Lifting of recollements and gluing of partial silting sets

TL;DR

This paper develops criteria for lifting recollements to TTF triples in triangulated categories and applies these results to construct silting sets in glued t-structures, with explicit methods for Artin algebras.

Contribution

It introduces new conditions for lifting recollements to TTF triples and provides explicit constructions for silting objects in glued t-structures, especially for Artin algebras.

Findings

Lifting of TTF triples from recollements is possible under specific criteria.

Recollements of derived categories of schemes can be lifted to ambient categories.

Explicit methods for gluing silting objects in Artin algebra derived categories.

Abstract

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.