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.Comment: 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 versio