# Flat ring epimorphisms of countable type

Research paper by Leonid Positselski

Indexed on: 02 Aug '18Published on: 02 Aug '18Published in: arXiv - Mathematics - Rings and Algebras

#### 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. 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.