regular scheme over a field $k$. Then for any $j\in\Z$ we have $H^i(X,W)=H^i(X,I^j)$ if $i\geq j+1$. Moreover, the natural homomorphism $H^j(X,I^j)\to H^j(X,W)$ is surjective.

In case $j=1$, one can say a bit more:

\[deg\_1\] Let $X$ be a regular scheme over a field $k$. Then $H^1(X,I)=H^1(X,W)$.

Using the long exact sequence in cohomology $$\xymatrix{H^0(X,W)\ar[r] & H^0(X,\overline W)\ar[r] & H^1(X,I)\ar[r] & H^1(X,W)\ar[r] & 0,}$$ we see that it suffices to prove that the homomorphism $H^0(X,W)\to H^0(X,\overline W)$ is surjective to conclude. Since the Gersten-Witt complex of a scheme is the direct sum of the Gersten-Witt complexes of its connected components, we can suppose that $X$ is connected. Let $p:X\to k$ be the projection. Then we have a commutative diagram $$\xymatrix{H^0(k,W)\ar[r]\ar[d]_-{p^*} & H^0(k,\overline W)\ar@{=}[d]\\
H^0(X,W)\ar[r] & H^0(X,\overline W)}$$ and the above horizontal homomorphism is clearly surjective.

Supports
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As in the previous section, let $X$ be an integral scheme of dimension $d$ over a field $k$. If $Y\subset X$ is a closed subscheme, one can consider the *Gersten-Witt complex with support on $Y$*: $$\xymatrix@C=0.9em{0\ar[r] & \displaystyle{\bigoplus_{x_0\in X^{(0)}\cap Y} W(\kappa(x_0))   }\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}\cap Y} W(\kappa(x_1))   }\ar[r] & \ldots  \ar[r] &  \displaystyle{\bigoplus_{x_d\in X^{(d)}\cap Y} W(\kappa(x_d))   }\ar[r] & 0}$$ and its filtered version $$\xymatrix@C=1.3em{0\ar[r] 