every projective module of rank $2$ is endowed with a non-degenerated antisymmetric form. In Section \[gwss\], we recall the definition and the basic properties of the Gersten-Witt complex. The last section shows that the group $\K A$ and the set $Um_4(A)/Sp_4(A)$ involved in the first theorem vanish when $A=S^3$ is the coordinate ring of the real algebraic sphere of dimension $d$. Since $\K {S^3}$ coincide with the derived Witt group $W^2(S^3)$ as defined by Balmer ([@BalSurvey]), we compute the latter. It must be pointed out that C. Walter announced the computation of the derived Witt groups of quadrics, and therefore we only compute the groups needed in our main result without going to much into the details. Our computation is rather technical, and probably one can find a better approach. We conclude with the proof of our theorem that all the finitely generated projective modules on $S^3$ are free.

Conventions
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All the rings considered are commutative and noetherian. The dimension of a ring will always be its Krull dimension. If $X$ is a scheme and $x\in X$, we denote by $\kappa(x)$ the residue field of $x$. If $X$ is integral, we simply denote by $\kappa(X)$ the residue field at 