Matlis category equivalences for a ring epimorphism

Abstract

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

LaTeX 2e with tikz-cd, 30 pages, 6 commutative diagrams. v.1: This is an improved, expanded version of Sections 16-18 of the long preprint arXiv:1807.10671v1, which was divided into three parts. v.2: Terminological change of "u-h-divisible" to "u-divisible"; Remark 1.2(1), Proposition 2.4, Lemma 3.4, and Remark 3.5 inserted; references added; v.3: Final version