The tilting-cotilting correspondence

Abstract

To a big n-tilting object in a complete, cocomplete abelian category A with an injective cogenerator we assign a big n-cotilting object in a complete, cocomplete abelian category B with a projective generator, and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of A and B. Under various assumptions on A, which cover a wide range of examples (for instance, if A is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that B is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom-functor.

LaTeX 2e with TikZ, 69 pages, 1 figure; v.2: improvement in Lemma 9.5, Remark 9.6, and Theorem 9.7, references added in Section 6.3; v.3: the presentation of the second half of the paper was restructured, a result on equivalences of contramodule categories was included, references were added and updated; v.4: small changes, the numbering of sections shifted to agree with the journal version

References in corpus (5)

• On exact categories and applications to triangulated adjoints and model structures
• Triangulated Matlis equivalence
• Flat ring epimorphisms of countable type
• The t-structure induced by an n-tilting module
• Exactness of direct limits for abelian categories with an injective cogenerator

Cited by in corpus (15)

• Cotilting Sheaves on Noetherian Schemes
• Contramodules over pro-perfect topological rings
• $t$-Structures with Grothendieck hearts via functor categories
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Covers and direct limits: a contramodule-based approach
• Differential graded Koszul duality: an introductory survey
• Projective covers of flat contramodules
• Remarks on derived complete modules and complexes
• Tilting classes over commutative rings
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Generalized periodicity theorems
• Tilting modules and tilting torsion pairs - filtrations induced by tilting modules
• Topologically semiperfect topological rings
• A bounded below, noncontractible, acyclic complex of projective modules
• A contramodule generalization of Neeman's flat and projective module theorem