Abstract. We use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type A, D and E respectively.

For any given symmetrizable Cartan matrix C with a symmetrizer D, Geiß et al. (2016) introduced a generalized preprojective algebra Π(C, D). We study tilting modules and support τtilting modules for the generalized preprojective algebra Π(C, D) and show that there is a bijection between the set of all cofinite tilting ideals of Π(C, D) and the corresponding Weyl group W (C) provided that C has no component of Dynkin type. When C is of Dynkin type, we also establish a bijection between the set of all basic support τ -tilting Π(C, D)-modules and the corresponding Weyl group W (C). These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035-1079, 2009) and Mizuno (Math. Zeit. 277(3), 665-690, 2014) over classical preprojective algebras.Recently, Geiß et al. [10] introduced a class of Iwanaga-Gorenstein algebras via quivers with relations for any symmetrizable Cartan matrices with symmetrizers, which generalizes the path algebras of quivers associated with symmetric Cartan matrices. They also introduced the corresponding generalized preprojective algebras. This new class of preprojective algebras reduces to the classical one provided that the Cartan matrix is symmetric and the symmetrizer is the identity matrix. Surprisingly, the generalized preprojective algebras still share many properties with the classical one. Since the classical preprojective algebras have many important applications in different fields of mathematics, it is an interesting question to find out which results or constructions for classical preprojective algebras can be generalized to the general setting. For example, if one can generalize the constructions of [4,9] to the new preprojective algebras, then one may obtain new categorifications for certain skew-symmetrizable cluster algebras. This note gives a first attempt to generalize certain classification results in tilting theory of preprojective algebras to this new setting. For a given algebra, a basic question in tilting theory is to classify all the tilting modules or support τ -tilting modules. For the classical preprojective algebras, the classification has been obtained by Buan et al. [4] for preprojective algebras of non-Dynkin type (cf.

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Let X be a weighted projective line of tubular type and coh X the category of coherent sheaves on X. The main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles of coh X is connected. This yields an alternative proof for the connectedness of the tilting graph of coh X. Our approach leads to the investigation of the change of slopes of a tilting sheaf in coh X under (co-)APR mutations, which may be of independent interest.

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