Geodesic
FLAT RING EPIMORPHISMS OF COUNTABLE TYPE
Glasgow mathematical journal, Tome 62 (2020) no. 2, pp. 383-439

Voir la notice de l'article provenant de la source Cambridge University Press

Let R→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 associative 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. 
POSITSELSKI, LEONID. FLAT RING EPIMORPHISMS OF COUNTABLE TYPE. Glasgow mathematical journal, Tome 62 (2020) no. 2, pp. 383-439. doi: 10.1017/S001708951900017X
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