### Citations

509 |
Higher algebraic K-theory I
- Quillen
- 1973
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Citation Context ...injective in U⊥, so we have Ext1A(N, τU) = 0, hence HomA(L, τU) = 0 and thus L is in U . Remark 3.7. Since U is closed under extensions in modA, it has a natural structure of an exact category, see =-=[25, 23]-=-. Then Proposition 3.6 says that admissible epimorphisms (resp. admissible monomorphisms) in U are exactly epimorphisms (resp. monomorphisms) in modA between modules in U . The next result is the main... |

347 | Elements of the Representation Theory of Associative Algebras - Assem, Simson, et al. - 2006 |

215 | Higher algebraic K-theory I, in: Algebraic K-theory I: Higher K-theories, - Quillen - 1972 |

176 | Quivers with potentials and their representations II: applications to cluster algebras
- Derksen, Weyman, et al.
- 2010
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Citation Context ...e algebra by the quiver Q = 2 1 3 x1 x2 x3 with relations x1x2 = 0, x2x3 = 0 and x3x1 = 0, see Example 2.19. Thus C is isomorphic to the Jacobian algebra of the quiver with potential (Q, x1x2x3), see =-=[13]-=- for definitions. By Theorem 3.8 we have that ⊥(τU) ∩U ⊥ is equivalent to modC. The Auslander-Reiten quiver of modC is the following, where each C-module is represented by its radical filtration: modC... |

150 | Applications of contravariantly finite subcategories - Auslander, Reiten - 1991 |

142 | Cluster-tilted algebras are Gorenstein and stably - Keller, Reiten - 2007 |

129 | Tilting theory and cluster combinatorics
- Buan, Marsh, et al.
- 2006
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Citation Context ...e conclude this section with an example illustrating the bijections in Theorem 4.23. Example 4.26. Let C be the cluster category of type D4. Recall that C is a 2-Calabi-Yau triangulated category, see =-=[10]-=-. The Auslander-Reiten quiver of C is the following, where the dashed edges are to be identified to form a cylinder, see [28, 6.4]: 24 G. JASSO C : • • T4 U [1] • T3 • U • • T2 • • T1 • • • T4 U [1] W... |

118 | Mutation in triangulated categories and rigid Cohen-Macaulay modules - Iyama, Yoshino |

102 | Coxeter functors and Gabriel’s theorem
- Bernstein, Gelfand, et al.
- 1973
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Citation Context ...tudy τ -tilting theory comes from various sources, the most important one is mutation of tilting modules. Mutation of tilting modules has its origin in Bernstein-Gelfand-Ponomarev reflection functors =-=[8]-=-, which were later generalized by Auslander, Reiten and Platzeck with the introduction of APR-tilting modules [5], which are obtained by replacing a simple direct summand of the tilting A-module A. Mu... |

91 | Cluster-tilted algebras
- Buan, Marsh, et al.
(Show Context)
Citation Context ...tilting modules. For example, let A be the path algebra of the quiver 2 1 3 x y z subject to the relations xy = 0, yz = 0 and zx = 0. Thus A is a self-injective cluster-tilted algebra of type A3, see =-=[11, 27]-=-. It follows from [1, Thm. 4.1] that basic support τ -tilting A-modules correspond bijectively with basic cluster-tilting objects in the cluster category of type A3. Hence there are 14 support τ -tilt... |

88 | Rigid modules over preprojective algebras - Geiß, Leclerc, et al. |

84 |
Perpendicular categories with applications to representations and sheaves
- Geigle, Lenzing
- 1991
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Citation Context ...associated to U . Namely, (FacU,U⊥) and (⊥(τU), Sub(τU)). We have the following result about the category ⊥(τU) ∩U⊥, which is an analog of the perpendicular category associated with U in the sense of =-=[14]-=-, see Example 3.4. Theorem 1.4 (Theorem 3.8). With the hypotheses of Theorem 1.1, the functor HomA(TU ,−) : modA→ mod (EndA(TU )) induces an equivalence of exact categories F : ⊥(τU) ∩ U⊥ −→ modC. It ... |

82 | Almost split sequences in subcategories - Auslander, Smalø - 1981 |

64 |
Coxeter functors without diagrams
- Auslander, Platzeck, et al.
- 1979
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Citation Context ...n of tilting modules has its origin in Bernstein-Gelfand-Ponomarev reflection functors [8], which were later generalized by Auslander, Reiten and Platzeck with the introduction of APR-tilting modules =-=[5]-=-, which are obtained by replacing a simple direct summand of the tilting A-module A. Mutation of tilting modules was introduced in full generality by Riedtmann and Schofield in their combinatorial stu... |

48 |
A.: On a simplicial complex associated with tilting modules
- Riedtmann, Schofield
- 1991
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Citation Context ...y replacing a simple direct summand of the tilting A-module A. Mutation of tilting modules was introduced in full generality by Riedtmann and Schofield in their combinatorial study of tilting modules =-=[26]-=-. Also, Happel and Unger showed in [16] that tilting mutation is intimately related to the partial order of tilting modules induced by the inclusion of the associated torsion classes. We note that one... |

41 | Derived categories and their uses
- Keller
- 1996
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Citation Context ...injective in U⊥, so we have Ext1A(N, τU) = 0, hence HomA(L, τU) = 0 and thus L is in U . Remark 3.7. Since U is closed under extensions in modA, it has a natural structure of an exact category, see =-=[25, 23]-=-. Then Proposition 3.6 says that admissible epimorphisms (resp. admissible monomorphisms) in U are exactly epimorphisms (resp. monomorphisms) in modA between modules in U . The next result is the main... |

38 | Derived categories and their uses, Handbook of algebra - Keller - 1996 |

36 | A geometric model for cluster categories of typeDn - Schiffler |

32 | Noncrossing partitions and representations of quivers
- Ingalls, Thomas
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Citation Context ...|. More generally, we say that M is a support τ-tilting A-module if there exists an idempotent e ∈ A such that M is a τ -tilting (A/〈e〉)-module. Support tilting A-modules are defined analogously, see =-=[18]-=-. Remark 2.8. Note that the zero-module is a support τ -tilting module (take e = 1A in Definition 2.7). Thus every non-zero finite dimensional algebra A admits at least two support τ -tilting A-module... |

28 | Silting mutation in triangulated categories - Aihara, Iyama |

28 |
L.: On a partial order of tilting modules
- Happel, Unger
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Citation Context ...the tilting A-module A. Mutation of tilting modules was introduced in full generality by Riedtmann and Schofield in their combinatorial study of tilting modules [26]. Also, Happel and Unger showed in =-=[16]-=- that tilting mutation is intimately related to the partial order of tilting modules induced by the inclusion of the associated torsion classes. We note that one limitation of mutation of tilting modu... |

21 | Stable categories of higher preprojective algebras - Iyama, Oppermann - 2013 |

20 | τ -tilting theory - Adachi, Iyama, et al. |

10 | The self-injective cluster-tilted algebras
- Ringel
- 2008
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Citation Context ...tilting modules. For example, let A be the path algebra of the quiver 2 1 3 x y z subject to the relations xy = 0, yz = 0 and zx = 0. Thus A is a self-injective cluster-tilted algebra of type A3, see =-=[11, 27]-=-. It follows from [1, Thm. 4.1] that basic support τ -tilting A-modules correspond bijectively with basic cluster-tilting objects in the cluster category of type A3. Hence there are 14 support τ -tilt... |

9 | Torsion theory and tilting modules - Smalø |

8 | Ordered Exchange graphs
- Brüstle, Yang
- 2013
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Citation Context ...atible with 2-Calabi-Yau reduction. These results enhance our understanding of the relationship between silting objects, cluster-tilting objects and support τ -tilting modules. We refer the reader to =-=[9]-=- for an in-depth survey of the relations between these objects and several other important concepts in representation theory. Finally, let us fix our conventions and notations, which we kindly ask the... |

5 | On a partial order of tilting modules.” Algebras and Representation Theory 8 - Happel, Unger - 2005 |

4 | Tilting-connected symmetric algebras. Algebr Represent Theor - Aihara - 2012 |

3 |
τ -tilting theory, arXiv: 1210.1036
- Adachi, Iyama, et al.
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Citation Context ... reduction 16 4.1. Silting reduction 17 4.2. Calabi-Yau reduction 21 References 25 1. Introduction Let A be a finite dimensional algebra over a field. Recently, Adachi, Iyama and Reiten introduced in =-=[1]-=- a generalization of classical tilting theory, which they called τ-tilting theory. Motivation to study τ -tilting theory comes from various sources, the most important one is mutation of tilting modul... |

3 | Intermediate co-t-structures, two-term silting objects, τ -tilting modules, and torsion classes.” (2013): preprint arXiv:1311.4891 - Iyama, Jorgensen, et al. |

2 | General presentations of algebras. arXiv:0911.4913v1 - Derksen, Fei |

2 |
Intermediate co-t-structures
- Iyama, Jørgensen, et al.
(Show Context)
Citation Context ...on-zero direct summands in addS[1]. We are interested in the subset of 2S- silt T given by 2S- siltU T := {M ∈ 2S- silt T | U ∈ addS} . The following theorem is similar to [1, Thm. 3.2]. Theorem 4.7. =-=[19]-=- With the hypotheses of Setting 4.1, the functor (13) induces an order-preserving bijection (−) : 2S- silt T −→ sτ- tiltA which induces a bijection (−) : 2S- siltU T −→ sτ- tiltU A. Silting reduction ... |

2 | Volume Reduction
- Yang, Raine
- 2005
(Show Context)
Citation Context ...n C, see [1, Thm. 4.1]. Reduction techniques were established (in greater generality) in [22, Thm. 4.9] for cluster-tilting objects and for silting objects in [3, Thm. 2.37] for a special case and in =-=[21]-=- for the general case. The aim of this section is to show that these reductions are compatible with τ -tilting reduction as established in Section 3. Given two subcategories X and Y of a triangulated ... |

2 | Almost split sequences in subcategories.” Journal of Algebra 69 - Auslander, Smalø - 1981 |

2 | The self-injective cluster-tilted algebras.” Archiv der Mathematik 91, no. 3 (2008): 218–25. at Pennsylvania State U niversity on Septem ber 12, 2016 http://im rn.oxfordjournals.org/ D ow nloaded from 48 - Ringel |

1 |
Tilting-connected symmetric algebras. arXiv:1012.3265
- Aihara
(Show Context)
Citation Context ...h ℓ > 0. Thus, by Proposition 4.3 we have that (11) M 6 S if and only if HomT ( ⊥(T 60),M) = 0 or equivalently, since (⊥(T 60), T 60) is a torsion pair by Proposition 4.3, M ∈ T 60 = (S ∗ S[1]) ∗ T 60=-=[2]-=-. By a similar argument, we have that (12) S[1] 6M if and only if HomT (M, T 60[2]) = 0. Then it follows from (11) and (12) that S[1] 6M 6 S if and only if M ∈ S ∗ S[1]. We need the following result... |

1 | Links of faithful partial tilting modules.” Algebras and Representation Theory 13 - Happel, Unger - 2010 |

1 | E-mail address: jasso.ahuja.gustavo@b.mbox.nagoya-u.ac.jp - Math - 2012 |

1 | General presentations of algebras.” (2009): preprint arXiv:0911.4913 - Derksen, Fei |

1 | Silting and Calabi–Yau reductions in triangulated categories.” (2014): preprint arXiv:1408.2678 - Iyama, Yang |

1 | Semi-tilting complexes.” Israel Journal of Mathematics 194, no. 2 (2012): 871–893. at Pennsylvania State U niversity on Septem ber 12, 2016 http://im rn.oxfordjournals.org/ D ow nloaded from - Wei |