the Gersten-Witt complex respect the powers of the fundamental ideal ([@Fa1 Theorem 9.2.4], or [@Gi2]), and therefore we get a complex $$\xymatrix@C=1.3em{0\ar[r] & I^j(\kappa(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} I^{j-1}(\kappa(x_1))   }\ar[r] & \ldots  \ar[r] &  \displaystyle{\bigoplus_{x_d\in X^{(d)}} I^{j-d}(\kappa(x_d))   }\ar[r] & 0}$$ for any $j$ (where $I^l(\kappa(x_p))$ is defined to be $W(\kappa(x_p))$ if $l\leq 0$). We denote by $H^i(X,I^j)$ the cohomology groups of this complex. We get $H^i(X,W)=H^i(X,I^j)$ when $j\leq 0$ by definition.

Let $\overline I^j(\kappa(x_p))=I^j(\kappa(x_p))/I^{j+1}(\kappa(x_p))$. Since the Gersten-Witt complex respects the powers of the fundamental ideal, we get a complex $$\xymatrix@C=1.3em{0\ar[r] &\overline I^j(\kappa(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}}\overline I^{j-1}(\kappa(x_1))   }\ar[r] & \ldots  \ar[r] &  \displaystyle{\bigoplus_{x_d\in X^{(d)}}\overline I^{j-d}(\kappa(x_d))   }\ar[r] & 0}$$ which coincides with the Gersten complex in étale cohomology with coefficient in $\mu_2$ by Voevodsky’s results ([@OVV] and [@Vo]). For any $i,j\in\Z$, let $H^i(X,\overline I^j)$ denote the cohomology groups of this complex. Observe that $H^i(X,\overline I^j)=0$ if $i>j$ and that $H^i(X,\overline I^i)=CH^i(X)/2$ ([@Gi2 Theorem 9.1]). By construction, we have for any $j$ a long exact sequence in cohomology $$\xymatrix@=1.3em{0\ar[r] & H^0(X,I^{j+1})\ar[r] & H^0(X,I^j)\ar[r] & H^0(X,\overline I^j)\ar[r] & H^1(X,I^{j+1})\ar[r] & \ldots }$$ This sequence (and the fact that $H^i(X,\overline I^j)=0$ if $i>j$) yields the following Lemma whose proof is obvious:

\[comparison\] Let $X$ be a 