The tilting-cotilting correspondence

TL;DR

This paper establishes a correspondence between big n-tilting and n-cotilting objects in abelian categories, leading to derived equivalences that generalize classical tilting theory.

Contribution

It introduces a duality between n-tilting and n-cotilting objects in different abelian categories and constructs derived equivalences via contramodule theory.

Findings

Constructs a bijective correspondence between n-tilting and n-cotilting objects.

Establishes derived equivalences between categories using contramodule-based functors.

Applies to a wide class of categories including module and Grothendieck categories.

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.