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

Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance