Auslander correspondence for proper connective DG-algebras via extended module categories

TL;DR

This paper establishes a one-to-one correspondence between certain $d$-truncated proper connective DG-algebras and their $d$-extended module categories, generalizing Auslander correspondence to a DG setting.

Contribution

It introduces a $d$-dimensional Auslander correspondence for proper connective DG-algebras using $d$-extended module categories, extending classical representation theory.

Findings

One-to-one correspondence between Auslander DG-algebras and their extended module categories.

Characterization of DG-endomorphism algebras of additive generators as Auslander.

Finiteness condition on indecomposable objects in the module category.

Abstract

We establish a $d$-dimensional Auslander correspondence for $d$-truncated proper connective DG-algebras via $d$-extended module categories. A $d$-truncated proper connective DG-algebra $\Gamma$ is called Auslander if its $d$-extended module category is a $d$-Auslander extriangulated category. Our main theorem gives a one-to-one correspondence: for any $d$-truncated proper connective DG-algebra $\Lambda$ whose $d$-extended module category has finitely many indecomposable objects, the DG-endomorphism algebra of an additive generator is Auslander, and conversely every Auslander $d$-truncated proper connective DG-algebra arises in this way.