The Stacks project

Lemma 30.9.10. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf with $\dim (\text{Supp}(\mathcal{F})) \leq 0$. Then $\mathcal{F}$ is generated by global sections and $H^ i(X, \mathcal{F}) = 0$ for $i > 0$.

Proof. By Lemma 30.9.7 we see that $\mathcal{F} = i_*\mathcal{G}$ where $i : Z \to X$ is the inclusion of the scheme theoretic support of $\mathcal{F}$ and where $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module. Since the dimension of $Z$ is $0$, we see $Z$ is a disjoint union of affines (Properties, Lemma 28.10.5). Hence $\mathcal{G}$ is globally generated and the higher cohomology groups of $\mathcal{G}$ are zero (Lemma 30.2.2). Hence $\mathcal{F} = i_*\mathcal{G}$ is globally generated. Since the cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree (Lemma 30.2.4 applies as a closed immersion is affine) we conclude that the higher cohomology groups of $\mathcal{F}$ are zero. $\square$

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