A ug 2 00 3 Representation dimension and Solomon zeta function

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 lat Γ of lat Λ. Formulating this correspondence Γ 7→ lat Γ 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.