The Stacks project

Theorem 59.83.11. Let $X$ be a finite type, dimension $1$ scheme over an algebraically closed field $k$. Let $\mathcal{F}$ be a torsion sheaf on $X_{\acute{e}tale}$. Then

\[ H_{\acute{e}tale}^ q(X, \mathcal{F}) = 0, \quad \forall q \geq 3. \]

If $X$ affine then also $H_{\acute{e}tale}^2(X, \mathcal{F}) = 0$.

Proof. If $X$ is separated, this follows immediately from the more precise Theorem 59.83.10. If $X$ is nonseparated, choose an affine open covering $X = X_1 \cup \ldots \cup X_ n$. By induction on $n$ we may assume the vanishing holds over $U = X_1 \cup \ldots \cup X_{n - 1}$. Then Mayer-Vietoris (Lemma 59.50.1) gives

\[ H^2_{\acute{e}tale}(U, \mathcal{F}) \oplus H^2_{\acute{e}tale}(X_ n, \mathcal{F}) \to H^2_{\acute{e}tale}(U \cap X_ n, \mathcal{F}) \to H^3_{\acute{e}tale}(X, \mathcal{F}) \to 0 \]

However, since $U \cap X_ n$ is an open of an affine scheme and hence affine by our dimension assumption, the group $H^2_{\acute{e}tale}(U \cap X_ n, \mathcal{F})$ vanishes by Theorem 59.83.10. $\square$

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