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

Abstract

We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.

45 pages; v.2: several misprints corrected, references updated -- this is intended as the final version

References in corpus (4)

• On exact categories and applications to triangulated adjoints and model structures
• New Approach to Arakelov Geometry
• On purity and applications to coderived and singularity categories
• Dedualizing complexes and MGM duality

Cited by in corpus (29)

• Contramodules over pro-perfect topological rings
• Infinity-tilting theory
• Contramodules
• Flat morphisms of finite presentation are very flat
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Matlis category equivalences for a ring epimorphism
• On strongly flat and weakly cotorsion modules
• Fp-projective periodicity
• Topologically semisimple and topologically perfect topological rings
• Abelian right perpendicular subcategories in module categories
• Covers and direct limits: a contramodule-based approach
• Closure properties of $\underrightarrow{\lim }\mathcal{C}$
• Projective covers of flat contramodules
• Flat ring epimorphisms of countable type
• Remarks on derived complete modules and complexes
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Relative nonhomogeneous Koszul duality
• Coderived and contraderived categories of locally presentable abelian DG-categories
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Exactness of direct limits for abelian categories with an injective cogenerator
• On the tensor product of well generated dg categories
• Exact categories of topological vector spaces with linear topology
• Nearly locally presentable categories
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Locally coherent exact categories
• Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
• Exponentiable Grothendieck categories in flat Algebraic Geometry
• Contraderived categories of CDG-modules
• A contramodule generalization of Neeman's flat and projective module theorem