On the universal regular homomorphism in codimension $2$

TL;DR

This paper identifies a gap in Murre's proof regarding the universal regular homomorphism for codimension 2 cycles and provides two methods to address this issue.

Contribution

The paper highlights a flaw in existing proof and proposes two solutions to establish the universal regular homomorphism in codimension 2.

Findings

Identified a gap in Murre's proof.

Proposed two methods to fill the gap.

Abstract

We point out a gap in Murre's proof of the existence of a universal regular homomorphism for codimension $2$ cycles on a smooth projective variety, and offer two arguments to fill this gap.