Lifting of recollements and gluing of partial silting sets

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.

Report on the finiteness of silting objects

We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$ -tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$ -tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$ -tilting-finite algebra is representation-finite.

Gluing Silting Objects

Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and cot-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.

Gluing Silting Objects

Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and cot-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.