Modules of bounded length in Auslander-Reiten components

Abstract

Let A be an artin algebra, and F a component of the Aaslander-Reiten qmver of A. it has been asked (12) whether the number of isomorphism classes of indecomposable A-modules in F of fixed length is finite. This question is answered affirmatively for certain types of regular components. More generally, we consider regular components of full additive subcategories with sink maps in a length category. Recall that a length category is an abelian category where every object X has a finite composition series; the length of such a composition series will be called the length of X and denoted by )XI. We denote by (X) the isomorphism class of the object X, and we also write I(x)l instead of lXl. Let ~ be a length category, and .~ a full additive subcategory. We consideras a Krull-Schmidt category with short exact sequences, the short exact sequences being those in d which belong to JU. Thus, the Auslander-Reiten quiver F (S) of cf is defined (see (111), but we consider F(cg') as a valued translation quiver, see (5), and we denote by z~e the Auslander-Reiten translation in 24#. An (Auslander-Reiten) component cg of S is, by definition, the full additive subcategory generated by the indecomposable objects X in cf whose isomorphism classes belong to a fixed component F' of F(2(C); of course, F' = F ((s A component cg of 2U is said to be regular provided F (c~) is a stable translation quiver. (Note that a component cg of 24# is regular if and only if the following condition is satisfied: if X is indecomposable in c6, then there exists a source map f and a sink map g for X in Y, and there are short exact sequences (ff') and (g', 9) which lie inside cg.) Let (g be a regular component of S. We assume in addition that there are indecomposable objects C1, C2 in cg and Xi, X2 in X\cg, with Horn (X 1, C1) ,I= 0 and Horn (C 2, Xz) + O. Finally, assume there is a constam d such that for any indecomposable object X of c~, we have I r~ X I < d (Xt. These conditions are obviously satisfied in the case that cg is a stable component of ~r = 2(( = A-rood. (For d, we may take (dim k A) 2, see (10)). They are also satisfied in the following situation: Let AT be a tilting module, ~f = (Y (AT) the full subcategory of all A-modules generated by A T, and sr = A-rood. (Here, take for X~ a suitable dirct summand of A T, for X 2 a suitable indecomposable injective A-module; also if zx X exists, it can be embedded into ~ X, see (7), thus, again let d = (dim k A)2.) A valued quiver A = (Ao, A1, d,d') is given by a quiver (A o, A~) without multiple arrows, and two functions d, d' defined on A ~ with values in the set NI of positive integers, if ~: a ~ b is an arrow, we write d~b instead of d(~) and d~ab instead of d'(~). A valued translation quiver F = (Fo, F1, d, d', ~) is given by a valued quiver (F o, ~, d, d') and a