Auslander-Reiten Triangles in Homotopy Categories

TL;DR

This paper demonstrates the existence of Auslander-Reiten triangles in the bounded homotopy categories of finitely generated modules over artin and Gorenstein algebras, providing new proofs and extending previous results.

Contribution

It establishes the presence of Auslander-Reiten triangles in these categories and offers an alternative proof of a key theorem, extending to Gorenstein modules.

Findings

Existence of Auslander-Reiten triangles in bounded homotopy categories of modules over artin algebras.

Extension of Auslander-Reiten theory to Gorenstein projective and injective modules.

Improvement of previous results by providing new proofs and broader applicability.

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].