Lemma 59.83.2. In Situation 59.83.1 assume $X$ is smooth and $\mathcal{F} = \underline{\mathbf{Z}/\ell \mathbf{Z}}$ for some prime number $\ell $. Then statements (1) – (8) hold for $\mathcal{F}$.
Proof. Since $X$ is smooth, we see that $X$ is a finite disjoint union of smooth curves. Hence we may assume $X$ is a smooth curve.
Case I: $\ell $ different from the characteristic of $k$. This case follows from Lemma 59.69.1 (projective case) and Lemma 59.69.3 (affine case). Statement (6) on cohomology and extension of algebraically closed ground field follows from the fact that the genus $g$ and the number of “punctures” $r$ do not change when passing from $k$ to $k'$. Statement (8) follows as $H^2_{\acute{e}tale}(U, \mathcal{F})$ is zero as soon as $U \not= X$, because then $U$ is affine (Varieties, Lemmas 33.43.2 and 33.43.10).
Case II: $\ell $ is equal to the characteristic of $k$. Vanishing by Lemma 59.63.4. Statements (5) and (7) follow from Lemma 59.63.5. $\square$
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