Lifting of recollements and gluing of partial silting sets

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.

Properties of triangulated and quotient categories arising from $n$-Calabi-Yau triples

“…Recall that precovers (respectively preenvelopes) are the same as right (respectively left) approximations. The following is a consequence of [15,Lemma 5.3], but we provide a proof for the benefit of the reader.…”

Transferring the TTF triples

“…We say that D is compactly generated when it has a generating set of compact objects. Denote by D c the subcategory of compact objects of D. The lemma we give below is implicit in [19,Theorem 3.3…”

“…To any recollement one canonically associates a TTF triple. Recently, M.Saorín and A.Zvonareva [19] gave a criterion for a recollement of triangulated subcategories to lift to a torsion torsionfree triple (TTF triple, simply) of ambient triangulated categories with coproducts. L.Angeleri Hügel, S.Koenig, Q.H.Liu and D.Yang [2] introduced and used ladders of recollements as the main tool to clarify the lifting and restriction of the different level of derived categories.…”