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

## TL;DR

This paper constructs a cycle class map connecting higher Chow groups with modulus to relative K-theory groups for smooth quasi-projective varieties, extending the understanding of algebraic cycles and K-theory in this context.

## Contribution

It introduces a new cycle class map from higher Chow groups with modulus to relative K-theory groups for pairs of varieties and divisors.

## Key findings

- Established the cycle class map for all n ≥ 0.
- Extended the relationship between Chow groups with modulus and K-theory.
- Provided a new tool for studying algebraic cycles with modulus.

## 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 $n\geq 0$.