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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
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.


References [Enhancements On Off] (What's this?)

  • [AR] M. Auslander and I. Reiten, Stable equivalence of dualizing $ R$-varieties, Adv. in Math. 12 (1974), 306-366. MR 0342505 (49:7251)
  • [B] A. Beilinson, Coherent sheaves on $ {\mathbb{P}^n}$ and problems of linear algebra, Funct. Anal. Appl. 12 (1979), 214-216.
  • [H1] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339-389. MR 910167 (89c:16029)
  • [H2] -, Triangulated categories in the representation theory of finite-dimensional algebras, Cambridge University Press 119 (1988). MR 935124 (89e:16035)
  • [KV] B. Keller and D. Vossieck, Sous les catégories dérivées, CR. Acad. Sci. Paris t. 305, Série I (1987), 225-228. MR 907948 (88m:18014)
  • [R] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer, 1984. MR 774589 (87f:16027)
  • [Ri1] J. Rickard, Morita theory for derived categories, J. London Math. Soc 39 (1989), 436-456. MR 1002456 (91b:18012)
  • [Ri2] -, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303-317. MR 1027750 (91a:16004)
  • [TW] H. Tachikawa and T. Wakamatsu, Carian matrices and Grothendieck groups of stable categories, preprint. MR 1140611 (92m:16014)
  • [V] J. L. Verdier, Catégories dérivées, état 0, Lecture Notes in Math., vol. 569, Springer, 1977, pp. 262-311. MR 0463174 (57:3132)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1045137-6
Keywords: Repetitive algebras, Auslander-Reiten triangles
Article copyright: © Copyright 1991 American Mathematical Society

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