N ov 2 00 3 Rejective subcategories of artin algebras and orders 1

We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function rΛ : modΛ → N≥0 which is a natural extension of Auslander’s representation dimension. In the representation theory of artin algebras and orders, it often plays an important role to study certain classes of subcategories of the module category. Typical examples are given by subcategories induced by morphisms of rings (§1.3), and subcategories induced by cotilting modules (§1.4). These subcategories are functorially finite in the sense of Auslander-Smalo [AS1]. One object of this paper is to study functorially finite subcategories from the viewpoint of its resolution dimension (§1.1). The subcategories of resolution dimension zero is often called bireflective [St], and we shall show that they correspond to ring epimorphisms (§1.6.1). We shall introduce a special class of bireflective subcategories called rejective subcategories (§1.5), which was well-known in the representation theory of orders and recently played a crucial role in the study of representation-finite orders [I1,2][Ru1,2]. They correspond to factor algebras of artin algebras, and overrings of orders (§1.6.1) which are non-commutative analogy of the normalization in the commutative ring theory. Another object of this paper is to study certain chains of rejective subcategories called rejective chains (§2.2), which give a method to construct rings of finite global dimension (§2.2.2). Recently, rejective chains were applied to give positive answers to two open problems in [I3,4]. One is Solomon’s conjecture on zeta functions of orders [S1,2], and another is the finiteness problem of the representation dimension of artin algebras [A1][Xi1] (see §4.1.1). We shall formulate these construction of rejective chains by using a certain functor FC (§2.3). Typical examples are given by preprojective partition of Auslander-Smalo [AS2,3], and Bass chains of Drozd-Kirichenko-Roiter [DKR] and Hijikata-Nishida [HN] (§2.3.4). It was first observed by Dlab-Ringel [DR4] that certain chains of subcategories are related to quasi-hereditary algebras, introduced by ClineParshall-Scott in the representation theory of Lie algebras and algebraic groups [CPS1,2]. In §3, we shall study the relationship between rejective chains and quasi-hereditary algebras (§3.5.1), and calculate the global dimension of rings with rejective chains (§3.3). We shall also relate neat algebras of Agoston-Dlab-Wakamatsu [ADW]. 2000 Mathematics Subject Classification. Primary 16E10; Secondary 16G10, 16G30