A correspondence between Hilbert polynomials and Chern polynomials over projective spaces

Abstract

We construct a map $\zeta$ from $\K(\mathbb P^d)$ to $(\mathbb Z[x]/x^{d+1})^{\times} \times \mathbb Z$, where $(\mathbb Z[x]/x^{d+1})^{\times}$ is a multiplicative Abelian group with identity $1$, and show that $\zeta$ induces an isomorphism between $\K(\mathbb P^d)$ and its image. This is inspired by a correspondence between Chern and Hilbert polynomials stated in Eisenbud~\cite[Exercise~19.18]{E}. The equivalence of these two polynomials over $\mathbb P^d$ is discussed in this paper.