Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend

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.

Cline-Parshall-Scott introduced the concept of quasi-hereditary algebras (§2.5) to study highest weight categories in the representation theory of Lie algebras and algebraic groups [CPS1,2]. Quasi-hereditary algebras were effectively applied in the representation theory of artin algebras as well by Dlab-Ringel [DR1,2,3] and many other authors. On the other hand, in the representation theory of orders, the concept of overorders and overrings (§1.1), a non-commutative analogy of the normalization in the commutative ring theory, plays a crucial role. From an overring Γ of an order Λ, we naturally obtain a full subcategory latΓ of lat Λ. Formulating this correspondence Γ ↦ → latΓ categorically, we obtain the concept of the rejection (§1,§2). Recently it was effectively applied to study orders of finite representation type by the author [I1,2,3] and Rump [Ru1,2,3]. Originally Drozd-Kirichenko-Roiter found the one-point rejection (§1.3) in their theory of Bass orders [DKR], and later Hijikata-Nishida applied the four-points rejection (§1.5) to local orders of finite representation type and suggested a possibility of generalization [HN1,2,3]. In this paper, we will show that there exists a close relationship between quasihereditary algebras and the rejection from the viewpoint of the approximation theory of Auslander-Smalo [AS2]. As an application, we will solve two open problems [I4,5]. One concerns the representation dimension of artin algebras introduced by M. Auslander about 30 years ago [A1], and another concerns the Solomon zeta functions of orders introduced by L. Solomon about 25 years ago [S1,2]. It will turn out that the rejection relates these two quite different problems with each other closely. orders (Krull dimension one) artin algebras (Krull dimension zero) overrings of an order Λ factor algebras of an artin algebra Λ

Abstract. In this paper we give a direct proof of the properties of the ZD∞ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author. 1.

Abstract. We explain the isomorphism between the G-Hilbert scheme and the F-blowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of D-modules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic. 1.