Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology

Research Article

Auslander-Reiten triangles in subcategories

Peter Jørgensena1

a1 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.

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

S175506960800056X_eqnU1

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

S175506960800056X_eqnU2

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

S175506960800056X_eqnU3

.

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

(Received May 2007)

Key Words

  • Auslander-Reiten sequences;
  • covers;
  • envelopes;
  • finite dimensional algebras;
  • homotopy categories;
  • minimal approximations;
  • precovers;
  • preenvelopes;
  • triangulated categories;
  • Wakamatsu's Lemma