A Cycle Class Map from Chow Groups with Modulus to Relative $K$-Theory

Abstract

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