Arakelov motivic cohomology I

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. In a sequel to this paper, we relate Arakelov motivic cohomology to classical constructions such as arithmetic K and Chow groups and the height pairing.