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

Abstract:

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