  In this paper, we formulate axioms of certain graded cohomology theory for
which Chern class maps from higher K-theory are defined, following the method
of Gillet [Gi1]. We will not include homotopy invariance nor purity in our
axioms. It will turn out that the Grothendieck-Riemann-Roch theorem and the
Riemann-Roch theorem without denominators hold for our higher Chern classes. We
will also give two applications of our Riemann-Roch results.
