Differential graded orders, their class groups and id\`eles

TL;DR

This paper extends classical order theory to differential graded algebras, defining orders, class groups, and idèles in this setting, and establishing foundational properties and relationships among these structures.

Contribution

It introduces the concept of differential graded orders, develops their classical ring theory analogues, and constructs class groups and idèles in this new context.

Findings

Any differential graded order is contained in a maximal one.

Established properties of Jacobson radical analogues in the differential graded setting.

Defined class groups and idèles, and proved their relationships and sequences similar to classical theory.

Abstract

For a Dedekind domain $R$ with field of fractions $K$ a classical $R$-order in a semisimple $K$-algebra $A$ is an $R$-projective $R$-subalgebra $\Lambda$ of $A$ such that $K\Lambda=A$. We study differential graded $K$-algebras which are semisimple as $K$-algebras and define differential graded $R$-orders as a differential graded $R$-subalgebras, which are in addition classical $R$-orders in $A$. We give a series of examples for such differential graded algebras and orders. We show that any differential graded $R$-order is contained in a maximal differential graded order. We develop parts of the classical ring theory in the differential graded setting, in particular the properties of analogues of the Jacobson radical. We further define class groups of differential graded orders as subgroups of the Grothendieck group of locally free differential graded modules. We define id\`eles in this setting showing that these id\`ele groups maps surjectively to the differential graded class group. Finally we give a homomorphism to the class group of the homology of the differential graded order and prove a Mayer-Vietoris like sequence for each central idempotent of $A$, including the analogous one for the kernel groups of these morphisms.