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Auslander-Reiten triangles in subcategories

  • Peter Jørgensen (a1)

This paper studies Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem shows that the Auslander-Reiten triangles in a subcategory are closely connected with the approximation properties of the subcategory. Namely, let C be an object in the subcategory C of the triangulated category T, and let

be an Auslander-Reiten triangle in T. Then under suitable assumptions, there is an Auslander-Reiten triangle

in C if and only if there is a minimal right-C-approximation of the form


The theory is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.

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1. F. W. Anderson and K. R. Fuller , “Rings and categories of modules”, Grad. Texts in Math. 13, Springer, Berlin, 1974

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9. H. Krause , Auslander-Reiten theory via Brown Representability, K-Theory 20 (2000), 331344

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15. C. M. Ringel , “Tame algebras and quadratic forms”, Lecture Notes in Math. 1099, Springer, Berlin, 1984

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Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
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