Theorem 59.83.10. If $k$ is an algebraically closed field, $X$ is a separated, finite type scheme of dimension $\leq 1$ over $k$, and $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$, then

1. $H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 2$,

2. $H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 1$ if $X$ is affine,

3. $H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 1$ if $p = \text{char}(k) > 0$ and $\mathcal{F}$ is $p$-power torsion,

4. $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is finite if $\mathcal{F}$ is constructible and torsion prime to $\text{char}(k)$,

5. $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is finite if $X$ is proper and $\mathcal{F}$ constructible,

6. $H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k'/k$ of algebraically closed fields if $\mathcal{F}$ is torsion prime to $\text{char}(k)$,

7. $H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k'/k$ of algebraically closed fields if $X$ is proper,

8. $H^2_{\acute{e}tale}(X, \mathcal{F}) \to H^2_{\acute{e}tale}(U, \mathcal{F})$ is surjective for all $U \subset X$ open.

Proof. The theorem says that in Situation 59.83.1 statements (1) – (8) hold. Our first step is to replace $X$ by its reduction, which is permissible by Proposition 59.45.4. By Lemma 59.73.2 we can write $\mathcal{F}$ as a filtered colimit of constructible abelian sheaves. Taking cohomology commutes with colimits, see Lemma 59.51.4. Moreover, pullback via $X_{k'} \to X$ commutes with colimits as a left adjoint. Thus it suffices to prove the statements for a constructible sheaf.

In this paragraph we use Lemma 59.83.4 without further mention. Writing $\mathcal{F} = \mathcal{F}_1 \oplus \ldots \oplus \mathcal{F}_ r$ where $\mathcal{F}_ i$ is $\ell _ i$-primary for some prime $\ell _ i$, we may assume that $\ell ^ n$ kills $\mathcal{F}$ for some prime $\ell$. Now consider the exact sequence

$0 \to \mathcal{F}[\ell ] \to \mathcal{F} \to \mathcal{F}/\mathcal{F}[\ell ] \to 0.$

Thus we see that it suffices to assume that $\mathcal{F}$ is $\ell$-torsion. This means that $\mathcal{F}$ is a constructible sheaf of $\mathbf{F}_\ell$-vector spaces for some prime number $\ell$.

By definition this means there is a dense open $U \subset X$ such that $\mathcal{F}|_ U$ is finite locally constant sheaf of $\mathbf{F}_\ell$-vector spaces. Since $\dim (X) \leq 1$ we may assume, after shrinking $U$, that $U = U_1 \amalg \ldots \amalg U_ n$ is a disjoint union of irreducible schemes (just remove the closed points which lie in the intersections of $\geq 2$ components of $U$). By Lemma 59.83.6 we reduce to the case $\mathcal{F} = j_!\mathcal{G}$ where $\mathcal{G}$ is a finite locally constant sheaf of $\mathbf{F}_\ell$-vector spaces on $U$.

Since we chose $U = U_1 \amalg \ldots \amalg U_ n$ with $U_ i$ irreducible we have

$j_!\mathcal{G} = j_{1!}(\mathcal{G}|_{U_1}) \oplus \ldots \oplus j_{n!}(\mathcal{G}|_{U_ n})$

where $j_ i : U_ i \to X$ is the inclusion morphism. The case of $j_{i!}(\mathcal{G}|_{U_ i})$ is handled in Lemma 59.83.9. $\square$

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).