1 Ju l 2 00 3 The relationship between homological properties and representation theoretic realization of artin algebras

We will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, τ -categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers. 0.1 There exists a bijection between equivalence classes of Krull-Schmidt categories C with additive generators M and Morita-equivalence classes of semiperfect rings Γ, which is given by C 7→ C(M,M) and the converse is given by Γ 7→ pr Γ for the category pr Γ of finitely generated projective Γ-modules. Although this bijection itself is rather formal, it will be very fruitful to study the relationship of (A)–(D) below. The object of this paper is to study it under the assumption that Γ is an artin algebra. (A) Homological properties for Γ. (B) Representation theoretic realization of C. (C) Categorical properties for C. (D) Combinatorial properties for the AR quiver A(C). For (A), we will study a property of the selfinjective resolution of Γ which is called the (l, n)-conditions (§1.1) and generalizes both the Auslander conditions [Bj] and the dominant dimension [T]. For (B), we will study the existence of an equivalence between C and a torsionfree class of modΛ over an artin algebra Λ (§1.2,§2.2), where such a subcategory is very popular in the representation theory of artin algebras [Ha][As]. For (C), we will treat a class of additive categories which are called τ -categories (§1.3) and introduced in [I3]. τ -categories are additive categories with generalized almost split sequences, and our motivation and definition were rather different from the work of Auslander and Smalo in [AS] since we aimed to treat categories which can be far from abelian, for example mesh categories of translation quivers (§1.3.2(3)). Nevertheless our result §2.1 asserts that some τ -categories are realized as torsionfree classes over artin algebras, and they form almost abelian categories (§1.5). For (D), we will study a combinatorial invariant A(C) of a τ -category C called the AR (=Auslander-Reiten) quiver (§4.1). Some results in this paper were already announced in [I6;7.4] without proof. 0.2 Background In [A], Auslander obtained a quite remarkable theorem which asserts that there exists a bijection between Morita-equivalence classes of representationfinite artin algebras Λ and those of Auslander algebras Γ, which is an artin algebra 2000 Mathematics Subject Classification. Primary 16E65; Secondary 16G70