Ideal mutations in triangulated categories and generalized Auslander-Reiten theory

TL;DR

This paper introduces ideal mutations in triangulated categories, generalizing existing theories, and provides new criteria for the existence of Auslander-Reiten triangles, extending classical Auslander-Reiten theory.

Contribution

It generalizes ideal mutations and Auslander-Reiten theory in triangulated categories, offering new methods to detect Auslander-Reiten triangles.

Findings

Generalized a theorem of Jorgensen to broader settings.

Provided criteria for the existence of Auslander-Reiten triangles based on the Jacobson radical.

Extended classical Auslander-Reiten theory using ideal mutations.

Abstract

We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino \cite{iyama2008mutation} 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 $\mathcal{T}$ over an algebraically closed field $K$. (1) We generalize a theorem of Jorgensen \cite[Theorem 3.3]{jorgensen2010quotients} to a more general setting; (2) We provide a method to detect whether $\mathcal{T}$ has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical $\mathcal{J}$: (i) $\mathcal{J}$ is functorially finite, (ii) Gh$_{\mathcal{J}}= {\rm CoGh}_{\mathcal{J}}$, and (iii) Gh$_{\mathcal{J}}$-source maps coincide with Gh$_{\mathcal{J}}$-sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.