Realisation functors in tilting theory

TL;DR

This paper develops a derived Morita theory using realisation functors linked to t-structures, generalising tilting theory to large objects, and applies it to classify recollements of derived categories.

Contribution

It introduces a theory of noncompact tilting and cotilting objects, extending derived Morita theory, and applies it to classify recollements and answer open questions.

Findings

Realisation functors can be identified with derived tensor products under certain conditions.

Recollements of derived categories correspond to tilting or cotilting t-structures.

The approach provides a standard form for recollements of derived module categories.

Abstract

Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (noncompact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.