Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories

2019

Glasgow Math. J.

Let R −→ U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology 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 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 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 G-separated G-complete left R-modules.

“…[16][17][18][19][20][21][22]). The material of Sections 2.7-2.8 was developed by the present author [20,21,22,26]. All topologies considered in this paper will be linear.…”

Section: Preliminaries On Topological Ringsmentioning

confidence: 99%

“…The assumption that the topology on R is right linear guarantees convergence [22, Section 2.1], [26,Section 5].…”

Section: 5mentioning

confidence: 99%

Section: F-systems and Pseudo-f-systemsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

See 3 more Smart Citations

Flat Ring Epimorphisms of Countable Type

2019

Glasgow Math. J.

Remarks on derived complete modules and complexes

Mathematische Nachrichten

2022

Let R be a commutative ring and I⊂R$I\subset R$ a finitely generated ideal. We discuss two definitions of derived I‐adically complete (also derived I‐torsion) complexes of R‐modules, which appear in the literature: the idealistic and the sequential ones. The two definitions are known to be equivalent for a weakly proregular ideal I; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider I‐adically flat R‐modules.

Comodules and Contramodules Over Graded Rings

Relative Nonhomogeneous Koszul Duality

2021

This paper is a follow-up to [38]. We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring C over a noncommutative ring A such that C is a flat right A-module, we show that all A-flat left C-comodules are < ℵ 1 -direct limits of A-countably presented A-flat C-comodules. In the context of a complete, separated topological ring R with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat R-contramodules are < ℵ 1 -direct limits of countably presented flat R-contramodules. We also describe short exact sequences and pure acyclic complexes of A-flat C-comodules and flat R-contramodules as certain < ℵ 1 -direct limits. Applications to cotorsion periodicity in the respective settings are discussed. In particular, in any acyclic complex of cotorsion R-contramodules, all the contramodules of cocycles are cotorsion.