we see that $\mathfrak p_1C^2_{\mathfrak p_1}$ is generated by $x_0^2+1$. We thus get an isomorphism (necessarily symmetric) of $\kappa(\mathfrak p_1)$-vector spaces $$\rho_{x_0^2+1}:\kappa(\mathfrak p_1)\to \exten 1{C^2_{\mathfrak p_1}}{\kappa(\mathfrak p_1)}{C^2_{\mathfrak p_1}}$$ defined by $1\mapsto Kos(x_0^2+1)$, where the latter is the Koszul complex associated to the regular parameter $x_0^2+1$.

Consider next $\mathfrak p_2=(x_2)$. As above, we get a symmetric isomorphism of $\kappa(\mathfrak p_2)$-vector spaces $$\rho_{x_2}:\kappa(\mathfrak p_2)\to \exten 1{C^2_{\mathfrak p_2}}{\kappa(\mathfrak p_2)}{C^2_{\mathfrak p_2}}$$ defined by $1\mapsto Kos(x_2)$, where the latter is the Koszul complex associated to the regular parameter $x_2$.

The isomorphism $H^1(C^2,I)\simeq H^1(C^2,\overline I)$ (and the fact that $\mathfrak p_2$ is principal) shows that the class of the cycle $(\kappa(\mathfrak p_1),\rho_{x_0^2+1})+(\kappa(\mathfrak p_2),x_1\rho_{x_2})$ is the generator of $H^1(C^2,I)$ (observe also that we could have considered $(\kappa(\mathfrak p_1),\rho_{x_0^2+1})$ alone as a generator, but we will need $(\kappa(\mathfrak p_2),\rho_{x_2})$ later and this is why we consider it).

Observe next that $B^1\simeq \R[x_0,x_0^{-1}]$ and $B^n/x_n=C^{n-2}[x_{n-1}]$ for $n\geq 2$, while $B^n_{x_n}\simeq \R[x_0,\ldots,x_{n-2},x_n,x_n^{-1}]=B^1[x_0,\ldots,x_{n-2}]$.

\[comput\_bn\] For $1\leq n\leq 3$, we have $$H^i(B^n,W)=\left\{ \begin{array}{cl} \Z\oplus \Z & \mbox{if $i=0$,}\\
0 & \mbox{if $i\geq 1$.}
\end{array}\right.$$

Since $B^1=\R[x_0,x_0^{-1}]$, the result for $B^1$ follows from [@Gi1 Corollary 5.2]. Since $B^n_{x_n}\simeq B^1[x_0,\ldots,x_{n-2}]$, the exact sequence of localization (and the homotopy invariance of the Witt cohomology, see 