Auslander-Reiten triangles in subcategories

Details Auslander-Reiten triangles in subcategories Auslander-Reiten triangles in subcategories

2006-06-26

arxiv.org/abs/math/0606647v1

Mathematics

Representation Theory

19 pages

Scientific paper

This paper introduces Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem of the paper shows that there is a close connection with covers and envelopes, also known as minimal right- and left-approximations. Namely, under suitable assumptions, if M is an object in the subcategory C of the triangulated category T and X --> Y --> M --> is an Auslander-Reiten triangle in T, then there is an Auslander-Reiten triangle K --> L --> M --> in C if and only if there is a C-cover of the form K --> X. The main theorem is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.

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