On the Auslander--Reiten Theory for Extended Hearts of Proper Connective DG Algebras

TL;DR

This paper explores the structure of extended hearts in derived categories of proper connective dg algebras, establishing their extriangulated nature and proving a version of the Brauer--Thrall Conjecture.

Contribution

It demonstrates that the extended heart forms an extriangulated category with almost-split conflations and proves a related version of the Brauer--Thrall Conjecture.

Findings

Extended hearts are extriangulated categories with almost-split conflations.

A version of the 1st Brauer--Thrall Conjecture is proved in this setting.

Provides new insights into the Auslander--Reiten theory for dg algebras.

Abstract

We prove that, for a proper connective dg algebra $A$ with cohomology concentrated in degrees between $1-d$ and $0$, the extended heart $\mathcal{D}^{fd}(A)^{(-d,0]}\subseteq \mathcal{D}^{fd}(A)$ is an extriangulated category with almost-split conflations. We also prove a version of the 1st Brauer--Thrall Conjecture in this context.