  Let $S$ be a finite dimensional noetherian scheme. For any proper morphism
between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher
algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis
neither on the schemes nor on the morphism. We also prove, without projective
assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher
$K$-theory and motivic cohomology as well as an analogue result for the
relative cohomology of a morphism.
  These results are obtained as corollaries of a motivic statement that is
valid for morphisms between oriented absolute spectra in the stable homotopy
category of $S$.
