much harder. For example, the question whether there exists a complex smooth algebra $A$ of dimension $3$ and a stably free non free projective $A$-module of rank $2$ has been solved only recently ([@Fa2 Theorem 5.4]). In this paper, we prove the following theorem (Theorem \[cancellation\] in the text):

Let $A$ be a ring of dimension $3$. Then every projective module of rank $2$ with trivial determinant is free if and only if the group $\K A$ and the set $Um_4(A)/\Sp A$ are both zero.

The group $\K A$ appearing in the theorem is just the Grothendieck-Witt group $GW^2(A)$ modulo the subgroup generated by the antisymmetric form $H(A)$ and $Um_4(A)/\Sp A$ is just the set of orbits of $Um_4(A)$ under the action of $\Sp A$. The first group is “computable” in the sense that it is part of a cohomology theory.

As an application of the above result, we get the following theorem (Theorem \[main\] in the text):

Every finitely generated projective module over the real algebraic sphere of dimension $3$ is free.

It should be pointed out that the stable structure of the algebraic vector bundles on spheres is well-known ([@Fo]; [@Sw2]), but that the set of isomorphism classes 