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## Stable categories of Cohen-Macaulay modules and cluster categories (2012)

Citations: | 18 - 8 self |

### Citations

520 | Triangulated Categories in the Representation Theory ofFinite Dimensional Algebras - Happel - 1988 |

187 | Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series 146 - Yoshino - 1990 |

185 | On triangulated orbit categories - Keller |

178 | Quivers with potentials and their representations I
- Derksen, Weyman, et al.
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Citation Context ... bipartite graph on a torus. We denote by Γ0 (resp. Γ1, and Γ2) the set of vertices (resp. edges and faces) of the graph. To such a graph we associate a quiver with a potential (Q,W ) in the sense of =-=[DWZ08]-=-. The quiver viewed as an oriented graph on the torus is the dual of the graph Γ. Faces of Q dual to white vertices are oriented clockwise and faces of Q dual to black vertices are oriented anti-clock... |

151 | Derived categories and stable equivalence - Rickard - 1989 |

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

129 | Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities, Algebra, arithmetic, and geometry: in honor of
- Orlov
- 2009
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Citation Context ...s (B2) since (n− a1) + (n− a2) + (n− a3) = n. Remark 5.4. (a) The equivalence F : Db(A)→ CMZ(SG) is obtained by Ueda [Ued08]. Our proof is very different since he uses a strong theorem due to Orlov =-=[Orl05]-=-. (b) The equivalence G : Cd−1(A) → CM (S G) is an analog of an independent result proved by Thanhoffer de Völcsey and Van den Bergh [TV10, Proposition 1.3]. They use generalized cluster categories a... |

120 | Cluster categories for algebras of global dimension 2 and quivers with potential, Annales de l’Institut Fourier 59
- Amiot
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Citation Context ...r n = 2 and an algebra Λ of global dimension 1, one gets the usual cluster category Db(Λ)/S2 constructed in [BMR+06]. • For n = 2, and an algebra Λ of global dimension 2, the construction is given in =-=[Ami09]-=- in the case where C2(Λ) is Hom-finite. • The generalization of results of [Ami09] from 2 to n ≥ 2 is described in [Guo10]. The functor pi : Db(Λ) → Cn(Λ) is also described by a universal property (cf... |

120 | Y.: Mutation in triangulated categories and rigid Cohen-Macaulay modules - Iyama, Yoshino - 2008 |

113 | Tilting theory and cluster combinatorics’, Adv - Buan, Marsh, et al. |

107 | Cluster structures for 2-Calabi-Yau categories and unipotent groups
- Buan, Iyama, et al.
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Citation Context ... in [IT10]. Results of a similar flavor have been shown in previous papers. In [Ami09, ART11, AIRT12], it was shown that the 2-Calabi-Yau categories Cw associated with elements w in Coxeter groups in =-=[BIRS09]-=- are triangle equivalent to generalized 2-cluster categories C2(A) for some algebras A of global dimension at most two. In [IO09], it was shown that the stable categories of modules over d-preprojecti... |

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

82 |
finite groups. In The Santa Cruz Conference on Finite Groups (Univ
- Graphs
- 1979
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Citation Context ...A4) in the previous section are satisfied in this case. We start with condition (A1*), and here we need some notation. First we give a concrete description of the McKay quiver Q of the cyclic group G =-=[McK80]-=-. The set Q0 of vertices is Z/nZ. The arrows are xij = xj : i→ i+ aj (i ∈ Z/nZ, 1 ≤ j ≤ d). Proposition 5.5. (a) A presentation of B is given by the McKay quiver with commutative relations x i+aj j′ x... |

80 | Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories
- Iyama
- 2007
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Citation Context ...(d−1)-Calabi-Yau. Recently they are studied from the viewpoint of higher analog of Auslander-Reiten theory, and the existence of (d − 1)-cluster tilting objects is shown for quotient singularities in =-=[Iya07a]-=- and for some three dimensional hypersurface singularities in [BIKR08]. They are further investigated in [IY08, KR08, KMV11]. It is of interest to understand the relationship between these two classes... |

67 |
Applications of morphisms determined by objects. In: Representation Theory of Algebras
- Auslander
- 1978
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Citation Context ...ical ones among n-Calabi-Yau triangulated categories having n-cluster tilting objects. On the other hand, a well-known example of Calabi-Yau triangulated categories was given in old work by Auslander =-=[Aus78]-=-, where the stable category of (maximal) CohenMacaulay modules over commutative isolated d-dimensional local Gorenstein singularities are shown to be (d−1)-Calabi-Yau. Recently they are studied from t... |

66 | Graded Calabi-Yau algebras of dimension 3 - Bocklandt - 2008 |

57 |
Rational singularities and almost split sequences
- Auslander
- 1986
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Citation Context ...bove setting are the following: • [Her78, Aus86] R is representation-finite in the sense that there are only finitely many indecomposable Cohen-Macaulay modules. More precisely CM (R) = addS holds. • =-=[Aus86]-=- The Auslander algebra EndR(S) is isomorphic to the skew group algebra S ∗ G. In particular, the AR quiver of CM (R̂) (where R̂ is the completion of the algebra R) is isomorphic to the McKay quiver of... |

57 | Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras”, - Iyama, Reiten - 2008 |

47 | Cluster algebra structures and semicanonical bases for unipotent groups, arXiv:math/0703039v4 [math.RT - Geiss, Leclerc, et al. |

40 | Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren CohenMacaulay-Moduln - Herzog - 1978 |

38 | I.: Cluster tilting for one-dimensional hypersurface singularities
- Burban, Iyama, et al.
- 2008
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Citation Context ...er analog of Auslander-Reiten theory, and the existence of (d − 1)-cluster tilting objects is shown for quotient singularities in [Iya07a] and for some three dimensional hypersurface singularities in =-=[BIKR08]-=-. They are further investigated in [IY08, KR08, KMV11]. It is of interest to understand the relationship between these two classes of Calabi-Yau triangulated categories, i.e. the stable categories of ... |

36 | Triangulated Calabi-Yau categories
- Keller
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Citation Context ... recovers a result due to Kajiura-SaitoTakahashi [KST07] and Lenzing-de la Peña [LP11] for d = 2 and due to Ueda [Ued08] for any d and G cyclic. Moreover the equivalence (0.0.2) was already shown in =-=[KR08]-=- for the case d = 3 and G = diag(ω, ω, ω) where ω is a primitive third root of unity. It would be interesting to generalize our result to non-cyclic quotient singularities. This could then be regarded... |

34 | Triangulated categories, trends in representation theory of algebras and related topics, - Keller, Calabi-Yau - 2008 |

30 | Cluster tilting for higher Auslander algebra - Iyama |

23 | Two-dimensional tame and maximal orders of finite representation type, - Reiten, Bergh - 1989 |

22 | Superpotentials and higher order derivations - Bocklandt, Schedler, et al. |

22 |
On rings with finite self-injective dimension
- Iwanaga
- 1979
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Citation Context ...l on cluster tilting subcategories and on generalized cluster categories. 1.1. Cohen-Macaulay modules over Iwanaga-Gorenstein algebras. The following class of noetherian algebras was given by Iwanaga =-=[Iwa79]-=-. Definition 1.1. A noetherian algebra C is called Iwanaga-Gorenstein if inj.dimCC <∞ and inj.dimCopC <∞. For example, commutative local Gorenstein algebras and finite dimensional selfinjective algebr... |

21 | Cluster equivalence and graded derived equivalence
- Amiot, Oppermann
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Citation Context ...the generalized cluster category C2(A ′) . Hence we get a triangle equivalence between the generalized cluster categories C2(A) ' C2(A ′) (the algebras A and A′ are cluster equivalent in the sense of =-=[AO10]-=-). However, one can show that the algebras A and A′ are not derived equivalent since they have different Coxeter polynomials. (One can also see this using results of [AO10].) Now we have two different... |

21 | Stable categories of higher preprojective algebras
- Iyama, Oppermann
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Citation Context ...bi-Yau categories Cw associated with elements w in Coxeter groups in [BIRS09] are triangle equivalent to generalized 2-cluster categories C2(A) for some algebras A of global dimension at most two. In =-=[IO09]-=-, it was shown that the stable categories of modules over d-preprojective algebras of (d − 1)-representationfinite algebras are triangle equivalent to generalized d-cluster categories of stable (d− 1)... |

20 | Dimer models and Calabi-Yau algebras - Broomhead |

20 | Dimer models and the special McKay correspondence - Ishii, Ueda |

20 |
Matrix factorization and representations of quivers
- Kajiura, Saito, et al.
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Citation Context ...valence CMZ(eBe) ' Db(B0/〈e〉) where CMZ(eBe) is the category of graded Cohen-Macaulay eBe-modules. This intermediate result in the case where B = S ∗ G recovers a result due to Kajiura-SaitoTakahashi =-=[KST07]-=- and Lenzing-de la Peña [LP11] for d = 2 and due to Ueda [Ued08] for any d and G cyclic. Moreover the equivalence (0.0.2) was already shown in [KR08] for the case d = 3 and G = diag(ω, ω, ω) where ω ... |

18 | les petites catégories triangulées - Amiot, Sur - 2008 |

18 | On two examples by Iyama and - Keller, Murfet, et al. |

18 |
la Peña. Extended canonical algebras and Fuchsian singularities
- Lenzing, de
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Citation Context ...ere CMZ(eBe) is the category of graded Cohen-Macaulay eBe-modules. This intermediate result in the case where B = S ∗ G recovers a result due to Kajiura-SaitoTakahashi [KST07] and Lenzing-de la Peña =-=[LP11]-=- for d = 2 and due to Ueda [Ued08] for any d and G cyclic. Moreover the equivalence (0.0.2) was already shown in [KR08] for the case d = 3 and G = diag(ω, ω, ω) where ω is a primitive third root of un... |

16 | Moduli of McKay quiver representations. II. Grobner basis techniques. - Craw, Maclagan, et al. - 2007 |

13 | Consistency conditions for brane tilings - Davison |

13 | Selfinjective quivers with potential and 2-representationfinite algebras
- Herschend, Iyama
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Citation Context ...contains exactly one arrow of degree 1, and then the potentialW is homogeneous of degree 1. Hence D induces a grading dD on the Jacobian algebra B. In other words D is a cut of (Q,W ) in the sense of =-=[HI11]-=-. Proposition 6.1. Let B be a Jacobian algebra coming from a consistent dimer model. Any perfect matching induces a grading on B making it bimodule 3-Calabi-Yau of Gorenstein parameter 1. Proof. We de... |

12 | On generalized cluster categories - Amiot - 2011 |

11 | Deformed Calabi-Yau completions (With an appendix by - Keller |

10 | Preprojective algebras and c-sortable words - Amiot, Iyama, et al. |

9 | Consistency conditions for dimer models, - Bocklandt - 2012 |

9 | Calabi-Yau algebras, arXiv:math/0612139v3 [math.AG] - Ginzburg - 2007 |

9 | Cluster Tilting Objects in Generalized Higher Cluster Categories,
- Guo
- 1005
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Citation Context ...r n = 2, and an algebra Λ of global dimension 2, the construction is given in [Ami09] in the case where C2(Λ) is Hom-finite. • The generalization of results of [Ami09] from 2 to n ≥ 2 is described in =-=[Guo10]-=-. The functor pi : Db(Λ) → Cn(Λ) is also described by a universal property (cf [Kel05, Ami09]). Here is the version we will use in this paper (see appendix [IO09]). Proposition 1.10. [Kel05, Ami09],[I... |

9 | Tilting and cluster tilting for quotient singularities
- Iyama, Takahashi
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Citation Context ...esting to generalize our result to non-cyclic quotient singularities. This could then be regarded as an analog of a triangle equivalence CMZ(SG) ' Db(Λ) for some finite dimensional algebra Λ given in =-=[IT10]-=-. Results of a similar flavor have been shown in previous papers. In [Ami09, ART11, AIRT12], it was shown that the 2-Calabi-Yau categories Cw associated with elements w in Coxeter groups in [BIRS09] a... |

8 | The structure of AS-Gorenstein algebras,
- Minamoto, Mori
- 2011
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Citation Context ...nt conditions for B to be the d-preprojective algebra of A = B0. In particular A is a (d− 1)-representation-infinite algebra in the sense of [HIO12] and a quasi extremely-Fano algebra in the sense of =-=[MM10]-=-. In section 4, we use results from sections 2 and 3 to prove our main result, which gives ha l-0 05 86 61 2,sv er sio ns2s- 1 1sJu l 2 01 2 STABLE CATEGORIES OF COHEN-MACAULAY MODULES AND CLUSTER CAT... |

4 | Triangulated categories of Gorenstein cyclic quotient singularities
- Ueda
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Citation Context ...raded Cohen-Macaulay eBe-modules. This intermediate result in the case where B = S ∗ G recovers a result due to Kajiura-SaitoTakahashi [KST07] and Lenzing-de la Peña [LP11] for d = 2 and due to Ueda =-=[Ued08]-=- for any d and G cyclic. Moreover the equivalence (0.0.2) was already shown in [KR08] for the case d = 3 and G = diag(ω, ω, ω) where ω is a primitive third root of unity. It would be interesting to ge... |

3 | Finite-dimensional algebras and singularities, Singularities, representation of algebras, and vector bundles - Reiten - 1985 |

2 |
n-representation infinite algebras, preprint, arXiv 1205.1272
- Herschend, Iyama, et al.
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Citation Context ... 2, we assume that B = ⊕ `≥0B` is graded, and give sufficient conditions for B to be the d-preprojective algebra of A = B0. In particular A is a (d− 1)-representation-infinite algebra in the sense of =-=[HIO12]-=- and a quasi extremely-Fano algebra in the sense of [MM10]. In section 4, we use results from sections 2 and 3 to prove our main result, which gives ha l-0 05 86 61 2,sv er sio ns2s- 1 1sJu l 2 01 2 S... |

1 | Buchweitz,Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished manuscript - O - 1987 |

1 |
Properly ordered dimers, RR-charges, and an efficient inverse algorithm
- Gulotta
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Citation Context ...enter C = Z(B) which is the coordinate ring of a Gorenstein affine toric threefold. Moreover the coordinate ring of any Gorenstein affine toric threefold can be obtained from a consistent dimer model =-=[Gul08, IU09]-=-. The following result gives an interpretation of the stable category of Cohen-Macaulay modules over certain Gorenstein affine toric threefold in terms of cluster categories. Theorem 6.3. Let Γ be a c... |

1 | Sous les catgories drives. (French) [Beneath the derived categories - Keller, Vossieck - 1987 |

1 |
den Bergh: Explicit models for some stable categories of maximal Cohen-Macaulay modules
- Völcsey, Van
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Citation Context ...sentationfinite algebras are triangle equivalent to generalized d-cluster categories of stable (d− 1)- Auslander algebras. We were able to use some of the ideas in these papers for d ≥ 2. We refer to =-=[TV10]-=- for similar independent results based on the language of quivers with potential. We thank Michel Van den Bergh for informing us about his work with Thanhoffer de Völcsey. Some results in this paper ... |