$S$-almost perfect commutative rings

Abstract

Given a multiplicative subset $S$ in a commutative ring $R$, we consider $S$-weakly cotorsion and $S$-strongly flat $R$-modules, and show that all $R$-modules have $S$-strongly flat covers if and only if all flat $R$-modules are $S$-strongly flat. These equivalent conditions hold if and only if the localization $R_S$ is a perfect ring and, for every element $s\in S$, the quotient ring $R/sR$ is a perfect ring, too. The multiplicative subset $S\subset R$ is allowed to contain zero-divisors.

29 pages; v.2: final version

References in corpus (4)

• Triangulated Matlis equivalence
• On varieties of groups generated by wreath products of abelian groups
• Flat morphisms of finite presentation are very flat
• On strongly flat and weakly cotorsion modules

Cited by in corpus (12)

• Contramodules over pro-perfect topological rings
• Contramodules
• Matlis category equivalences for a ring epimorphism
• On strongly flat and weakly cotorsion modules
• Topologically semisimple and topologically perfect topological rings
• Covers and direct limits: a contramodule-based approach
• Flat commutative ring epimorphisms of almost Krull dimension zero
• Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity
• Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
• Countably generated flat modules are quite flat
• Characterizing $S$-projective modules and $S$-semisimple rings by uniformity
• Enveloping Classes over Commutative Rings