## Chern Class and Riemann-Roch Theorem for Cohomology Theory without Homotopy Invariance

J. Math. Sci. Univ. Tokyo
Vol. 26 (2019), No. 3, Page 249-334.

Asakura, Masanori; Sato, Kanetomo; with Appendix B by Hagihara, Kei
Chern Class and Riemann-Roch Theorem for Cohomology Theory without Homotopy Invariance

Abstract:
In this paper, we formulate axioms of certain graded cohomology theory and define higher Chern class maps following the method of Gillet [Gi1]. We will not include homotopy invariance nor purity in our axioms. It will turn out that the Riemann-Roch theorem without denominators holds for our higher Chern classes. We will give two applications of our Riemann-Roch results in $\S\S$11-13.

Keywords: $K$-theory, cohomology theory, Chern class, Riemann-Roch theorem.

Mathematics Subject Classification (2010): 14C40, 19E20.
Mathematical Reviews Number: MR4246746