Flat ring epimorphisms of countable type

TL;DR

This paper investigates flat ring epimorphisms of countable type, establishing bounds on projective dimension, embedding properties of contramodule categories, and characterizations of strongly flat modules within a homological framework.

Contribution

It introduces new bounds on projective dimension for flat ring epimorphisms with countable Gabriel topologies and explores the relationships between associated abelian categories and module completions.

Findings

Projective dimension of $U$ is at most 1.

Full embedding of contramodule categories into perpendicular subcategories.

Characterization of $U$-strongly flat modules via Ext-orthogonality.

Abstract

Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$-module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$-module $U$ has projective dimension at most $1$. Furthermore, the abelian category of left contramodules over the completion of $R$ at $\mathbb G$ fully faithfully embeds into the Geigle-Lenzing right perpendicular subcategory to $U$ in the category of left $R$-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an assocative ring $R$, we consider the induced topology on every left $R$-module, and for a perfect Gabriel topology $\mathbb G$ compare the completion of a module with an appropriate Ext module. Finally, we characterize the $U$-strongly flat left $R$-modules by the two conditions of left positive-degree Ext-orthogonality to all left $U$-modules and all $\mathbb G$-separated $\mathbb G$-complete left $R$-modules.