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FLAT RING EPIMORPHISMS OF COUNTABLE TYPE
Glasgow Mathematical Journal ( IF 0.5 ) Pub Date : 2019-05-07 , DOI: 10.1017/s001708951900017x LEONID POSITSELSKI
Glasgow Mathematical Journal ( IF 0.5 ) Pub Date : 2019-05-07 , DOI: 10.1017/s001708951900017x LEONID POSITSELSKI
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:
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。
更新日期:2019-05-07
中文翻译:
可数类型的平环表象
让