The Stacks project

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

  1. 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 $H^ q_{\acute{e}tale}(Z', \underline{M}) = 0$.

  2. 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 $H^ q_{\acute{e}tale}(X' \times _ X Z, \underline{M})$.

Proof. Write $X = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathbf{Z}[x_ i]/J$ for some ideal $J$. Let $R$ be the integral closure of $\mathbf{Z}[x_ i]$ in an algebraic closure of the fraction field of $\mathbf{Z}[x_ i]$. Let $A' = R/JR$ and set $X' = \mathop{\mathrm{Spec}}(A')$. This gives an example as in (1) by Lemma 57.78.7.

Proof of (2). Let $X' \to X$ be the integral surjective morphism we found above. Certainly, $\xi $ maps to zero in $H^ q_{\acute{e}tale}(X' \times _ X Z, \underline{M})$. We may write $X'$ as a limit $X' = \mathop{\mathrm{lim}}\nolimits X'_ i$ of schemes finite and of finite presentation over $X$; this is easy to do in our current affine case, but it is a special case of the more general Limits, Lemma 32.7.2. By Lemma 57.51.5 we see that $\xi $ maps to zero in $H^ q_{\acute{e}tale}(X'_ i \times _ X Z, \underline{M})$ for some $i$ large enough. $\square$


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