Locally finite triangulated categories ✩

A k-linear triangulated category A is called locally finite provided ∑ X∈indA dimk HomA(X,Y ) < ∞ 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 L̂n: · · · · ̂ n n− 1 2 1 By using this, we prove that the Auslander–Reiten quiver of any locally finite triangulated category A is of the form Z −→ ∆/G, where ∆ is a Dynkin diagram and G is an automorphism group of Z −→ ∆. For most automorphism groups G, the triangulated categories with Z −→ ∆/G as their Auslander–Reiten quivers are constructed. In particular, a triangulated category with L̂n as its Auslander–Reiten quiver is constructed.  2005 Elsevier Inc. All rights reserved.