Abelian right perpendicular subcategories in module categories

Abstract

We show that an abelian category can be exactly, fully faithfully embedded into a module category as the right perpendicular subcategory to a set of modules or module morphisms if and only if it is a locally presentable abelian category with a projective generator, or in other words, the category of models of an additive algebraic theory of possibly infinite bounded arity. This includes the categories of contramodules over topological rings and other examples. Various versions of the definition of the right perpendicular subcategory are considered, all leading to the same class of abelian categories. We also discuss sufficient conditions under which the natural forgetful functors from the categories of contramodules to the related categories of modules are fully faithful.

LaTeX 2e with pb-diagram and xy-pic, 57 pages, 2 commutative diagrams; v.5: several sentences inserted in examples 5.6(1-2), references added and updated; v.6: section 7 added, references added and updated, several misprints corrected; v.7: details added in the proof of proposition 2.1, references added and updated, several misprints corrected; v.8: references updated

References in corpus (8)

• Triangulated Matlis equivalence
• Contramodules over pro-perfect topological rings
• Infinity-tilting theory
• Flat morphisms of finite presentation are very flat
• On strongly flat and weakly cotorsion modules
• Derived Equivalence induced by $n$-tilting modules
• Smooth duality and co-contra correspondence
• Errata: On the Homology of Completion and Torsion

Cited by in corpus (10)

• Contramodules over pro-perfect topological rings
• Infinity-tilting theory
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Flat morphisms of finite presentation are very flat
• On strongly flat and weakly cotorsion modules
• Matlis category equivalences for a ring epimorphism
• Adversarial Example Games
• Remarks on derived complete modules and complexes
• Exactness of direct limits for abelian categories with an injective cogenerator
• Enveloping Classes over Commutative Rings