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

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.