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

## TL;DR

This paper constructs a cycle class map linking higher Chow groups of 0-cycles to relative K-theory in the context of modulus pairs, establishing a pro-isomorphism in a specific algebraic setting.

## Contribution

It introduces a new cycle class map from higher Chow groups to relative K-theory and proves a pro-isomorphism for additive higher Chow groups of relative 0-cycles.

## Key findings

- Cycle class map from higher Chow groups to relative K-theory constructed.
- Pro-isomorphism established between additive higher Chow groups and relative K-theory.
- Results hold over regular semi-local rings of finite type over a characteristic zero field.

## Abstract

We construct a cycle class map from the higher Chow groups of 0-cycles to the relative $K$-theory of a modulus pair. We show that this induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and relative $K$-theory of truncated polynomial rings over a regular semi-local ring, essentially of finite type over a characteristic zero field.