Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-30T22:52:54.635Z Has data issue: false hasContentIssue false
  • English
  • Français

Auslander-Reiten triangles in subcategories

Published online by Cambridge University Press:  14 November 2008

Peter Jørgensen
Peter Jørgensen
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom, http://www.staff.ncl.ac.uk/peter.jorgensen, [email protected].
Get access

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.

Keywords

Auslander-Reiten sequencescoversenvelopesfinite dimensional algebrashomotopy categoriesminimal approximationsprecoverspreenvelopestriangulated categoriesWakamatsu's Lemma
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 3 , June 2009 , pp. 583 - 601
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Anderson, F. W. and Fuller, K. R., “Rings and categories of modules”, Grad. Texts in Math. 13, Springer, Berlin, 1974Google Scholar
2.Auslander, M., Representation theory of Artin algebras I, Comm. Algebra 1 (1974), 177–268CrossRefGoogle Scholar
3.Auslander, M. and Reiten, I., Representation theory of Artin algebras III, Comm. Algebra 3 (1975), 239–294CrossRefGoogle Scholar
4.Auslander, M., Reiten, I., and Smalø, S. O., “Representation theory of Artin algebras”, Cambridge Stud. Adv. Math. 36, Cambridge University Press, Cambridge, 1997, (the first paperback edition with corrections)Google Scholar
5.Auslander, M. and Smalø, S. O., Almost split sequences in subcategories, J. Algebra 69 (1981), 426–454CrossRefGoogle Scholar
6.Auslander, M. and Smalø, S. O., Addendum to “Almost split sequences in subcategories”, J. Algebra 71 (1981), 592–594CrossRefGoogle Scholar
7.Happel, D., On the derived category of a finite dimensional algebra, Comment. Math. Helv. 62 (1987), 339–389CrossRefGoogle Scholar
8.Kleiner, M., Approximations and almost split sequences in homologically finite subcategories, J. Algebra 198 (1997), 135–163CrossRefGoogle Scholar
9.Krause, H., Auslander-Reiten theory via Brown Representability, K-Theory 20 (2000), 331–344CrossRefGoogle Scholar
10.Krause, H., Auslander-Reiten triangles and a theorem of Zimmermann, Bull. London Math. Soc. 37 (2005), 361–372CrossRefGoogle Scholar
11.Krause, H., The stable derived category of a noetherian scheme, Compositio Math. 141 (2005), 1128–1162CrossRefGoogle Scholar
12.Krause, H. and Le, J., The Auslander-Reiten formula for complexes of modules, Adv. Math. 207 (2006), 133–148CrossRefGoogle Scholar
13.Neeman, A., “Triangulated categories”, Ann. of Math. Stud. 148, Princeton University Press, Princeton, 2001Google Scholar
14.Reiten, I. and Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295–366CrossRefGoogle Scholar
15.Ringel, C. M., “Tame algebras and quadratic forms”, Lecture Notes in Math. 1099, Springer, Berlin, 1984Google Scholar