# A remark on 0-cycles as modules over algebras of finite correspondences

## TL;DR

This paper studies the structure of 0-cycles on a smooth projective variety as modules over the algebra of finite correspondences, revealing that rationally trivial 0-cycles form a simple and essential submodule.

## Contribution

It establishes that the submodule of rationally trivial 0-cycles is absolutely simple and essential within the module of all 0-cycles over the algebra of finite correspondences.

## Key findings

- Rationally trivial 0-cycles form an absolutely simple submodule.
- This submodule is essential in the module of all 0-cycles.
- The structure reveals intrinsic properties of 0-cycles under algebraic correspondences.

## Abstract

Given a smooth projective variety $X$ over a field, consider the $\mathbb Q$-vector space $Z_0(X)$ of 0-cycles (i.e. formal finite $\mathbb Q$-linear combinations of the closed points of $X$) as a module over the algebra of finite correspondences. Then the rationally trivial 0-cycles on $X$ form an absolutely simple and essential submodule of $Z_0(X)$.