Zero-cycles with modulus and relative $K$-theory

Abstract

Let $D$ be an effective Cartier divisor on a regular quasiprojective scheme $X$ of dimension $d\ge 1$ over a field. For an integer $n\ge 0$, we construct a cycle class map from the higher Chow groups with modulus ${\left\{{CH}^{n+d}\left(X|mD,n\right)\right\}}_{m\ge 1}$ to the relative $K$-groups ${\left\{K_n\left(X,mD\right)\right\}}_{m\ge 1}$ in the category of pro-abelian groups. We show that this induces a proisomorphism between the additive higher Chow groups of relative $0$-cycles and the reduced algebraic $K$-groups of truncated polynomial rings over a regular semilocal ring which is essentially of finite type over a characteristic zero field.