Auslander-Reiten triangles in derived categories of finite-dimensional algebras

Abstract

Let A A be a finite-dimensional algebra. The category b m o d A bmod A of finitely generated left A A -modules canonically embeds into the derived category D b ( A ) {D^b}\left ( A \right ) of bounded complexes over b m o d A bmod A and the stable category mod _ Z T ( A ) {\underline {\bmod } ^\mathbb {Z}}T\left ( A \right ) of Z \mathbb {Z} -graded modules over the trivial extension algebra of A A by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from D b ( A ) {D^b}\left ( A \right ) to mod _ Z T ( A ) \underline {\bmod }^{\mathbb {Z}}T\left ( A \right ) . Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only if A A has finite global dimension.