maximal ideal $(1-x_2,x_1,x_0)$ and then $H^2_{1-x_2,x_1}(S^2,W)\simeq \Z$. Since $1-x_2$ is a regular parameter in $S^2_{x_1}$, we get $$H^1_{1-x_2}(S^2_{x_1},W)\simeq H^0(S^2_{x_1}/(1-x_2),W)\simeq H^0(\C[x_1,x_1^{-1}],W)\simeq \Z/2 \oplus \Z/2,$$ the second factor being generated by the class of the form $\langle x_1\rangle$ ([@Gi1 Theorem 5.6]). We also get $H^2_{1-x_2}(S^2_{x_1},W)=0$ and therefore $H^2_{1-x_2}(S^2,W)\simeq \Z$.

Finally, we can compute the Witt cohomology of $S^2$:

\[s2\] Let $S^2$ be the algebraic real $2$-sphere. Then $$H^i(S^2,W)\simeq \left\{ \begin{array}{ll} \Z\oplus \Z/2 & \mbox{if $i=0$}\\
\Z/2 & \mbox{if $i=1$}\\
\Z & \mbox{if $i=2$}\\
0 & \mbox{if $i\geq 3$}
\end{array}\right.$$

We just have to examine the exact sequence $$\xymatrix@C=1.4em{0\ar[r] & H^0(S^2,W)\ar[r] & H^0(S^2_{1-x_2},W)\ar[r] & H^1_{1-x_2}(S^2,W)\ar[r] & H^1(S^2,W)\ar[r] & \ldots}$$ In light of the above computations, the only difficulty is to understand the map $H^0(S^2_{1-x_2},W)\to H^1_{1-x_2}(S^2,W)$. But it is not hard to see that the form $\langle 1-x_2\rangle$ generating the factor $\Z/4$ in $H^0(S^2_{1-x_2},W)$ is sent to the form $\langle 1\rangle$ generating the first $\Z/2$ factor in $H^1_{1-x_2}(S^2,W)$.

\[explicit\_s2\] Let $\mathfrak q$ be the prime ideal of $S^2$ generated by $(1-x_2)$. The Koszul complex $Kos(1-x_2)$ associated to $(1-x_2)$ can be taken as a generator of $\exten 1{S^2_{\mathfrak q}}{\kappa(\mathfrak q)}{S^2_{\mathfrak q}}$ and we get a symmetric isomorphism $$\rho_{1-x_2}:\kappa(\mathfrak q)\to \exten 1{S^2_{\mathfrak q}}{\kappa(\mathfrak q)}{S^2_{\mathfrak q}}$$ defined 