Lemma 57.78.8. Let $X$ be an affine scheme.

There exists an integral surjective morphism $X' \to X$ such that for every closed subscheme $Z' \subset X'$, every finite abelian group $M$, and every $q \geq 1$ we have $H^ q_{\acute{e}tale}(Z', \underline{M}) = 0$.

For any closed subscheme $Z \subset X$, finite abelian group $M$, $q \geq 1$, and $\xi \in H^ q_{\acute{e}tale}(Z, \underline{M})$ there exists a finite surjective morphism $X' \to X$ of finite presentation such that $\xi $ pulls back to zero in $H^ q_{\acute{e}tale}(X' \times _ X Z, \underline{M})$.

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