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## SILTING REDUCTION AND CALABI–YAU REDUCTION OF TRIANGULATED CATEGORIES

### Citations

500 |
Residues and Duality
- Hartshorne
- 1966
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Citation Context ...nd modA ∩ Z = modA ∩ ⊥T S<0 = CMA. It is enough to show Z ⊂ modA. Since A is Iwanaga–Gorenstein, we have a duality (−)∗ := RHomA(−, A) : T → T ′ (e.g. [36, Corollary 2.11]) as in the commutative case =-=[18]-=-. Since S<0 = (S ′>0) ∗ holds clearly, we have ⊥T S<0 = (S ′ >0 ⊥T ′ )∗ (3.4.1) = (D≤0(modAop))∗ ⊂ D≥0(modA). Therefore Z = (⊥T S<0) ∩ (S>0⊥T ) ⊂ D≤0(modA) ∩ D≥0(modA) = modA holds. Another applicat... |

201 |
Faisceaux pervers
- Beilinson, Bernstein, et al.
- 1983
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Citation Context ... → M → YM → X [1] in T with XM ∈ X and YM ∈ Y . It is elementary that the condition (T1) can be replaced by the following condition: (T1′) HomT (X ,Y) = 0, X = addX and Y = addY . A t-structure on T (=-=[7]-=-) is a pair (T ≤0, T ≥0) of full subcategories of T such that T ≥1 ⊂ T ≥0 and (T ≤0, T ≥1) is a torsion pair. Here for an integer n we denote T ≤n = T ≤0[−n] and T ≥n = T ≥0[−n]. In this case, the tri... |

186 | Triangulated categories - Neeman - 2001 |

160 | Maximal Cohen-Macaulay Modules and Tate-Cohomology over Gorenstein Rings
- Buchweitz
- 1986
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Citation Context ...that silting reduction can also be realised as a subfactor category of T (Theorems 3.1 and 3.7). We recover, as a special case of this realisation, the well-known triangle equivalence due to Buchweitz=-=[11]-=- CMA ≃ −→ Db(modA)/Kb(projA) for an Iwanaga–Gorenstein ring A (Example 3.11). Moreover, there is a natural bijection between silting subcategories of T containing P and silting subcategories of U (The... |

151 | DG quotients of DG categories
- Drinfeld
- 2004
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Citation Context ...with H i(A) = 0 (i > 0) together with a triangle equivalence D(A) → DM(M) which restricts to a triangle equivalence Dfd(A) → KbM(M). Sketch. We sketch the construction of A. We take the dg quotient N =-=[13]-=- of M by the category projE of projective objects of E . By construction, the morphism complexes of N have cohomologies concentrated in non-positive degrees. Further, objects of projE are homotopic to... |

150 | Calabi-Yau algebras
- Ginzburg
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Citation Context ...ng reduction of T with respect to P to the Calabi–Yau reduction of C with respect to π(P ). Remark 4.14. Let (Q,W ) be a quiver with potential and Γ = Γ(Q,W ) be its complete Ginzburg dg algebra, see =-=[12, 15, 33]-=-. Assume that H0(Γ) is finite-dimensional. Then the triple (per(Γ),Dfd(Γ),Γ) is a 3-Calabi–Yau triple. The triangle quotient C(Q,W ) = per(Γ)/Dfd(Γ) 32 OSAMU IYAMA AND DONG YANG is called the cluster ... |

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

118 | Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l’institut Fourier
- Amiot
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Citation Context ...: (i) HomT (X [2], U [d+ 1]) ≃ HomT (W,U [1]); (ii) HomT (W,U [1]) ≃ DHomΛ(F (U), F (L)); (iii) DHomΛ(F (U), F (L)) ≃ HomT (L, U [d+ 1]). By the triangle (A.0.2), we have an exact sequnece HomT (X≥3−d=-=[3]-=-,−) // HomT (W [d],−) // HomT (X [2],−) // HomT (X≥3−d[2],−). Evaluated at U [d + 1], this gives the functorial isomorphism (i), since HomT (X≥3−d[≤ 3], U [d + 1]) = 0. SILTING REDUCTION AND CY REDUCT... |

101 |
Relative Homological Algebra, in: de Gruyter Expositions in
- Enochs, Jenda
- 2000
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Citation Context ...have the assertion. 3.4. A theorem of Buchweitz. Recall that a noetherian ring A is called Iwanaga– Gorenstein if A has finite injective dimension as an A-module and also as an Aop-module (see e.g. =-=[14]-=-). In this case, we define the category of Cohen–Macaulay A-modules (also often called modules of Gorenstein dimension zero, Gorenstein projective modules, or totally reflexive modules) by CMA := {X ∈... |

85 |
Deriving DG categories, Ann. Sci. École Norm
- Keller
- 1994
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Citation Context ...MX) // H0(X) // H0(NX [1]), X 0 SILTING REDUCTION AND CY REDUCTION 13 showing thatM0 is a projective generator ofH, sinceH0(MX) ∈ addH0(M) = addM0. 2.5. Derived categories of dg algebras. We follow =-=[24, 26]-=-. Let k be a field and A be a dg (k-)algebra, that is, a graded algebra endowed with a compatible structure of a complex. A (right) dg A-module is a (right) graded A-module endowed with a compatible s... |

84 | Derived equivalences from mutations of quivers with potential
- Keller, Yang
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Citation Context ...ng reduction of T with respect to P to the Calabi–Yau reduction of C with respect to π(P ). Remark 4.14. Let (Q,W ) be a quiver with potential and Γ = Γ(Q,W ) be its complete Ginzburg dg algebra, see =-=[12, 15, 33]-=-. Assume that H0(Γ) is finite-dimensional. Then the triple (per(Γ),Dfd(Γ),Γ) is a 3-Calabi–Yau triple. The triangle quotient C(Q,W ) = per(Γ)/Dfd(Γ) 32 OSAMU IYAMA AND DONG YANG is called the cluster ... |

82 |
Almost split sequences in subcategories
- Auslander, Smalø
- 1981
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Citation Context ...a right S-approximation. Dually, we define left S-approximations and covariantly finite subcategories. We say that S is functorially finite if it is both contravariantly finite and covariantly finite =-=[6]-=-. Denote by addT S (or simply addS) the smallest full subcategory of T which contains S and which is closed under taking isomorphisms, finite direct sums and direct summands. Denote by [S] the ideal o... |

56 | structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general
- Bondarko, Weight
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Citation Context ...unctors. The heart H := T ≤0 ∩T ≥0 is always an abelian category. The t-structure (T ≤0, T ≥0) is said to be bounded if⋃ n∈Z T ≤n = T = ⋃ n∈Z T ≥n, equivalently, if T = thickH. A co-t-structure on T (=-=[39, 9]-=-) is a pair (T≥0, T≤0) of full subcategories of T such that T≥1 ⊂ T≥0 and (T≥1, T≤0) is a torsion pair. Here for an integer n we denote T≥n = T≥0[−n] and T≤n = T≤0[−n]. It is easy to see that the co-h... |

55 | Aisles in derived categories
- Keller, Vossieck
- 1988
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Citation Context ... using Lemma 2.1. 2.4. From silting objects to t-structures. Let T be a triangulated category. In this section, we show that under certain conditions silting objects yield t-structures. We refer to =-=[32, 34, 28, 10, 4, 40]-=- for more on this subject. For a silting subcategory P in T , we consider subcategories of T : P[<0]⊥T = {X ∈ T | HomT (P[<0], X) = 0}, P[>0]⊥T = {X ∈ T | HomT (P[>0], X) = 0}. We adopt the notation i... |

38 |
Andrei Zelevinsky, Quivers with potentials and their representations
- Derksen, Weyman
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Citation Context ...ng reduction of T with respect to P to the Calabi–Yau reduction of C with respect to π(P ). Remark 4.14. Let (Q,W ) be a quiver with potential and Γ = Γ(Q,W ) be its complete Ginzburg dg algebra, see =-=[12, 15, 33]-=-. Assume that H0(Γ) is finite-dimensional. Then the triple (per(Γ),Dfd(Γ),Γ) is a 3-Calabi–Yau triple. The triangle quotient C(Q,W ) = per(Γ)/Dfd(Γ) 32 OSAMU IYAMA AND DONG YANG is called the cluster ... |

33 | Compact corigid objects in triangulated categories and co-t-structures
- Pauksztello
- 2008
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Citation Context ...unctors. The heart H := T ≤0 ∩T ≥0 is always an abelian category. The t-structure (T ≤0, T ≥0) is said to be bounded if⋃ n∈Z T ≤n = T = ⋃ n∈Z T ≥n, equivalently, if T = thickH. A co-t-structure on T (=-=[39, 9]-=-) is a pair (T≥0, T≤0) of full subcategories of T such that T≥1 ⊂ T≥0 and (T≥1, T≤0) is a torsion pair. Here for an integer n we denote T≥n = T≥0[−n] and T≤n = T≤0[−n]. It is easy to see that the co-h... |

30 |
On a generalized version of the Nakayama conjecture,
- Auslander, Reiten
- 1975
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Citation Context ...irwise non-isomorphic simple A-modules. Motivated by Tachikawa’s study [41] on the famous Nakayama conjecture, Auslander and Reiten proposed the following conjecture: The Auslander–Reiten Conjecture (=-=[5]-=-) If X ∈ modA satisfies ExtiA(X,X ⊕ A) = 0 for all i > 0, then X is a projective A-module. Now we pose the following conjectures in the context of silting theory. 22 OSAMU IYAMA AND DONG YANG Conjectu... |

17 |
Reduction techniques for homological conjectures,
- Happel
- 1993
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Citation Context ...bcategory T of Db(modA) containing Kb(projA) such that the Grothendieck group K0(T ) is a free abelian group with rank strictly bigger than n. We have the following observation (see also Section 4 of =-=[17]-=-). Theorem 3.15. Conjecture 3.14⇒ Conjecture 3.13⇒ the Auslander–Reiten Conjecture. Proof. To prove the first implication, assume that a non-projective A-module X satisfies ExtiA(X,X ⊕ A) = 0 for all ... |

17 |
differential graded categories
- On
(Show Context)
Citation Context ...MX) // H0(X) // H0(NX [1]), X 0 SILTING REDUCTION AND CY REDUCTION 13 showing thatM0 is a projective generator ofH, sinceH0(MX) ∈ addH0(M) = addM0. 2.5. Derived categories of dg algebras. We follow =-=[24, 26]-=-. Let k be a field and A be a dg (k-)algebra, that is, a graded algebra endowed with a compatible structure of a complex. A (right) dg A-module is a (right) graded A-module endowed with a compatible s... |

14 |
Weight structures and simple dg modules for positive dg algebras
- Keller, Nicolás
(Show Context)
Citation Context ...t any silting subcategory gives a co-t-structure on T . The following proposition is well-known, which was proved as [35, Theorem 5.5], see also [2, Proposition 2.22], [9, proof of Theorem 4.3.2] and =-=[28]-=-. Proposition 2.8. Let P be a silting subcategory of T with P = addP. (a) Then (T≥0, T≤0) is a bounded co-t-structure on T , where T≥0 := ⋃ i≥0 P[−i] ∗ · · · ∗ P[−1] ∗ P and T≤0 := ⋃ i≥0 P ∗ P[1] ∗ · ... |

9 | Grothendieck group and generalized mutation rule for 2-Calabi-Yau triangulated categories - Palu |

9 |
Quasi-Frobenius rings and generalizations. QF-3 and QF-1 rings. Notes by Claus Michael Ringel
- Tachikawa
- 1973
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Citation Context ...of Tachikawa and that of Auslander–Reiten. Let k be a field and A be a finite-dimensional k-algebra and let n be the number of pairwise non-isomorphic simple A-modules. Motivated by Tachikawa’s study =-=[41]-=- on the famous Nakayama conjecture, Auslander and Reiten proposed the following conjecture: The Auslander–Reiten Conjecture ([5]) If X ∈ modA satisfies ExtiA(X,X ⊕ A) = 0 for all i > 0, then X is a pr... |

8 | Ordered Exchange graphs
- Brüstle, Yang
- 2013
(Show Context)
Citation Context ... using Lemma 2.1. 2.4. From silting objects to t-structures. Let T be a triangulated category. In this section, we show that under certain conditions silting objects yield t-structures. We refer to =-=[32, 34, 28, 10, 4, 40]-=- for more on this subject. For a silting subcategory P in T , we consider subcategories of T : P[<0]⊥T = {X ∈ T | HomT (P[<0], X) = 0}, P[>0]⊥T = {X ∈ T | HomT (P[>0], X) = 0}. We adopt the notation i... |

8 | Cluster tilting objects in generalized higher cluster categories, arXiv:1005.3564
- Guo
(Show Context)
Citation Context ...an drop this assumption thanks to the realisation of U as a subfactor category of T . The second main result of this paper is to compare these two reduction processes using the Amiot–Guo construction =-=[3, 16]-=- (based on Keller’s work [25, 27]). Let T be a triangulated category, M ∈ T an object and T fd ⊂ T a triangulated subcategory such that (T , T fd,M) is a (d+1)-Calabi–Yau triple (see Section 4.1 for t... |

8 | Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras
- Koenig, Yang
(Show Context)
Citation Context ... using Lemma 2.1. 2.4. From silting objects to t-structures. Let T be a triangulated category. In this section, we show that under certain conditions silting objects yield t-structures. We refer to =-=[32, 34, 28, 10, 4, 40]-=- for more on this subject. For a silting subcategory P in T , we consider subcategories of T : P[<0]⊥T = {X ∈ T | HomT (P[<0], X) = 0}, P[>0]⊥T = {X ∈ T | HomT (P[>0], X) = 0}. We adopt the notation i... |

8 | Duality for derived categories and cotilting bimodules - Miyachi - 1996 |

6 |
125–180, With an appendix by Michel Van den
- Math
- 2011
(Show Context)
Citation Context ...o the realisation of U as a subfactor category of T . The second main result of this paper is to compare these two reduction processes using the Amiot–Guo construction [3, 16] (based on Keller’s work =-=[25, 27]-=-). Let T be a triangulated category, M ∈ T an object and T fd ⊂ T a triangulated subcategory such that (T , T fd,M) is a (d+1)-Calabi–Yau triple (see Section 4.1 for the precise definition). Let P be ... |

4 | Relative singularity categories I: Auslander resolutions - Kalck, Yang |

3 | María José Souto Salorio, Auslander-Buchweitz context and co-t-structures - Mendoza, Sáenz, et al. |

2 | Vitória: Silting modules
- Angeleri-Hügel, Marks, et al.
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Citation Context |

1 |
Osamu Iyama and Idun Reiten, τ-tilting theory
- Adachi
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Citation Context ...subcategories ⊥T S := {X ∈ T | HomT (X,S) = 0}, S⊥T := {X ∈ T | HomT (S, X) = 0}. When it does not cause confusion, we will simply write ⊥S and S⊥. Let T be a triangulated category (we will denote by =-=[1]-=- the shift functor of any triangulated category unless otherwise stated). For two objects X and Y of T and an integer n, by HomT (X, Y [>n]) = 0 (respectively, HomT (X, Y [≥n]) = 0, HomT (X, Y [<n]) =... |

1 | Yoshiaki Kato and Jun-ichi Miyachi, On t -structures and torsion theories induced by compact objects - Hoshino |

1 | Jørgensen and Dong Yang, Intermediate co-t-structures, two-term silting objects, τ-tilting modules, and torsion classes - Iyama, Peter |

1 |
hearts and cluster tilting objects
- Cluster
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Citation Context ...e the two left-going maps are bijections due to properties of silting reduction and Calabi–Yau reduction and the two right-going maps are surjections when d = 1 and when d = 2 (due to Keller–Nicolás =-=[29]-=- in the algebraic setting) (Corollary 4.10). Acknowledgement: The first-named author acknowledges financial support from JSPS Grant-in-Aid for Scientific Research (B) 24340004, (C) 23540045 and (S) 22... |