Lemma 59.83.6. In Situation 59.83.1 assume $\mathcal{F}$ constructible. Let $j : X' \to X$ be the inclusion of a dense open subscheme. Then statements (1) – (8) hold for $\mathcal{F}$ if and only if they hold for $j_!j^{-1}\mathcal{F}$.

Proof. Since $X'$ is dense, we see that $Z = X \setminus X'$ has dimension $0$ and hence is a finite set $Z = \{ x_1, \ldots , x_ n\}$ of $k$-rational points. Consider the short exact sequence

$0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0$

of Lemma 59.70.8. Observe that $H^ q_{\acute{e}tale}(X, i_*i^{-1}\mathcal{F}) = H^ q_{\acute{e}tale}(Z, i^*\mathcal{F})$. Namely, $i : Z \to X$ is a closed immersion, hence finite, hence we have the vanishing of $R^ qi_*$ for $q > 0$ by Proposition 59.55.2, and hence the equality follows from the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). Since $Z$ is a disjoint union of spectra of algebraically closed fields, we conclude that $H^ q_{\acute{e}tale}(Z, i^*\mathcal{F}) = 0$ for $q > 0$ and

$H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) = \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{F}_{x_ i}$

which is finite as $\mathcal{F}_{x_ i}$ is finite due to the assumption that $\mathcal{F}$ is constructible. The long exact cohomology sequence gives an exact sequence

$0 \to H^0_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^0_{\acute{e}tale}(X, \mathcal{F}) \to H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) \to H^1_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^1_{\acute{e}tale}(X, \mathcal{F}) \to 0$

and isomorphisms $H^ q_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^ q_{\acute{e}tale}(X, \mathcal{F})$ for $q > 1$.

At this point it is easy to deduce each of (1) – (8) holds for $\mathcal{F}$ if and only if it holds for $j_!j^{-1}\mathcal{F}$. We make a few small remarks to help the reader: (a) if $\mathcal{F}$ is torsion prime to the characteristic of $k$, then so is $j_!j^{-1}\mathcal{F}$, (b) the sheaf $j_!j^{-1}\mathcal{F}$ is constructible, (c) we have $H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) = H^0_{\acute{e}tale}(Z_{k'}, i^{-1}\mathcal{F}|_{Z_{k'}})$, and (d) if $U \subset X$ is an open, then $U' = U \cap X'$ is dense in $U$. $\square$

Comment #6842 by Patrick Herter on

Right after the long exact sequence it should be "and isomorphisms $H^q_{étale} \rightarrow H^q_{étale}$ for $q>1$". At the moment it is "$H^0_{étale} \rightarrow H^0_{étale}$".

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