Locally finite triangulated categories

摘要

A k-linear triangulated category A is called locally finite provided Sigma(X is an element of indA) dim(k) Hom(A)(X, Y) < infinity for any indecomposable object Y in A. It has Auslander-Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander-Reiten triangles and contains loops, then its Auslander-Reiten quiver is of the form L-n:[GRAPHICS]By using this, we prove that the Auslander-Reiten quiver of any locally finite triangulated category A is of the form Z Delta/G, where Delta is a Dynkin diagram and G is an automorphism group of Z Delta. For most automorphism groups G, the triangulated categories with Z Delta/G as their Auslander-Reiten quivers are constructed. In particular, a triangulated category with L, as its Auslander-Reiten quiver is constructed. (c) 2005 Elsevier Inc. All rights reserved.