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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Dieter Happel
Journal: Proc. Amer. Math. Soc. 112 (1991), 641-648
MSC: Primary 16G70; Secondary 16D90
DOI: https://doi.org/10.1090/S0002-9939-1991-1045137-6
MathSciNet review: 1045137
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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.


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Keywords: Repetitive algebras, Auslander-Reiten triangles
Article copyright: © Copyright 1991 American Mathematical Society