Auslander Correspondence for Triangulated Categories

TL;DR

This paper extends the Auslander correspondence to certain triangulated categories, providing homological characterizations and exploring algebraic triangle structures, thus deepening the understanding of their algebraic and categorical properties.

Contribution

It introduces analogues of Auslander correspondence for two classes of triangulated categories with finiteness conditions, including graded endomorphism algebras and homological characterizations.

Findings

Homological characterization of Auslander algebras in triangulated categories

Uniqueness of algebraic triangle structures on homotopy categories

Introduction of $[1]$-additive generators and their graded endomorphism algebras

Abstract

We give analogues of the Auslander correspondence for two classes of triangulated categories satisfying certain finiteness conditions. The first class is triangulated categories with additive generators and we consider their endomorphism algebras as the Auslander algebras. For the second one, we introduce the notion of $[1]$-additive generators and consider their graded endormorphism algebras as the Auslander algebras. We give a homological characterization of the Auslander algebras for each class. Along the way, we also show that the algebraic triangle structures on the homotopy categories are unique up to equivalence.