# Matlis category equivalences for a ring epimorphism

## TL;DR

This paper explores the construction of Matlis category equivalences for ring epimorphisms, especially focusing on homological cases of flat or projective dimension 1, and describes derived category recollements involving comodules and contramodules.

## Contribution

It introduces new Matlis category equivalences for associative ring epimorphisms and constructs derived category recollements under specific homological conditions.

## Key findings

- Constructs Matlis equivalences for ring epimorphisms.
- Describes recollements of derived categories involving comodules and contramodules.
- Proves flatness of homological epimorphisms of projective dimension 1 for commutative rings.

## 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.