## On triangulated orbit categories (2005)

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Venue: | DOC. MATH |

Citations: | 80 - 5 self |

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@ARTICLE{Keller05ontriangulated,

author = {Bernhard Keller},

title = {On triangulated orbit categories},

journal = {DOC. MATH},

year = {2005},

pages = {551--581}

}

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### Abstract

### Citations

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Model categories, Mathematical surveys and monographs
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- 1998
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Citation Context ...all dg categories is the localization Ho(dgcat) of dgcat with respect to the class of quasi-equivalences. According to [56], the category dgcat admits a structure of Quillen model category (cf. [22], =-=[31]-=-) whose weak equivalences are the quasi-equivalences. This implies in particular that for A, B ∈ dgcat, the morphisms from A to B in the localization Ho(dgcat) form a set. 9.2. The bimodule bicategory... |

123 |
Homotopy Theories and Model Categories, Handbook of Algebraic Topology
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- 1995
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Citation Context ... of small dg categories is the localization Ho(dgcat) of dgcat with respect to the class of quasi-equivalences. According to [54], the category dgcat admits a structure of Quillen model category (cf. =-=[22]-=-, [31]) whose weak equivalences are the quasi-equivalences. This implies in particular that for A, B ∈ dgcat, the morphisms from A to B in the localization Ho(dgcat) form a set.� �� ON TRIANGULATED O... |

97 |
Enhanced triangulated categories
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- 1991
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Citation Context ...clusion of the full subcategory of (small) exact dg categories into Ho(dgcat) admits a left adjoint, namely the functor A ↦→ pretr(A) which maps a dg category to its ‘pretriangulated hull’ defined in =-=[14]-=-, cf. also [33, 2.2]. More precisely, the adjunction morphism A → pretr(A) induces an equivalence of categories rep (pretr(A), B) → rep(A, B) for each exact dg category B, cf. [55]. The bicategory enh... |

89 |
Graded algebras of global dimension 3
- Artin, Schelter
- 1987
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Citation Context ...ove this for M = B and X = k. Then one checks it using the fact that RHomB(k, B) ∼ → k[−n]. These arguments still work for certain non-commutative algebras B: If B is an ArtinSchelter regular algebra =-=[2]-=- [1] of global dimension 3 and type A and T the localizing subcategory of the derived category D(B) of non graded B-modules generated by the trivial module, then T c is Calabi-Yau and one even has the... |

83 | Generators and representability of functors in commutative and noncommutative geometry, Mosc
- Bondal, Bergh
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Citation Context ...X) ∼ → HomT (ν ′ , ?) , X ∈ T . The category T has Serre duality if it has both a left and a right Serre functor, or equivalently, if it has a onesided Serre functor which is an equivalence, cf. [47] =-=[13]-=-. The following lemma is used in [9]. Lemma. Suppose that T has a left Serre functor ν ′ . Let U ⊂ T be a thick triangulated subcategory and L : T → T /U the localization functor. a) If L admits a rig... |

80 | Quivers with relations arising from clusters (An case
- Caldero, Chapoton, et al.
- 2006
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Citation Context ... This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also =-=[16]-=-) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. T... |

80 | Topological conformal field theories and Calabi-Yau categories
- Costello
(Show Context)
Citation Context ...ory of a category of modules or sheaves), then it often comes from a Calabi-Yau A∞category. These are of considerable interest in mathematical physics, since, as Kontsevich shows [38], [37], cf. also =-=[18]-=-, a topological quantum field theory is associated with each Calabi-Yau A∞-category satisfying some additional assumptions 1 . 8.3. Examples. (1) If A is a finite-dimensional k-algebra, then the homot... |

78 |
Derived equivalences as derived functors
- Rickard
- 1991
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Citation Context ...d, and T is the bounded derived category D b (mod A) of the category of finite-dimensional (right) modules mod A over a finite-dimensional k-algebra A. Assume that F : T → T is a standard equivalence =-=[48]-=-, i.e. F is isomorphic to the derived tensor product ? L ⊗A X : D b (mod A) → D b (mod A) for some complex X of A-A-bimodules. All autoequivalences with an ‘algebraic construction’ are of this form, c... |

77 | Quotient of dg-categories
- Drinfeld
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Citation Context ...ditary algebras. We give a first, abstract, construction of the triangulated hull of an orbit category in section 5. This construction is based on the formalism of dg categories as developped in [32] =-=[21]-=- [57]. Using the natural t-structure on the derived category of a hereditary category we prove the main theorem in section 6. We give a more concrete construction of the triangulated hull of the orbit... |

71 |
Some algebras associated to automorphisms of elliptic curves
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Citation Context ...nsional A-modules such that M p = 0 for all large |p| and morphisms are obtained from morphisms of complexes by formally inverting all quasi-isomorphisms. The suspension functor S is defined by SM = M=-=[1]-=-, where M[1] p = M p+1 and d M[1] = −dM, and the triangles are constructed from short exact sequences of complexes. Suppose that A is hereditary. Then the orbit category D b (A)/S 2 , first introduced... |

70 |
Deriving DG categories, Ann
- Keller
- 1994
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Citation Context ... hereditary algebras. We give a first, abstract, construction of the triangulated hull of an orbit category in section 5. This construction is based on the formalism of dg categories as developped in =-=[32]-=- [21] [57]. Using the natural t-structure on the derived category of a hereditary category we prove the main theorem in section 6. We give a more concrete construction of the triangulated hull of the ... |

64 |
Triangulated categories, Annals of Mathematics Studies
- Neeman
- 2001
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Citation Context ...angulated categories turn out to be ‘self-reproducing’: Let T be a triangulated category. Then the category mod T of finitely generated functors from T op to Mod k is abelian and Frobenius, cf. [24], =-=[42]-=-. If we denote by Σ the exact functor mod T → mod T which takes Hom(?, X) to Hom(?, SX), then it is not hard to show [24] [25] that we have Σ ∼ → S 3 as triangle functors modT → modT . One deduces the... |

62 |
On the derived category of a finite-dimensional algebra
- Happel
- 1987
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Citation Context ...M p+1 and d M[1] = −dM, and the triangles are constructed from short exact sequences of complexes. Suppose that A is hereditary. Then the orbit category D b (A)/S 2 , first introduced by D. Happel in =-=[27]-=-, is triangulated. This result is due to Peng and Xiao [43], who show that the orbit category is equivalent to the homotopy category of the category of 2-periodic complexes of projective A-modules. On... |

60 | The homotopy theory of dg-categories and derived Morita theory
- Toën
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Citation Context ...y algebras. We give a first, abstract, construction of the triangulated hull of an orbit category in section 5. This construction is based on the formalism of dg categories as developped in [32] [21] =-=[57]-=-. Using the natural t-structure on the derived category of a hereditary category we prove the main theorem in section 6. We give a more concrete construction of the triangulated hull of the orbit cate... |

46 |
Sous les catégories dérivées
- Keller, Vossieck
- 1987
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Citation Context ...onal T A-module admits a grading. Proof. We have a natural functor mod A → grmod T A given by viewing an A-module as a graded T A-module concentrated in degree 0. As shown by D. Happel [27], cf. also =-=[34]-=-, this functor extends to a triangle equivalence Φ from D b (A) to the stable category grmod T A, obtained from grmod T A by killing all morphisms factoring through projective-injectives. We would lik... |

43 | Cluster mutation via quiver representations
- Buan, Marsh, et al.
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Citation Context ...ategory T has Serre duality if it has both a left and a right Serre functor, or equivalently, if it has a onesided Serre functor which is an equivalence, cf. [44] [13]. The following lemma is used in =-=[9]-=-. Lemma. Suppose that T has a left Serre functor ν ′ . Let U ⊂ T be a thick triangulated subcategory and L : T → T /U the localization functor. a) If L admits a right adjoint R, then Lν ′ R is a left ... |

34 | The stable derived category of a Noetherian scheme
- Krause
(Show Context)
Citation Context ...ne degree. In the cases considered by Buan et al. [8] and Caldero-Chapoton-Schiffler [16], this also yields an interesting new description of the orbit category itself in terms of the stable category =-=[40]-=- of a differential graded algebra. In section 8, we observe that triangulated orbit categories provide easily constructed examples of triangulated categories with the Calabi-Yau property. Finally, in ... |

33 |
Stable homotopy
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- 1966
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Citation Context ...au triangulated categories turn out to be ‘self-reproducing’: Let T be a triangulated category. Then the category mod T of finitely generated functors from T op to Mod k is abelian and Frobenius, cf. =-=[24]-=-, [42]. If we denote by Σ the exact functor mod T → mod T which takes Hom(?, X) to Hom(?, SX), then it is not hard to show [24] [25] that we have Σ ∼ → S 3 as triangle functors modT → modT . One deduc... |

31 |
Dimensions of triangulated categories
- Rouquier
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Citation Context ... sometimes says that T is Calabi-Yau of fractional dimension d/e. Note that d ∈ Z is only determined up to a multiple of the order of S. It would be interesting to link the CY-dimension to Rouquier’s =-=[51]-=- notion of dimension of a triangulated category. The terminology has its origin in the following example: Let X be a smooth projective variety of dimension d and let ωX = ΛdT ∗ X be the canonical bund... |

30 |
Analyse et topologie sur les espaces singuliers
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- 1982
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Citation Context ....3, the morphism g is invertible and we are done. 6.1. Extension of t-structures to unbounded categories. Let T be a triangulated category and U an aisle [35] in T . Denote the associated t-structure =-=[10]-=- by (U≤0, U≥0), its heart by U0, its homology functors by Hn U : T → U0 and its truncation functors by τ≤n and τ>n. Suppose that U is dominant, i.e. the following two conditions hold: 1) a morphism s ... |

30 |
Floer cohomology, and braid group actions
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Citation Context ...ing ungraded algebra of B is the quadratic dual of the preprojective algebra associated with An, cf. [15]. The algebra B viewed as a differential graded algebra was investigated by Khovanov-Seidel in =-=[36]-=-. Here the authors show that D b (B) admits a canonical action by the braid group on n + 1 strings, a result which was obtained independently in a similar context by Zimmermann-Rouquier [52]. The cano... |

30 |
Une structure de catégorie de modèles de Quillen sur la catégorie des dgcatégories, Comptes Rendus de l’Acadmie de Sciences de Paris 340
- Tabuada
- 2005
(Show Context)
Citation Context ... denote by dgcat the category of small dg categories. The homotopy category of small dg categories is the localization Ho(dgcat) of dgcat with respect to the class of quasi-equivalences. According to =-=[54]-=-, the category dgcat admits a structure of Quillen model category (cf. [22], [31]) whose weak equivalences are the quasi-equivalences. This implies in particular that for A, B ∈ dgcat, the morphisms f... |

21 |
Skew category, Galois covering and smash product of a kcategory
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Citation Context ...y definition, the orbit category T /F = T /F Z has the same objects as T and its morphisms from X to Y are in bijection with ⊕ HomT (X, F n Y ). n∈Z The composition is defined in the natural way (cf. =-=[17]-=-, where this category is called the skew category). The canonical projection functor π : T → T /F is endowed with a natural isomorphism π ◦ F ∼ → π and 2-universal among such functors. Clearly T /F is... |

21 |
Ueberlagerungen und zurück
- Riedtmann, Algebren
- 1980
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Citation Context ...f a quiver of type A2 under the Nakayama autoequivalence ν. Thus, we obtain examples of triangulated categories whose Auslander-Reiten quiver contains a loop. It has been known since Riedtmann’s work =-=[49]-=- that this cannot occur in the stable category (cf. below) of a selfinjective finite-dimensional algebra. It may therefore seem surprising, cf. [58], that loops do occur in this more general context. ... |

20 |
den Bergh: Noetherian hereditary abelian categories satisfying Serre duality
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Citation Context ...Yau property 8.1. Serre functors and localizations. Let k be a field and T a k-linear triangulated category with finite-dimensional Hom-spaces. We denote the suspension functor of T by S. Recall from =-=[44]-=- that a right Serre functor for T is the datum of a triangle functor ν : T → T together with bifunctor isomorphisms D HomT (X,?) ∼ → HomT (?,νX) , X ∈ T , where D = Homk(?,k). If ν exists, it is uniqu... |

18 | Realizability of modules over Tate cohomology
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Citation Context ... (B)/ per(B) where B = A ⊕ (DA)[−2]. This equivalence yields in fact more than just a triangulated structure: it shows that proj C is endowed with a canonical Hochschild 3-cocycle m3, cf. for example =-=[11]-=-. It would be interesting to identify this cocycle in the description given in [23]. 7.4. Projectives over Λ(Ln). The category of projective modules over the algebra k[ε]/(ε 2 ) of dual numbers is tri... |

17 |
On Hochschild cohomology of preprojective algebras
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Citation Context ... a triangulated structure: it shows that proj C is endowed with a canonical Hochschild 3-cocycle m3, cf. for example [11]. It would be interesting to identify this cocycle in the description given in =-=[23]-=-. 7.4. Projectives over Λ(Ln). The category of projective modules over the algebra k[ε]/(ε 2 ) of dual numbers is triangulated. Indeed, it is equivalent to the orbit category of the derived category o... |

17 | Picard groups for derived module categories
- Rouquier, Zimmermann
- 2003
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Citation Context ...Seidel in [36]. Here the authors show that D b (B) admits a canonical action by the braid group on n + 1 strings, a result which was obtained independently in a similar context by Zimmermann-Rouquier =-=[52]-=-. The canonical generators� �� �� � 10 BERNHARD KELLER of the braid group act by triangle functors Ti endowed with morphisms φi : Ti → 1. The cone on each φi belongs to per(B) and per(B) is in fact e... |

17 | Idun Reiten, and Gordana Todorov. Tilting theory and cluster combinatorics - Buan, Marsh, et al. |

15 |
The preprojective algebra of a quiver, in: Algebras and modules
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Citation Context ...projective algebra. Let A be the path algebra of a Dynkin quiver, i.e. a quiver whose underlying graph is a Dynkin diagram of type A, D or E. Let C be the associated preprojective algebra [26], [20], =-=[50]-=-. In Proposition 3.3 of [7], Auslander-Reiten show that the category of projective modules over C is equivalent to the stable category of maximal Cohen-Macaulay modules over a representation-finite is... |

12 |
A∞-algebras for ring theorists
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Citation Context ... graded B-modules generated by the trivial module, then T c is Calabi-Yau and one even has the isomorphism (∗) for each perfect complex of B-modules M and each X ∈ T c , cf. for example section 12 of =-=[41]-=-. 8.4. Orbit categories with the Calabi-Yau property. The main theorem yields the following Corollary. If d ∈ Z and Q is a quiver whose underlying graph is Dynkin of type A, D or E, then T = D b (kQ)/... |

11 |
Two-dimensional tame and maximal orders of finite representation type
- Reiten, Bergh
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Citation Context ...flexive modules over certain non commutative generalizations of local rings of rational double points, as shown by Auslander-Reiten in [6]. These were completely classified by Reiten-Van den Bergh in =-=[46]-=-. In particular, the example of the dual numbers and its generalization below are among the cases covered by [46]. The example of the dual numbers generalizes as follows: Let n ≥ 1 be an integer. Foll... |

10 |
Periodic algebras which are almost Koszul
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Citation Context ... if A is the quiver algebra of an alternating quiver whose underlying graph is An, then the underlying ungraded algebra of B is the quadratic dual of the preprojective algebra associated with An, cf. =-=[15]-=-. The algebra B viewed as a differential graded algebra was investigated by Khovanov-Seidel in [36]. Here the authors show that D b (B) admits a canonical action by the braid group on n + 1 strings, a... |

10 |
Model algebras and representations of graphs, Funktsional. Anal. i Prilozhen
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Citation Context ...over the preprojective algebra. Let A be the path algebra of a Dynkin quiver, i.e. a quiver whose underlying graph is a Dynkin diagram of type A, D or E. Let C be the associated preprojective algebra =-=[26]-=-, [20], [50]. In Proposition 3.3 of [7], Auslander-Reiten show that the category of projective modules over C is equivalent to the stable category of maximal Cohen-Macaulay modules over a representati... |

10 |
diagrams and low-dimensional topology, First European
- Feynman
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Citation Context ...le, the derived category of a category of modules or sheaves), then it often comes from a Calabi-Yau A∞category. These are of considerable interest in mathematical physics, since, as Kontsevich shows =-=[38]-=-, [37], cf. also [18], a topological quantum field theory is associated with each Calabi-Yau A∞-category satisfying some additional assumptions 1 . 8.3. Examples. (1) If A is a finite-dimensional k-al... |

7 |
Stable equivalence of Artin algebras
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Citation Context ...S 3 as triangle functors modT → modT . One deduces the following lemma, which is a variant of a result which Auslander-Reiten [7] obtained using dualizing R-varieties [4] and their functor categories =-=[3]-=-, cf. also [5] [45]. A similar result is due to Geiss [25]. Lemma. If T is Calabi-Yau of CY-dimension d, then the stable category mod T is CalabiYau of CY-dimension 3d − 1. Moreover, if the suspension... |

7 | On Gorenstein algebras, In: Representation theory of finite groups and finitedimensional algebras (Proc. Conf. at - Happel - 1991 |

6 |
Invariants additifs de DG-catégories
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Citation Context ...ed hull’ defined in [14], cf. also [33, 2.2]. More precisely, the adjunction morphism A → pretr(A) induces an equivalence of categories rep (pretr(A), B) → rep(A, B) for each exact dg category B, cf. =-=[55]-=-. The bicategory enh of enhanced [14] triangulated categories, cf. [32] [21], has as objects all small exact dg categories; the morphism category between two objects A, B is rep(A, B); the composition... |

3 |
The preprojective algebra of a modulated graph, Representation theory
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Citation Context ...he preprojective algebra. Let A be the path algebra of a Dynkin quiver, i.e. a quiver whose underlying graph is a Dynkin diagram of type A, D or E. Let C be the associated preprojective algebra [26], =-=[20]-=-, [50]. In Proposition 3.3 of [7], Auslander-Reiten show that the category of projective modules over C is equivalent to the stable category of maximal Cohen-Macaulay modules over a representation-fin... |

3 |
Binary polyhedral groups and Euclidean diagrams
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Citation Context ...of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel =-=[29]-=-, whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7]. 1. Introduction Let T be an additive category and F : T → T an automorphism (a standard construction allows one to r... |

3 |
additifs de dg-catégories, Int
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Citation Context ...ed hull’ defined in [14], cf. also [33, 2.2]. More precisely, the adjunction morphism A → pretr(A) induces an equivalence of categories rep (pretr(A), B) → rep(A, B) for each exact dg category B, cf. =-=[55]-=-. The bicategory enh of enhanced [14] triangulated categories, cf. [32] [21], has as objects all small exact dg categories; the morphism category between two objects A, B is rep(A, B); the composition... |

3 |
Relations for the Grothendieck groups of triangulated categories
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Citation Context ...ns a loop. It has been known since Riedtmann’s work [49] that this cannot occur in the stable category (cf. below) of a selfinjective finite-dimensional algebra. It may therefore seem surprising, cf. =-=[58]-=-, that loops do occur in this more general context. However, loops already do occur in stable categories of finitely generated reflexive modules over certain non commutative generalizations of local r... |

2 |
Andrzej Skowroński, Deformed preprojective algebras of generalized Dynkin type
- Bia̷lkowski, Erdmann
- 2004
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Citation Context ...lso have an isomorphism DA ⊗A DA ∼ → A in the stable category of bimodules, by the remark following definition 4.6 in [15]. For the algebra Λ(Ln), the analogous result follows from Proposition 2.3 of =-=[12]-=-.ON TRIANGULATED ORBIT CATEGORIES 15 9. Universal properties 9.1. The homotopy category of small dg categories. Let k be a field. A differential graded (=dg) k-module is a Z-graded vector space V = ⊕... |

2 |
Topological field theory for triangulated categories. Talk at the conference on K-theory and Noncommutative Geometry, Institut Henri Poincaré
- Kontsevich
- 2004
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Citation Context ...e derived category of a category of modules or sheaves), then it often comes from a Calabi-Yau A∞category. These are of considerable interest in mathematical physics, since, as Kontsevich shows [38], =-=[37]-=-, cf. also [18], a topological quantum field theory is associated with each Calabi-Yau A∞-category satisfying some additional assumptions 1 . 8.3. Examples. (1) If A is a finite-dimensional k-algebra,... |

2 |
Finite-dimensional algebras and singularities, Singularities, representation of algebras, and vector bundles
- Reiten
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Citation Context ...tegory of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], =-=[44]-=-, [7]. 1. Introduction Let T be an additive category and F : T → T an automorphism (a standard construction allows one to replace a category with autoequivalence by a category with automorphism). Let ... |

2 |
structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories
- Une
(Show Context)
Citation Context ... denote by dgcat the category of small dg categories. The homotopy category of small dg categories is the localization Ho(dgcat) of dgcat with respect to the class of quasi-equivalences. According to =-=[56]-=-, the category dgcat admits a structure of Quillen model category (cf. [22], [31]) whose weak equivalences are the quasi-equivalences. This implies in particular that for A, B ∈ dgcat, the morphisms f... |

1 |
equivalence of dualizing R-varieties
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- 1974
(Show Context)
Citation Context ...how [24] [25] that we have Σ ∼ → S 3 as triangle functors modT → modT . One deduces the following lemma, which is a variant of a result which Auslander-Reiten [7] obtained using dualizing R-varieties =-=[4]-=- and their functor categories [3], cf. also [5] [45]. A similar result is due to Geiss [25]. Lemma. If T is Calabi-Yau of CY-dimension d, then the stable category mod T is CalabiYau of CY-dimension 3d... |

1 |
equivalence of dualizing R-varieties. II. Hereditary dualizing R-varieties
- Stable
- 1975
(Show Context)
Citation Context ...e functors modT → modT . One deduces the following lemma, which is a variant of a result which Auslander-Reiten [7] obtained using dualizing R-varieties [4] and their functor categories [3], cf. also =-=[5]-=- [45]. A similar result is due to Geiss [25]. Lemma. If T is Calabi-Yau of CY-dimension d, then the stable category mod T is CalabiYau of CY-dimension 3d − 1. Moreover, if the suspension of T is of or... |

1 |
split sequences for rational double points
- Almost
- 1987
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Citation Context ...he category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work =-=[6]-=-, [44], [7]. 1. Introduction Let T be an additive category and F : T → T an automorphism (a standard construction allows one to replace a category with autoequivalence by a category with automorphism)... |

1 |
Cohomologie à supports propres, pp. vi+640
- Deligne
- 1973
(Show Context)
Citation Context ...L admits a right adjoint R, then Lν ′ R is a left Serre functor for T /U. b) More generally, if the functor ν ′ : T → T admits a total right derived functor Rν ′ : T /U → T /U in the sense of Deligne =-=[19]-=- with respect to the localization T → T /U, then Rν ′ is a left Serre functor for T /U. Proof. a) For X, Y in T , we have Hom T /U(Lν ′ RX, Y ) = HomT (ν ′ RX, RY ) = D HomT (RY, RX) = D Hom T /U(Y, X... |