# K-theory and 0-cycles on schemes

## TL;DR

This paper proves Bloch's formula for 0-cycles on certain schemes over algebraically closed fields, extending its validity and deriving applications like 0-cycles with modulus.

## Contribution

It establishes Bloch's formula for affine schemes and regular codimension-one schemes, and applies it to 0-cycles with modulus.

## Key findings

- Bloch's formula proven for affine schemes over algebraically closed fields.
- Extended Bloch's formula to projective regular schemes in codimension one.
- Derived applications including Bloch's formula for 0-cycles with modulus.

## Abstract

We prove Bloch's formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in codimension one. Several applications, including Bloch's formula for 0-cycles with modulus, are derived.