Ideal mutations in triangulated categories and generalized Auslander-Reiten theory

Abstract

We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino [15] by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom-finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen [17, Theorem 3.3] to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J: (i) J is functorially finite, (ii) GhJ=CoGhJ, and (iii) GhJ-source maps coincide with GhJ-sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.