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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, peter.jorgensen@ncl.ac.uk.
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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

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