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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
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

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