---
abstract: 'Let $A$ be a commutative noetherian ring of Krull dimension $3$. We give a necessary and sufficient condition for $A$-projective modules of rank $2$ to be free. Using this, we show that all the finitely generated projective modules over the algebraic real $3$-sphere are free.'
address: |
    Jean Fasel\
    EPFL SB SMA-GE\
    MA C3 595 (Bâtiment MA)\
    Station 8\
    CH-1015 Lausanne 
author:
- 'J. Fasel'
bibliography:
- 'biblio\_cancellation.bib'
title: Projective modules over the real algebraic sphere of dimension $3$
---

Introduction
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Let $A$ be a noetherian commutative ring of Krull dimension $d$ and let $P$ be a projective $A$-module of rank $r$. Finding “computable” necessary and sufficient conditions for $P$ to be free is in general a very hard question. The first obvious obstruction is the class of $P$ in $\widetilde K_0(A)$ and, if $r>d$ its vanishing is also a sufficient condition by Bass-Schanuel cancellation theorem. Thus one is reduced to the case of stably free modules of rank $r\leq d$. In the critical case $r=d$, the situation is rather well understood. Observe first that a stably free module of rank $r=d$ can be seen as the kernel of a surjection $A^{d+1}\to A$ and that such surjections 