Lifting and restricting recollement data

TL;DR

This paper investigates the conditions under which recollement data for triangulated categories can be lifted or restricted, providing criteria for recognizing and constructing recollements in derived categories of dg categories and algebras.

Contribution

It introduces new criteria for lifting and restricting TTF triples in triangulated categories, extending known recollements from bounded to unbounded derived categories.

Findings

Recollements of right bounded derived categories can be viewed as restrictions of unbounded recollements.

Necessary and sufficient conditions are established for a dg category's derived category to be a recollement of other derived categories.

The paper provides criteria to detect recollements in general right bounded derived categories.

Abstract

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in "Parametrizing recollement data", allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a 'right bounded' derived category of a differential graded(=dg) category to be a recollement of 'right bounded' derived categories of dg categories. In Theorem 2 we consider the particular case in which those dg categories are just ordinary algebras.