Auslander-Reiten Triangles in Homotopy Categories

Abstract

Let $A$ be an artin algebra. We show that the bounded homotopy category of finitely generated right $A$-modules has Auslander-Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [H2]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules admits Auslander-Reiten triangles, which improves a main result in [G].