  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.

Rings and Algebras;Category Theory