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## SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES FOR FINITE-DIMENSIONAL ALGEBRAS

Citations: | 8 - 2 self |

### Citations

518 | Triangulated categories in representation theory of finite dimensional algebras - Happel - 1988 |

280 | Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 - Kontsevich, Soibelman |

251 | Braid group actions on derived categories of coherent sheaves
- Seidel, Thomas
(Show Context)
Citation Context ...ts of shifts of P1(n), n ≥ 1. The Auslander–Reiten translation τ takes P1(n) to Σ −1P1(n). It is straightforward to check that P1 is a 0-spherical object of Db(modΛ) in the sense of Seidel and Thomas =-=[45]-=-. The additive closure of this component is the triangulated subcategory generated by P1. This component will be referred to as the 0-spherical component. 28 STEFFEN KOENIG AND DONG YANG The ZA∞∞ comp... |

157 | Tilting in abelian categories and quasitilted algebras - Happel, Reiten, et al. - 1996 |

118 | Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l’institut Fourier
- Amiot
(Show Context)
Citation Context ... a graded H∗(A)-module structure, and hence a graded Ā = H0(A)-module structure. In particular, a stalk dg A-module concentrated in degree 0 is an Ā-module. 4.1. The standard t-structure. We follow =-=[22, 4, 34]-=-, where the dg algebra is not necessarily finite-dimensional. Let M = . . . → M i−1 di−1→ M i di→ M i+1 → . . . be a dg A-module. Consider the standard truncation functors τ≤0 and τ>0: τ≤0M = τ>0M = .... |

101 |
Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. Representation theory
- Brenner, Butler
- 1979
(Show Context)
Citation Context ... if Λ/Λ(1− ei)Λ is projective as a Λ-module. When Λ/Λ(1− ei)Λ is a division algebra (i.e. there are no loops in the quiver of Λ at the vertex i), this specialises to the ‘classical’ BB tilting module =-=[13]-=- and APR tilting module [6]. The following proposition generalises [1, Theorem 2.53]. Proposition 7.4. (a) T is isomorphic to the left mutation µ+i (Λ) of Λ. (b) T is a tilting Λ-module of projective ... |

85 |
Deriving DG categories, Ann. Sci. École Norm
- Keller
- 1994
(Show Context)
Citation Context ...) denote the category of (right) dg modules over A and K(A) the homotopy category. Let D(A) denote the derived category of dg Amodules, i.e. the triangle quotient of K(A) by acyclic dg A-modules, cf. =-=[29, 30]-=-, and let Dfd(A) denote its full subcategory of dg A-modules whose total cohomology is finite-dimensional. The category C(A) is abelian and the other three categories are triangulated with suspension ... |

84 | Derived equivalences from mutations of quivers with potential
- Keller, Yang
(Show Context)
Citation Context ... a graded H∗(A)-module structure, and hence a graded Ā = H0(A)-module structure. In particular, a stalk dg A-module concentrated in degree 0 is an Ā-module. 4.1. The standard t-structure. We follow =-=[22, 4, 34]-=-, where the dg algebra is not necessarily finite-dimensional. Let M = . . . → M i−1 di−1→ M i di→ M i+1 → . . . be a dg A-module. Consider the standard truncation functors τ≤0 and τ>0: τ≤0M = τ>0M = .... |

56 | structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general
- Bondarko, Weight
(Show Context)
Citation Context ...etting) and bounded t-structures. The correspondence between silting objects and co-t-structures appears implicitly on various levels of generality in the work of Aihara and Iyama [1] and of Bondarko =-=[12]-=- and explicitly in full generality in the work of Mendoza, Sáenz, Santiago and Souto Salorio [39] and of Keller and Nicolás [31]. For homologically smooth non-positive dg algebras, all the bijection... |

56 | T-structures on some local Calabi-Yau varieties
- Bridgeland
(Show Context)
Citation Context ...F}. Similarly one defines the right mutation µ−i (C≤0, C≥0). These mutations provide an effective method to compute the space of Bridgeland’s stability conditions on C by gluing different charts, see =-=[14, 48]-=-. Proposition 7.10. The pairs µ+i (C≤0, C≥0) and µ−i (C≤0, C≥0) are bounded t-structures of C. The heart of µ+i (C≤0, C≥0) has a torsion pair (ΣF ,T ) and the heart of µ−i (C≤0, C≥0) has a torsion pai... |

55 | Aisles in derived categories
- Keller, Vossieck
- 1988
(Show Context)
Citation Context ...he end of the article demonstrates one practical use of these bijections and their properties. Finally we give some remarks on the literature. For path algebras of Dynkin quivers, Keller and Vossieck =-=[33]-=- have already given a bijection between bounded t-structures and silting objects. The bijection between silting objects and t-structures with length heart has been established by Keller and Nicolás [... |

52 |
Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics,
- Buan, Marsh, et al.
- 2006
(Show Context)
Citation Context ...ation theory, geometry and topology. They are also closely related to fundamental concepts in cluster theory such as clusters ([20]), c-matrices and g-matrices ([21, 40]) and cluster-tilting objects (=-=[7]-=-). We refer to the survey paper [16] for more details. A concrete example to be given at the end of the article demonstrates one practical use of these bijections and their properties. Finally we give... |

46 |
Analyse et topologie sur les espaces singulares, Astirisque 100, Sot
- Beilinson, Bernstein, et al.
- 1982
(Show Context)
Citation Context ...odules is a simple-minded collection in Db(modΛ). A natural question is: do any two simple-minded collections have the same collection of endomorphism algebras? 3.3. t-structures. A t-structure on C (=-=[8]-=-) is a pair (C≤0, C≥0) of strict (that is, closed under isomorphisms) and full subcategories of C such that · ΣC≤0 ⊆ C≤0 and Σ−1C≥0 ⊆ C≥0; · Hom(M,Σ−1N) = 0 for M ∈ C≤0 and N ∈ C≥0, · for each M ∈ C t... |

33 | Compact corigid objects in triangulated categories and co-t-structures
- Pauksztello
- 2008
(Show Context)
Citation Context ...extension closure of Σm{Si | i ∈ I} for m ≥ 0 respectively for m ≤ 0. (e) C = thick(Si, i ∈ I). (f) If I is finite, then {Si | i ∈ I} is a simple-minded collection. 3.4. Co-t-structures. According to =-=[41]-=-, a co-t-structure on C (or weight structure in [12]) is a pair (C≥0, C≤0) of strict and full subcategories of C such that · both C≥0 and C≤0 are additive and closed under taking direct summands, · Σ−... |

32 | Equivalences of derived categories for symmetric algebras
- Rickard
(Show Context)
Citation Context ...j, · X1, . . . ,Xr generate C (i.e. C = thick(X1, . . . ,Xr)). 6 STEFFEN KOENIG AND DONG YANG Simple-minded collections are variants of simple-minded systems in [36] and were first studied by Rickard =-=[43]-=- in the context of derived equivalences of symmetric algebras. For a finitedimensional algebra Λ, a complete collection of pairwise non-isomorphic simple modules is a simple-minded collection in Db(mo... |

24 | Cluster characters for cluster categories with infinite-dimensional morphism spaces - Plamondon |

20 | On t-structures and torsion theories induced by compact objects - Hoshino, Kato, et al. |

17 |
differential graded categories
- On
(Show Context)
Citation Context ...) denote the category of (right) dg modules over A and K(A) the homotopy category. Let D(A) denote the derived category of dg Amodules, i.e. the triangle quotient of K(A) by acyclic dg A-modules, cf. =-=[29, 30]-=-, and let Dfd(A) denote its full subcategory of dg A-modules whose total cohomology is finite-dimensional. The category C(A) is abelian and the other three categories are triangulated with suspension ... |

17 | Exchange graphs of acyclic Calabi-Yau categories
- King, Qiu
(Show Context)
Citation Context ...proximation Σ−1Xj gj // Xij . Similarly one defines the right mutation µ−i (X1, . . . ,Xr). This generalises Kontsevich–Soibelman’s mutation of spherical collections [38, Section 8.1] and appeared in =-=[35]-=- in the case of derived categories of acyclic quivers. Proposition 7.6. (a) µ+i ◦ µ−i (X1, . . . ,Xr) ∼= (X1, . . . ,Xr) ∼= µ−i ◦ µ+i (X1, . . . ,Xr). (b) Assume that · for any j 6= i the object Σ−1Xj... |

14 |
María Inés Platzeck, and Idun Reiten. Coxeter functors without diagrams
- Auslander
- 1979
(Show Context)
Citation Context ...e as a Λ-module. When Λ/Λ(1− ei)Λ is a division algebra (i.e. there are no loops in the quiver of Λ at the vertex i), this specialises to the ‘classical’ BB tilting module [13] and APR tilting module =-=[6]-=-. The following proposition generalises [1, Theorem 2.53]. Proposition 7.4. (a) T is isomorphic to the left mutation µ+i (Λ) of Λ. (b) T is a tilting Λ-module of projective dimension at most 1. Proof.... |

14 | Stability conditions, torsion theories and tilting
- Woolf
- 2010
(Show Context)
Citation Context ...F}. Similarly one defines the right mutation µ−i (C≤0, C≥0). These mutations provide an effective method to compute the space of Bridgeland’s stability conditions on C by gluing different charts, see =-=[14, 48]-=-. Proposition 7.10. The pairs µ+i (C≤0, C≥0) and µ−i (C≤0, C≥0) are bounded t-structures of C. The heart of µ+i (C≤0, C≥0) has a torsion pair (ΣF ,T ) and the heart of µ−i (C≤0, C≥0) has a torsion pai... |

13 | Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields, Doc. Math., extra volume: Andrei Suslin’s Sixtieth Birthday
- Bondarko
- 2010
(Show Context)
Citation Context ...bounded t-structure on Db(modΛ) with length heart. Proof. Because (C≤0, C≥0) = φ31 ◦ φ14(C≥0, C≤0). √ By definition (C≤0, C≥0) is right orthogonal to the given co-t-structure in the sense of Bondarko =-=[11]-=-. Define φ34(C≥0, C≤0) = (C≤0, C≥0). If Λ has finite global dimension, then Kb(projΛ) is identified with Db(modΛ). As a consequence, C≤0 = C≤0 and C≥0 = νC≥0. Thus the t-structure (C≤0, C≥0) is right ... |

11 |
conditions on triangulated categories
- Stability
(Show Context)
Citation Context ...the heart and for any m < 0. The t-structure (C≤0, C≥0) is said to be bounded if ⋃ n∈Z ΣnC≤0 = C = ⋃ n∈Z ΣnC≥0. A bounded t-structure is one of the two ingredients of a Bridgeland stability condition =-=[15]-=-. A typical example of a t-structure is the pair (D≤0,D≥0) for the derived category D(ModΛ) of an (ordinary) algebra Λ, where D≤0 consists of complexes with vanishing cohomologies in positive degrees,... |

11 |
algebras IV: Coefficients, Compositio Mathematica 143
- Cluster
- 2007
(Show Context)
Citation Context ... four concepts are crucial in representation theory, geometry and topology. They are also closely related to fundamental concepts in cluster theory such as clusters ([20]), c-matrices and g-matrices (=-=[21, 40]-=-) and cluster-tilting objects ([7]). We refer to the survey paper [16] for more details. A concrete example to be given at the end of the article demonstrates one practical use of these bijections and... |

11 | Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object - Holm, Jørgensen, et al. |

9 | From m-clusters to m-noncrossing partitions via exceptional sequences
- Buan, Reiten, et al.
(Show Context)
Citation Context ...al algebras in our preprint [37], which has been partly incorporated into the present article, and partially in the work [44] of Rickard and Rouquier. For hereditary algebras, Buan, Reiten and Thomas =-=[17]-=- studied the bijections between silting objects, simple-minded collections (=Hom≤0-configurations in their setting) and bounded t-structures. The correspondence between silting objects and co-t-struct... |

8 | Equivalences of derived categories for selfinjective algebras - Al-Nofayee |

8 | Ordered Exchange graphs
- Brüstle, Yang
- 2013
(Show Context)
Citation Context .... They are also closely related to fundamental concepts in cluster theory such as clusters ([20]), c-matrices and g-matrices ([21, 40]) and cluster-tilting objects ([7]). We refer to the survey paper =-=[16]-=- for more details. A concrete example to be given at the end of the article demonstrates one practical use of these bijections and their properties. Finally we give some remarks on the literature. For... |

7 | The co-stability manifold of a triangulated category, arXiv:1109.4006
- Jørgensen, Pauksztello
(Show Context)
Citation Context ...> 0. The co-t-structure (C≤0, C≥0) is said to be bounded [12] if ⋃ n∈Z ΣnC≤0 = C = ⋃ n∈Z ΣnC≥0. A bounded co-t-structure is one of the two ingredients of a Jørgensen–Pauksztello costability condition =-=[27]-=-. A typical example of a co-t-structure is the pair (K≥0,K≤0) for the homotopy category Kb(projΛ) of a finite-dimensional algebra Λ, where K≥0 consists of complexes which are homotopy equivalent to a ... |

7 |
by the anaphase-promoting complex.
- Wei, Ayad, et al.
- 2004
(Show Context)
Citation Context ...of C. This notion was introduced by Keller and Vossieck in [33] to study t-structures on the bounded derived category of representations over a Dynkin quiver. Recently it has also been studied by Wei =-=[47]-=- (who uses the terminology semi-tilting complexes) from the perspective of classical tilting theory. A tilting object is a silting object M such that Hom(M,ΣmM) = 0 for m < 0. For an algebra Λ, a tilt... |

4 | categories and reconstruction
- Rickard, Rouquier
- 2010
(Show Context)
Citation Context ...positive dg algebras in Keller and Nicolás’ work [32] and for finite-dimensional algebras in our preprint [37], which has been partly incorporated into the present article, and partially in the work =-=[44]-=- of Rickard and Rouquier. For hereditary algebras, Buan, Reiten and Thomas [17] studied the bijections between silting objects, simple-minded collections (=Hom≤0-configurations in their setting) and b... |

3 |
Maŕıa José Souto Salorio, and Sonia Trepode, Ext-projectives in suspended subcategories
- Assem
(Show Context)
Citation Context ...ished by Keller and Nicolás [32] for homologically smooth non-positive dg algebras, by Assem, Souto SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES 3 Salorio and Trepode =-=[5]-=- and by Vitória [46], who are focussing on piecewise hereditary algebras. An unbounded version of this bijection has been studied by Aihara and Iyama [1]. The bijection between simple-minded collecti... |

3 |
Cluster hearts and cluster tilting objects, in preparation
- Keller, Nicolás
(Show Context)
Citation Context ...els of generality in the work of Aihara and Iyama [1] and of Bondarko [12] and explicitly in full generality in the work of Mendoza, Sáenz, Santiago and Souto Salorio [39] and of Keller and Nicolás =-=[31]-=-. For homologically smooth non-positive dg algebras, all the bijections are due to Keller and Nicolás [31]. The intersection of our results with those of Keller and Nicolás is the case of finite-dim... |

3 |
On tilting complexes providing derived equivalences that send simpleminded objects to simple objects
- Koenig
(Show Context)
Citation Context ...stablished implicitely in Al-Nofayee’s work [3] and explicitely for homologically smooth non-positive dg algebras in Keller and Nicolás’ work [32] and for finite-dimensional algebras in our preprint =-=[37]-=-, which has been partly incorporated into the present article, and partially in the work [44] of Rickard and Rouquier. For hereditary algebras, Buan, Reiten and Thomas [17] studied the bijections betw... |

3 |
María José Souto Salorio, Auslander-Buchweitz context and co-t-structures
- Mendoza, Sáenz, et al.
(Show Context)
Citation Context ...ppears implicitly on various levels of generality in the work of Aihara and Iyama [1] and of Bondarko [12] and explicitly in full generality in the work of Mendoza, Sáenz, Santiago and Souto Salorio =-=[39]-=- and of Keller and Nicolás [31]. For homologically smooth non-positive dg algebras, all the bijections are due to Keller and Nicolás [31]. The intersection of our results with those of Keller and Ni... |

3 |
On tropical dualities in cluster algebras, Algebraic groups and quantum groups
- Nakanishi, Zelevinsky
- 2012
(Show Context)
Citation Context ... four concepts are crucial in representation theory, geometry and topology. They are also closely related to fundamental concepts in cluster theory such as clusters ([20]), c-matrices and g-matrices (=-=[21, 40]-=-) and cluster-tilting objects ([7]). We refer to the survey paper [16] for more details. A concrete example to be given at the end of the article demonstrates one practical use of these bijections and... |

2 |
Christof Geiß, and Andrzej Skowroński, Classification of discrete derived categories, Cent
- Bobiński
(Show Context)
Citation Context ...ightmost components have been put in degree 0. 8.2. The Auslander–Reiten quiver. The Auslander–Reiten quiver of Db(modΛ) consists of three components: two ZA∞ components and one ZA ∞ ∞ component (see =-=[10, 28]-=-) ◦ ◦ Σ−1P1 ◦ ◦ ◦ P1 ◦ ◦ ΣP1 ?? ?? ?? ❄ ❄ ?? ❄ ?? ❄ ❄ ?? ❄ ❄ ❄ ◦ ◦ I2 ◦ S1 ΣS1 ◦ P2 ΣP2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ❄ ❄ ?? ❄ ❄ ?? ?? ❄ ❄ ?? ?? ?? ?? ❄ ?? ❄ ?? ❄ ❄ ... |

2 | Derived categories for nodal rings and projective configurations, Noncommutative algebra and geometry
- Burban, Drozd
(Show Context)
Citation Context ... projective Λ-modules corresponding to the vertices 1 and 2. Then up to isomorphism and up to shift an indecomposable object in Db(modΛ) belongs to one of the following four families (see for example =-=[19, 9]-=-) · P1(n) = P1 → P1 → . . .→ P1 → P1, n ≥ 1, · R(n) = P1 → P1 → . . .→ P1 → P1 → P2, n ≥ 0, · L(n) = P2 → P1 → P1 → . . .→ P1 → P1, n ≥ 0, · B(n) = P2 → P1 → P1 → . . .→ P1 → P1 → P2, n ≥ 1, where the... |

2 |
Derived categories of graded gentle one-cycle algebras, in preparation
- Kalck, Yang
(Show Context)
Citation Context ...ightmost components have been put in degree 0. 8.2. The Auslander–Reiten quiver. The Auslander–Reiten quiver of Db(modΛ) consists of three components: two ZA∞ components and one ZA ∞ ∞ component (see =-=[10, 28]-=-) ◦ ◦ Σ−1P1 ◦ ◦ ◦ P1 ◦ ◦ ΣP1 ?? ?? ?? ❄ ❄ ?? ❄ ?? ❄ ❄ ?? ❄ ❄ ❄ ◦ ◦ I2 ◦ S1 ΣS1 ◦ P2 ΣP2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ❄ ❄ ?? ❄ ❄ ?? ?? ❄ ❄ ?? ?? ?? ?? ❄ ?? ❄ ?? ❄ ❄ ... |

2 | structures and simple dg modules for positive dg algebras - Weight |

1 |
objects in the heart of a t-structure
- Simple
(Show Context)
Citation Context ...nded version of this bijection has been studied by Aihara and Iyama [1]. The bijection between simple-minded collections and bounded t-structures has been established implicitely in Al-Nofayee’s work =-=[3]-=- and explicitely for homologically smooth non-positive dg algebras in Keller and Nicolás’ work [32] and for finite-dimensional algebras in our preprint [37], which has been partly incorporated into t... |

1 |
Aisles, recollements and dg categories
- Fu
- 2006
(Show Context)
Citation Context ... a graded H∗(A)-module structure, and hence a graded Ā = H0(A)-module structure. In particular, a stalk dg A-module concentrated in degree 0 is an Ā-module. 4.1. The standard t-structure. We follow =-=[22, 4, 34]-=-, where the dg algebra is not necessarily finite-dimensional. Let M = . . . → M i−1 di−1→ M i di→ M i+1 → . . . be a dg A-module. Consider the standard truncation functors τ≤0 and τ>0: τ≤0M = τ>0M = .... |

1 |
and Yuming Liu, Gluing of idempotents, radical embeddings and two classes of stable equivalences
- Koenig
(Show Context)
Citation Context ...gebra and Hom(Xi,Xj) vanishes for i 6= j, · X1, . . . ,Xr generate C (i.e. C = thick(X1, . . . ,Xr)). 6 STEFFEN KOENIG AND DONG YANG Simple-minded collections are variants of simple-minded systems in =-=[36]-=- and were first studied by Rickard [43] in the context of derived equivalences of symmetric algebras. For a finitedimensional algebra Λ, a complete collection of pairwise non-isomorphic simple modules... |

1 |
Silting objects on derived module categories
- Vitória
(Show Context)
Citation Context ...Nicolás [32] for homologically smooth non-positive dg algebras, by Assem, Souto SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES 3 Salorio and Trepode [5] and by Vitória =-=[46]-=-, who are focussing on piecewise hereditary algebras. An unbounded version of this bijection has been studied by Aihara and Iyama [1]. The bijection between simple-minded collections and bounded t-str... |