  We construct a motivic Eilenberg-MacLane spectrum with a highly structured
multiplication over smooth schemes over Dedekind domains which represents
Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes.
Our method is by gluing p-completed and rational parts along an arithmetic
square. Hereby the finite coefficient spectra are obtained by truncated \'etale
sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of
Geisser's version of syntomic cohomology, and the rational spectra are the ones
which represent Beilinson motivic cohomology. As an application the arithmetic
motivic cohomology groups can be realized as Ext-groups in a triangulated
category of Tate sheaves with integral coefficients. These can be modelled as
representations of derived fundamental groups. Our spectrum is compatible with
base change giving rise to a formalism of six functors for triangulated
categories of motivic sheaves over general base schemes including the
localization triangle. Further applications include a generalization of the
Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod
algebra in the case where the coefficient characteristic is invertible on the
base scheme.
