The Stacks project

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

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

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

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 $U$. $\square$