Recollements and tilting objects

Hügel, Lidia Angeleri; Koenig, Steffen; Liu, Qunhua

doi:10.1016/j.jpaa.2010.04.027

Cited by 63 publications

(36 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, the question of when we can glue derived equivalencesthat is, tilting objects-comes as a particular setting of the general context of gluing silting. Similar constructions of tilting objects have been discussed in [19] and [2]. In particular, we will show that the construction in [2] is a particular case of the preceding construction.…”

Section: §1 Introductionsupporting

confidence: 55%

See 1 more Smart Citation

Gluing Silting Objects

Liu¹,

Vitória²,

Yang³

2014

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: §1 Introductionsupporting

confidence: 55%

“…Similar constructions of tilting objects have been discussed in [19] and [2]. In particular, we will show that the construction in [2] is a particular case of the preceding construction. The following is our main theorem concerning tilting (see Theorem 4.5).…”

Section: §1 Introductionsupporting

confidence: 55%

Gluing Silting Objects

Liu¹,

Vitória²,

Yang³

2014

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this sense, a recollement of derived categories can be viewed as a natural generalization of a derived equivalence. The relation between tilting theory and recollements of derived categories has been further studied in [1] [6]. The dg version of Rickard's theorem was developed by Keller in [10], and the result of Koenig was generalized to the dg setting by Jørgensen [9] and Nicolás-Saorín [17], where the role of partial tilting complexes is played by compact objects.…”

mentioning

confidence: 99%

Recollements from generalized tilting

Yang

2011

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let A be a small dg category over a field k and let U be a small full subcategory of the derived category DA which generate all free dg A-modules. Let (B, X) be a standard lift of U. We show that there is a recollement such that its middle term is DB, its right term is DA, and the three functors on its right side are constructed from X. This applies to the pair (A, T ), where A is a k-algebra and T is a good n-tilting module, and we obtain a result of Bazzoni-Mantese-Tonolo. This also applies to the pair (A, U), where A is an augmented dg category and U is the category of 'simple' modules, e.g. A is a finite-dimensional algebra or the Kontsevich-Soibelman A∞-category associated to a quiver with potential. Mathematics Subject Classifications: 18E30, 16E45.A recollement of triangulated categories is a diagram of triangulated categories and triangle• (i * , i * , i ! ) and (j ! , j * , j * ) are adjoint triples;• i * , j * , j ! are fully faithful;• j * • i * = 0;• for every object X of T there are two triangleswhere the four morphisms are the units and counits.We also say that this is a recollement of T in terms of T ′ and T ′′ . This notion was introduced

show abstract

“…Since t-structures can be glued via a recollement (see [10]) another natural question is the following: given a recollement (1) of triangulated categories and silting sets and in and , is it possible to construct a silting set in corresponding to the glued t-structure? This problem was studied in the context of tilting objects in [4] under some restrictions and in the context of gluing with respect to co-t-structures in [35]. The second goal of this paper is to show that the process of gluing t-structures allows to construct partial silting sets in the central category of a recollement out of partial silting sets in its outer categories.…”

Section: Introductionmentioning

confidence: 99%

Lifting of recollements and gluing of partial silting sets

Saorı́n¹,

Zvonareva²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

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.

show abstract

Recollements and tilting objects

Cited by 63 publications

References 28 publications

Gluing Silting Objects

Gluing Silting Objects

Recollements from generalized tilting

Lifting of recollements and gluing of partial silting sets

Contact Info

Product

Resources

About