On strongly flat and weakly cotorsion modules

Abstract

The aim of this paper is to describe the classes of strongly flat and weakly cotorsion modules with respect to a multiplicative subset or a finite collection of multiplicative subsets in a commutative ring. The strongly flat modules are characterized by a set of conditions, while the weakly cotorsion modules are produced by a generation procedure. Passing to the collection of all countable multiplicative subsets, we define quite flat and almost cotorsion modules, and show that, over a Noetherian ring with countable spectrum, all flat modules are quite flat and all almost cotorsion modules are cotorsion.

LaTeX 2e, 51 pages; v.3: section 0.0 inserted in the introduction, some details added in the proofs of theorems 2.6 and 7.7, several references added; v.4: small changes in section 0.1, a reference added at the end of section 0.2; v.5: several misprints corrected, references updated, the numbering of sections shifted to agree with the journal version

References in corpus (5)

• $S$-almost perfect commutative rings
• Triangulated Matlis equivalence
• The countable Telescope Conjecture for module categories
• Flat morphisms of finite presentation are very flat
• Abelian right perpendicular subcategories in module categories

Cited by in corpus (7)

• $S$-almost perfect commutative rings
• Flat morphisms of finite presentation are very flat
• Abelian right perpendicular subcategories in module categories
• Flat ring epimorphisms of countable type
• An explicit self-dual construction of complete cotorsion pairs in the relative context
• Flat commutative ring epimorphisms of almost Krull dimension zero
• Countably generated flat modules are quite flat