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Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Cluster categories and selfinjective algebras: type A
, 2006
"... We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories. ..."
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Cited by 7 (2 self)
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We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories.
QUOTIENTS OF CLUSTER CATEGORIES
"... Abstract. Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2cluster categories, the “ ..."
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Cited by 2 (1 self)
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Abstract. Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2cluster categories, the “lowest” type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2cluster categories. In contrast, the same is not true in Dynkin type E. This paper is about the connection between quotient categories and cluster categories, so let me start by explaining these two notions. Quotient categories come in a number of different flavours. The one
ACYCLIC CALABIYAU CATEGORIES BERNHARD KELLER AND IDUN REITEN
"... Abstract. We prove a structure theorem for triangulated CalabiYau categories: An algebraic 2CalabiYau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterizati ..."
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Abstract. We prove a structure theorem for triangulated CalabiYau categories: An algebraic 2CalabiYau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to
unknown title
, 2006
"... Abstract. We introduce the CalabiYau (CY) objects in a Homfinite KrullSchmidt triangulated kcategory, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and AuslanderReiten triangles is provided. Finall ..."
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Abstract. We introduce the CalabiYau (CY) objects in a Homfinite KrullSchmidt triangulated kcategory, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and AuslanderReiten triangles is provided. Finally we classify all the CY modules of selfinjective Nakayama algebras, determining this way the selfinjective Nakayama algebras admitting indecomposable CY modules. In particular, this result recovers the algebras whose stable categories are CalabiYau, which have been obtained in [BS]. 1.
STABLE CALABIYAU DIMENSION FOR FINITE TYPE SELFINJECTIVE ALGEBRAS
, 2007
"... Abstract. We show that the CalabiYau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent results of C. Amiot, and hence apply more generally ..."
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Abstract. We show that the CalabiYau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent results of C. Amiot, and hence apply more generally to triangulated categories having only finitely many indecomposable objects. Throughout, k is an algebraically closed field, and all kalgebras are finite dimensional. A selfinjective kalgebra A has a stable module category stab A. This is a triangulated category with suspension Σ given by taking the first syzygy Ω −1 A in an injective resolution. There is also a Serre functor given by S = ΩA ◦ νA where νA: = DA ⊗ − is the Nakayama functor, see [6, prop. 1.2(ii)]. By definition, the CalabiYau dimension of a triangulated category with Serre functor S is the smallest integer d ≥ 0 such that there is an equivalence of functors S ≃ Σ d, or ∞ if no d exists. CalabiYau dimensions of stable module categories of selfinjective algebras have been of considerable interest recently, see for instance [2], [3], [5], [6], [7]. In finite representation type, these dimensions occur in connection with ucluster categories of Dynkin type [9], [10], [11]. According to the definition, for determining the precise value of the CalabiYau dimension of the stable module category of a selfinjective algebra A, one needs to find the minimal d ≥ 0 such that there is an equivalence of functors νA ≃ Ω −d−1 A. In some situations it is not too difficult to show that these functors agree on objects (i.e. νA(M) ∼ = (M) for every Amodule M). On the other hand, it can be a
Derived equivalences for clustertilted algebras of Dynkin type E
, 906
"... We study the question of when clustertilted algebras of Dynkin type E are derived equivalent and obtain a farreaching, but unfortunately not complete, classification. It turns out that a useful invariant for distinguishing clustertilted algebras of type E up to derived equivalence are the unimodu ..."
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We study the question of when clustertilted algebras of Dynkin type E are derived equivalent and obtain a farreaching, but unfortunately not complete, classification. It turns out that a useful invariant for distinguishing clustertilted algebras of type E up to derived equivalence are the unimodular equivalence classes of their Cartan matrices. For type E6 all details are given in the paper, for types E7 and E8 (where 112 and 391 algebras are involved) we present the results in a concise form from which our findings should easily be verifiable. 1
Derived equivalence classification of clustertilted algebras of Dynkin type E
, 906
"... We address the question of when clustertilted algebras of Dynkin type E are derived equivalent and as main result obtain a complete derived equivalence classification. It turns out that two clustertilted algebras of type E are derived equivalent if and only if their Cartan matrices represent equiva ..."
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We address the question of when clustertilted algebras of Dynkin type E are derived equivalent and as main result obtain a complete derived equivalence classification. It turns out that two clustertilted algebras of type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers. For type E6 all details are given in the paper, for types E7 and E8 we present the results in a concise form from which our findings should easily be verifiable. 1
THE IMAGE OF THE DERIVED CATEGORY IN THE CLUSTER CATEGORY
"... Abstract. Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for nonhereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In ..."
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Abstract. Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for nonhereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In the more general setup of algebras of global dimension at most 2, cluster categories are defined to be these triangulated hulls of the orbit categories. In this paper we study the image of the natural functor from