Mathematics > Rings and Algebras
[Submitted on 2 Aug 2018 (v1), last revised 16 Sep 2021 (this version, v8)]
Title:Flat ring epimorphisms of countable type
View PDFAbstract: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.
Submission history
From: Leonid Positselski [view email][v1] Thu, 2 Aug 2018 17:42:56 UTC (50 KB)
[v2] Sun, 19 Aug 2018 21:40:49 UTC (52 KB)
[v3] Thu, 30 Aug 2018 17:25:49 UTC (57 KB)
[v4] Wed, 20 Mar 2019 16:52:48 UTC (59 KB)
[v5] Wed, 24 Apr 2019 16:26:23 UTC (58 KB)
[v6] Fri, 17 May 2019 11:08:52 UTC (65 KB)
[v7] Wed, 15 Apr 2020 23:12:05 UTC (67 KB)
[v8] Thu, 16 Sep 2021 16:16:30 UTC (69 KB)
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