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.

中文翻译：

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可数类型的平环表象

让R→ü是一个结合环外同态，使得ü是平左R-模块。假设相关的 Gabriel 拓扑$\mathbb{G}$正确的理想在R有可数基数。然后我们证明左边R-模块ü射影维数最多为 1。此外，左反模的阿贝尔范畴超过R在$\mathbb{G}$完全忠实地嵌入到 Geigle-Lenzing 右垂子类别中ü在左的类别中R-modules，而后一个阿贝尔范畴的每个对象都是前一个范畴的两个对象的扩展。我们讨论两个阿贝尔范畴等价的条件。给定关联环上的右线性拓扑R，我们考虑每个左边的诱导拓扑R-module 并且，对于一个完美的 Gabriel 拓扑$\mathbb{G}$, 将一个模块的完成情况与适当的 Ext 模块进行比较。最后，我们表征ü- 左强平R-由左正度Ext-正交性两个条件到所有左边的模ü-模块和所有$\mathbb{G}$- 分离的$\mathbb{G}$- 完成左侧R-模块。