Abstract. In this article we study Cohen-Macaulay modules over one-dimensional 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 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

Abstract. We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually u-cluster categories. Since their introduction in [6], [7], cluster categories have become a central topic in representation theory. They provide the framework for the representation-theoretic approach to the highly successful theory of cluster algebras, as introduced by Fomin and Zelevinsky [11].

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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 2-cluster 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 2-cluster 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

Abstract. We prove a structure theorem for triangulated Calabi-Yau categories: An algebraic 2-Calabi-Yau 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

Abstract. We introduce the Calabi-Yau (CY) objects in a Hom-finite Krull-Schmidt triangulated k-category, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and Auslander-Reiten triangles is provided. Finally we classify all the CY modules of selfinjective Nakayama algebras, determining this way the self-injective Nakayama algebras admitting indecomposable CY modules. In particular, this result recovers the algebras whose stable categories are Calabi-Yau, which have been obtained in [BS]. 1.

Abstract. We show that the Calabi-Yau 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 k-algebras are finite dimensional. A selfinjective k-algebra 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 Calabi-Yau 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. Calabi-Yau 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 u-cluster categories of Dynkin type [9], [10], [11]. According to the definition, for determining the precise value of the Calabi-Yau 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 A-module M). On the other hand, it can be a

We study the question of when cluster-tilted algebras of Dynkin type E are derived equivalent and obtain a far-reaching, but unfortunately not complete, classification. It turns out that a useful invariant for distinguishing cluster-tilted 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

We address the question of when cluster-tilted 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