The Stacks project

Lemma 54.9.2. In Situation 54.9.1. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then

  1. $H^ p(X, \mathcal{F}) = 0$ for $p \not\in \{ 0, 1\} $, and

  2. $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ is globally generated.

Proof. Part (1) follows from Cohomology of Schemes, Lemma 30.20.9. If $\mathcal{F}$ is globally generated, then there is a surjection $\bigoplus _{i \in I} \mathcal{O}_ X \to \mathcal{F}$. By part (1) and the long exact sequence of cohomology this induces a surjection on $H^1$. Since $H^1(X, \mathcal{O}_ X) = 0$ as $S$ has a rational singularity, and since $H^1(X, -)$ commutes with direct sums (Cohomology, Lemma 20.19.1) we conclude. $\square$

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