# The cancellation of projective modules of rank 2 with a trivial determinant

## TL;DR

This paper investigates when projective modules of rank 2 with trivial determinant over certain rings are cancellative and free, linking this property to Hermitian K-theory groups.

## Contribution

It establishes a criterion for the freeness of stably free modules of rank 2 over smooth affine algebras of dimension 4, based on the triviality of a Hermitian K-theory group.

## Key findings

- Stably free modules of rank 2 are free iff a specific Hermitian K-theory group is trivial.
- The result applies to smooth affine algebras over algebraically closed fields with 6 invertible.
- Provides a new connection between module cancellation and Hermitian K-theory.

## Abstract

We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove that stably free $R$-modules of rank $2$ are free if and only if a Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ is trivial.