Differential graded division algebras, their modules, and dg-simple algebras

TL;DR

This paper introduces the concept of dg-division algebras, classifies them, and explores their properties, including their relation to dg-simple algebras and a Jacobson-Chevalley density theorem for acyclic cases.

Contribution

It defines dg-division algebras, classifies them completely, and establishes their connections to dg-simple algebras and modules, providing new structural insights.

Findings

dg-division algebras are either acyclic or have zero differential

the graded centre of dg-simple dg-algebras is a dg-division algebra

dg-endomorphism rings of dg-simple modules are dg-division algebras

Abstract

We give the definition of a dg-division algebra, that is a concept of a differential graded algebra which may serve as an analogue of a division algebra. We classify them completely, and show that they are either acyclic or have differential $0$. Further, we prove that the graded centre of dg-simple dg-algebras is a dg-division algebra, and also the dg-endomorphism ring of a dg-simple module is a dg-division algebra. We also shall give a Jacobson-Chevalley density theorem for acyclic dg-algebras.