Given a closed immersion $i :Z \hookrightarrow X$ of regular noetherian schemes, of pure codimension $c$. Let $\Lambda$ be the locally constant sheaf $Z/nZ$ and assume $n$ is invertible on $X$. Gabber purity tells us when the following morphism of etale cohomology groups
$H^{r-2c}(Z, \Lambda (-c)) \rightarrow H_{Z}^{r}(X, \Lambda)$
is an isomorphism, where $\Lambda (-c)$ is the tensor product of sheaf of the $n^{th}$ roots of unity $c$ times, for $c < 0$ and for $c > 0$, one takes the dual. I know the existence of above isomorphism in following cases
- $X$ and $Y$ are smooth over a field $k$.
- $X$ is a local ring of dimension 1 and $Y$ is the closed point.
I want to know if the theorem is more generally true. I am particularly interested in the case where $X$ is a local ring of any dimension greater than 1 and $Y$ is the closed point or in fact any other closed scheme (need not be smooth).
P.S (There is another way of stating this theorem in terms of right derived functors of $i^!$, which I have eschewed.)