  This paper introduces a new cohomology theory for schemes of finite type over
an arithmetic ring. The main motivation for this Arakelov-theoretic version of
motivic cohomology is the conjecture on special values of $L$-functions and
zeta functions formulated by the second author. Taking advantage of the six
functors formalism in motivic stable homotopy theory, we establish a number of
formal properties, including pullbacks for arbitrary morphisms, pushforwards
for projective morphisms between regular schemes, localization sequences,
$h$-descent. We round off the picture with a purity result and a higher
arithmetic Riemann-Roch theorem.
