### Citations

283 | Morita theory for Derived Categories - Rickard - 1989 |

201 |
Faisceaux pervers
- Beilinson, Bernstein, et al.
- 1983
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Citation Context ...tand its structure. Two kinds of torsion pairs have been considered with particular emphasis in the literature. These are the notions of t-structure (introduced by Beilinson, Bernstein and Deligne in =-=[6]-=-) and co-tstructure (introduced independently by Bondarko in [9] and Pauksztello in [23]). These are torsion pairs with an additional property concerning the suspension functor of the underlying trian... |

172 |
Representable functors, Serre functors, and reconstructions
- Bondal, Kapranov
- 1989
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Citation Context ...of equivalence classes of silting objects in Kb(pro j-R); • the set of bounded co-t-structures in Kb(pro j-R). Recall that R is of finite global dimension if and only if Db(R) admits a Serre functor (=-=[8]-=-, [24]). Also, in this case, we have Kb(pro j-R) ∼= Db(R) and, thus, the correspondence between t-structures and co-t-structures occurs in the same category. Under this assumption, the bijections of t... |

157 |
Tilting in abelian categories and quasitilted algebras
- Happel, Reiten, et al.
- 1996
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Citation Context .... Among these silting objects, Z0 = S2⊕P2 and Z1 = S2⊕S1[1] are tilting objects. 6. HRS-TILTS AND RECOLLEMENTS In this section we show that HRS-tilts of t-structures with respect to torsion theories (=-=[15]-=-) are compatible with the glueing of t-structures via recollements. The main results of this section are theorem 6.4 and proposition 6.5. Our notation is fixed as follows. • D is a triangulated catego... |

62 | M.: Noetherian hereditary abelian categories satisfying Serre duality
- Reiten, Bergh
- 2002
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Citation Context ...uivalence classes of silting objects in Kb(pro j-R); • the set of bounded co-t-structures in Kb(pro j-R). Recall that R is of finite global dimension if and only if Db(R) admits a Serre functor ([8], =-=[24]-=-). Also, in this case, we have Kb(pro j-R) ∼= Db(R) and, thus, the correspondence between t-structures and co-t-structures occurs in the same category. Under this assumption, the bijections of the the... |

56 | structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general
- Bondarko, Weight
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Citation Context ...red with particular emphasis in the literature. These are the notions of t-structure (introduced by Beilinson, Bernstein and Deligne in [6]) and co-tstructure (introduced independently by Bondarko in =-=[9]-=- and Pauksztello in [23]). These are torsion pairs with an additional property concerning the suspension functor of the underlying triangulated category and they give rise to additive (or even abelian... |

56 | T-structures on some local Calabi-Yau varieties - Bridgeland |

55 | Aisles in derived categories
- Keller, Vossieck
- 1988
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Citation Context ...subcategories which are of interest. In this paper, we work with correspondences that classify these torsion pairs in terms of objects of the triangulated category. Keller and Vossieck established in =-=[18]-=- a bijection between bounded t-structures and equivalence classes of silting objects in the bounded derived category of modules over the path algebra of a Dynkin quiver over a field. Recently, this bi... |

33 | Compact corigid objects in triangulated categories and co-t-structures
- Pauksztello
- 2008
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Citation Context ...hasis in the literature. These are the notions of t-structure (introduced by Beilinson, Bernstein and Deligne in [6]) and co-tstructure (introduced independently by Bondarko in [9] and Pauksztello in =-=[23]-=-). These are torsion pairs with an additional property concerning the suspension functor of the underlying triangulated category and they give rise to additive (or even abelian in the case of t-struct... |

24 | Recollements and tilting objects,
- Hügel, Koenig, et al.
- 2011
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Citation Context ...can glue derived equivalences, i.e., tilting objects, comes as a particular setting of the general context of glueing silting. Similar constructions of tilting objects have been discussed in [20] and =-=[3]-=-. In particular, we will show that the construction in [3] is a particular case of the construction above. The following is our main theorem concerning tilting. Theorem (Theorem 4.5) Let R be a recoll... |

23 | Algebraic stratification in representation categories - Cline, Parshall, et al. |

14 | Stability conditions, torsion theories and tilting - Woolf - 2010 |

13 |
Ana Jeremías López, and María José Souto Salorio. Construction of t-structures and equivalences of derived categories
- Tarrío
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Citation Context ...collement there. Let (D≤0,D≥0) be the standard t-structure on Db(R), which is depicted in the Auslander-Reiten quiver of Db(R) as · · · P2[−2] S2[−2] S1[−1] P2[−1] S2[−1] S1 P2 S2 S1[1] P2[1] S2[1] S1=-=[2]-=- P2[2] · · · ❄ ❄❄ ❄❄ ?? ❄ ❄❄ ❄❄ ?? ❄ ❄❄ ❄❄ ?? ❄ ❄❄ ❄❄ ?? ❄ ❄❄ ❄❄ ?? ❄ ❄❄ ❄❄ ?? where the objects in the boxes belong to the aisle D≤0. This t-structure restri... |

8 | Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras
- Koenig, Yang
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Citation Context ... bijection has been extended by Keller and Nicolás in [17] for the bounded derived categories of homologically homologically smooth non-positive differential graded algebras and by Koenig and Yang in =-=[19]-=- for bounded derived categories of finite dimensional algebras over a field. Indeed, they show that in such a category, there is a bijection between silting objects and bounded t-structures whose hear... |

7 | Three kinds of mutation
- Buan, Reiten, et al.
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Citation Context ...0 (add(S),S⊥)[−1]) 6 The torsion pairs (add(S),S⊥) and ( ⊥S,add(S)) will be called mutation torsion pairs. On the other hand, silting mutation was introduced and studied by Buan, Reiten and Thomas in =-=[11]-=- and, independently, by Aihara and Iyama in [1]. We recall its definition. Definition 2.10. Let M = X ⊕Y be a silting object in Db(R). The left mutation of M at X , denoted by µ−X (M), is defined as t... |

6 |
Cocovers and tilting modules
- Rickard, Schofield
- 1989
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Citation Context ...1 is a partial tilting module of projective dimension 2. It has a minimal projective resolution over R given by the exact sequence 0 // P2 β // P3 δ // P1 // S1 // 0 . Rickard and Schofield showed in =-=[26]-=- that S1 cannot be completed to a tilting module over R. We will strengthen this result by showing that S1 cannot be completed to a tilting object in Db(R). If e is the idempotent e2 + e3 then, as a r... |

5 | Derived equivalences of triangular matrix rings arising from extensions of tilting modules’, Algebr. Represent. Theory 14
- Ladkani
- 2011
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Citation Context ... when we can glue derived equivalences, i.e., tilting objects, comes as a particular setting of the general context of glueing silting. Similar constructions of tilting objects have been discussed in =-=[20]-=- and [3]. In particular, we will show that the construction in [3] is a particular case of the construction above. The following is our main theorem concerning tilting. Theorem (Theorem 4.5) Let R be ... |

3 | Reflecting recollements - Jørgensen |

3 |
Cluster hearts and cluster tilting objects, in preparation
- Keller, Nicolás
(Show Context)
Citation Context ...alence classes of silting objects in the bounded derived category of modules over the path algebra of a Dynkin quiver over a field. Recently, this bijection has been extended by Keller and Nicolás in =-=[17]-=- for the bounded derived categories of homologically homologically smooth non-positive differential graded algebras and by Koenig and Yang in [19] for bounded derived categories of finite dimensional ... |

3 | t-structures via recollements for piecewise hereditary algebras
- Liu, Vitória
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Citation Context ...d, recent work by the two first authors has shown that, in the piecewise hereditary case, all bounded t-structures whose heart is a length category are glued with respect to a non-trivial recollement =-=[21]-=-. In this setting it is then clear that every silting object can be decomposed by this process into as many pieces as derived simple factors of the algebra (check [4] and [5] for terminology). It turn... |

3 |
María José Souto Salorio, Auslander-Buchweitz context and co-t-structures
- Mendoza, Sáenz, et al.
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Citation Context ...lary, since for algebras of finite representation type all hearts of bounded t-structures are length categories. A new correspondence between silting objects and bounded co-t-structures was proved in =-=[22, 17]-=-. This bijection will be central in our approach. Silting objects play, thus, a more general role than tilting objects. They describe all hearts which are length categories and these turn out to be pr... |

2 |
theorems for derived module categories of piecewise hereditary algebras
- Jordan-Hölder
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Citation Context ...ect to a non-trivial recollement [21]. In this setting it is then clear that every silting object can be decomposed by this process into as many pieces as derived simple factors of the algebra (check =-=[4]-=- and [5] for terminology). It turns out, however, that an answer to the problem of glueing silting objects can be given more easily when the focus is on co-t-structures rather than on t-structures. Ou... |

2 |
Franjou and Teimuraz Pirashvili. Comparison of abelian categories recollements
- Vincent
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Citation Context ...). Then the functors in the recollement (6.1) are given by pi∗ = H0Y ◦ i ∗ ◦ εD , pi! = H0Y ◦ i ! ◦ εD , pi∗ = H0D ◦ i∗ ◦ εY , p j! = H0D ◦ j! ◦ εX , p j∗ = H0D ◦ j∗ ◦ εX , p j∗ = H0X ◦ j∗ ◦ εD . See =-=[14]-=- for more on recollements of abelian categories. Remark 6.1. Since for our fixed t-structures, i∗ and j∗ are t-exact (see, for example, [21] for details), we have that pi∗ = i∗ ◦ εY and p j∗ = j∗ ◦ εD... |

2 |
Nondegeneration and boundedness of t-structure induced by recollement, Xiamen Daxue Xuebao Ziran Kexue Ban 45
- Wang
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Citation Context ...ructures, then (D ′,D ′′[−1]) is a t-structure in D; (3) ([9, theorem 8.2.3]) If (X ′,X ′′[−1]) and (Y ′,Y ′′[−1]) are co-t-structures, then (D ′,D ′′[1]) is a co-t-structure in D . 4 (4) ([9], [21], =-=[28]-=-) The glueing of bounded t-structures whose heart is a length category is a bounded t-structure whose heart is a length category. Also, the glueing of bounded co-tstructures is still bounded. If D adm... |

1 | E-mail address: jorge.vitoria@univr.it - LIU |