract Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u : R ⟶ U . Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U-modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required.

中文翻译：

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环同胚的 Matlis 范畴等价

摘要 在温和的假设下，我们为关联环表同态 u : R ⟶ U 构造了两个 Matlis 加性范畴等价。假设环同胚是平面/射影维1的同调，我们讨论u-余模和u-反模的阿贝尔范畴，并构造R-模、U-模和R-模的复合的无界派生范畴的重新集合与 u-co/contramodule 上同调。进一步的假设允许将 recollement 中的第三个类别描述为 u-comodules 和 u-contramodules 的阿贝尔类别的无界派生类别。对于交换环，我们还证明了射影维数为 1 的任何同构同胚是平的。不需要映射 u 的注入性。