Derived equivalences induced by big cotilting modules

TL;DR

This paper establishes a connection between tilting objects in Grothendieck categories and big cotilting modules through derived equivalences, also relating these to model category structures.

Contribution

It shows that triangle equivalences from tilting objects induce big cotilting modules and constructs a correspondence that is essentially unique, linking derived categories and model structures.

Findings

Triangle equivalences send injective cogenerators to big cotilting modules.

Every big cotilting module arises from a tilting object in a Grothendieck category.

Derived equivalences are induced by Quillen equivalences between model categories.

Abstract

We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique way. We also prove that the triangle equivalence is at the base of an equivalence of derivators, which in turn is induced by a Quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes.