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arXiv · 1907.04973

Matlis category equivalences for a ring epimorphism

Silvana Bazzoni·Leonid Positselski
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Abstract

Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism $u\colon R\to U$. 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.

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Silvana Bazzoni, Leonid Positselski. 2019-07-11. Matlis category equivalences for a ring epimorphism. https://doi.org/10.1016/j.jpaa.2020.106398

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