The Stacks project

Lemma 54.78.7. Let $X$ be a normal integral affine scheme with separably closed function field. Let $Z \subset X$ be a closed subscheme. For any finite abelian group $M$ we have $H^ q_{\acute{e}tale}(Z, \underline{M}) = 0$ for $q \geq 1$.

Proof. We have seen that the result is true for $H^1$ in Lemma 54.78.6. We will prove the result for $q \geq 2$ by induction on $q$. Let $\xi \in H^ q_{\acute{e}tale}(Z, \underline{M})$.

Let $X = \mathop{\mathrm{Spec}}(R)$. Let $I \subset R$ be the set of elements $f \in R$ sch that $\xi |_{Z \cap D(f)} = 0$. All local rings of $Z$ are strictly henselian by Lemma 54.78.3 and Algebra, Lemma 10.150.16. Hence étale cohomology on $Z$ or open subschemes of $Z$ is equal to Zariski cohomology, see Lemma 54.78.1. In particular $\xi $ is Zariski locally trivial. It follows that for every prime $\mathfrak p$ of $R$ there exists an $f \in I$ with $f \not\in \mathfrak p$. Thus if we can show that $I$ is an ideal, then $1 \in I$ and we're done. It is clear that $f \in I$, $r \in R$ implies $rf \in I$. Thus we now assume that $f, g \in I$ and we show that $f + g \in I$. Note that

\[ D(f + g) \cap Z = D(f(f + g)) \cap Z \cup D(g(f + g)) \cap Z \]

By Mayer-Vietoris (Cohomology, Lemma 20.9.2 which applies as étale cohomology on open subschemes of $Z$ equals Zariski cohomology) we have an exact sequence

\[ \xymatrix{ H^{q - 1}_{\acute{e}tale}(D(fg(f + g)) \cap Z, \underline{M}) \ar[d] \\ H^ q_{\acute{e}tale}(D(f + g) \cap Z, \underline{M}) \ar[d] \\ H^ q_{\acute{e}tale}(D(f(f + g)) \cap Z, \underline{M}) \oplus H^ q_{\acute{e}tale}(D(g(f + g)) \cap Z, \underline{M}) } \]

and the result follows as the first group is zero by induction. $\square$


Comments (2)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09ZC. Beware of the difference between the letter 'O' and the digit '0'.