& \displaystyle{\bigoplus_{x_0\in X^{(0)}\cap Y} I^j(\kappa(x_0))   }\ar[r] & \ldots  \ar[r] &  \displaystyle{\bigoplus_{x_d\in X^{(d)}\cap Y} I^{j-d}(\kappa(x_d))   }\ar[r] & 0.}$$ We denote by $H^i_Y(X,I^j)$ the cohomology groups of this complex. In case $Y$ is the zero locus of a global section $f\in \O_X(X)$, we simply denote by $H^i_f(X,I^j)$ the group $H^i_Y(X,I^j)$. All the results of Section \[gwss\] remains true for groups with support. In addition, there is a long exact sequence $$\xymatrix@C=1.4em{\ldots\ar[r] & H^i_Y(X,I^j)\ar[r] & H^i(X,I^j)\ar[r] & H^i(U,I^j)\ar[r] & H^{i+1}_Y(X,I^j)\ar[r] & \ldots}$$ for any $j\in\Z$, where $U=X-Y$. If $Y\to X$ is smooth, one can compare the cohomology groups with support on $Y$ with the cohomology groups of $Y$ ([@Fa1 Remark 9.3.5]). We will need only the following particular result ([@Gi1 Theorem 4.1] and [@Fa1 Remark 9.3.5]) :

\[reg\_param\] Let $X=\spec A$ be a regular affine scheme over a field $k$. Let $f\in A$ be a regular element such that $Y=\spec {A/f}$ is regular. Then $$H^i_f(X,I^j)\simeq H^{i-1}(Y,I^{j-1})$$ for any $i,j\in\Z$.

The real sphere of dimension $3$ {#witt}
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Our goal of the section is to compute the group $\K {S^3}$, which coincide with the Witt group $W^2(S^3)$ (see Corollary \[s3\]). As $S^3$ is of dimension $3$, $W^2(S^3)=H^2(S^3,W)$ which is really the group that we 