Flat ring epimorphisms of countable type

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.