Skip to main content
×
Home
    • Aa
    • Aa

Auslander-Reiten triangles in subcategories

  • Peter Jørgensen (a1)
Abstract
Abstract

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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

2. M. Auslander , Representation theory of Artin algebras I, Comm. Algebra 1 (1974), 177268

3. M. Auslander and I. Reiten , Representation theory of Artin algebras III, Comm. Algebra 3 (1975), 239294

5. M. Auslander and S. O. Smalø , Almost split sequences in subcategories, J. Algebra 69 (1981), 426454

6. M. Auslander and S. O. Smalø , Addendum to “Almost split sequences in subcategories”, J. Algebra 71 (1981), 592594

7. D. Happel , On the derived category of a finite dimensional algebra, Comment. Math. Helv. 62 (1987), 339389

8. M. Kleiner , Approximations and almost split sequences in homologically finite subcategories, J. Algebra 198 (1997), 135163

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

12. H. Krause and J. Le , The Auslander-Reiten formula for complexes of modules, Adv. Math. 207 (2006), 133148

14. I. Reiten and M. Van den Bergh , Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295366

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

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 76 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd May 2017. This data will be updated every 24 hours.