A cycle class map from Chow groups with modulus to relative $K$-theory

Abstract

Let $\bar{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective Cartier divisor on it. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair $(\bar{X},D)$ in the range $(d+n,n)$ to the relative $K$-groups $K_n(\bar{X},D)$ for every $n\ge 0$.

24 pages. Final version to appear in Documenta Math