Differential graded orders, their class groups and idèles

For a Dedekind domain [Formula: see text] with field of fractions [Formula: see text], a classical [Formula: see text]-order in a semisimple [Formula: see text]-algebra [Formula: see text] is an [Formula: see text]-projective [Formula: see text]-subalgebra [Formula: see text] of [Formula: see text] such that [Formula: see text]. We study differential graded [Formula: see text]-algebras which are semisimple as [Formula: see text]-algebras and define differential graded [Formula: see text]-orders as a differential graded [Formula: see text]-subalgebras, which are in addition classical [Formula: see text]-orders in [Formula: see text]. We give a series of examples for such differential graded algebras and orders. We show that any differential graded [Formula: see text]-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 analogs 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 idèles in this setting showing that these idèle groups map 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 [Formula: see text], including the analogous one for the kernel groups of these morphisms.