The Stacks project

Lemma 37.69.3. Let $f : X \to Y$ be a proper morphism of schemes such that $\dim (X_ y) \leq 1$ and $H^1(X_ y, \mathcal{O}_{X_ y}) = 0$ for all $y \in Y$. Let $\mathcal{F}$ be quasi-coherent on $X$. Then

  1. $R^ pf_*\mathcal{F} = 0$ for $p > 1$, and

  2. $R^1f_*\mathcal{F} = 0$ if there is a surjection $f^*\mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ quasi-coherent on $Y$.

If $Y$ is affine, then we also have

  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. The vanishing in (1) is Limits, Lemma 32.18.2. To prove (2) we may work locally on $Y$ and assume $Y$ is affine. Then $R^1f_*\mathcal{F}$ is the quasi-coherent module on $Y$ associated to the module $H^1(X, \mathcal{F})$. Here we use that $Y$ is affine, quasi-coherence of higher direct images (Cohomology of Schemes, Lemma 30.4.5), and Cohomology of Schemes, Lemma 30.4.6. Since $Y$ is affine, the quasi-coherent module $\mathcal{G}$ is globally generated, and hence so is $f^*\mathcal{G}$ and $\mathcal{F}$. In this way we see that (4) implies (2). Part (3) follows from (1) as well as the remarks on quasi-coherence of direct images just made. Thus all that remains is the prove (4). 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 $H^1(X, \mathcal{O}_ X) = 0$ because $R^1f_*\mathcal{O}_ X = 0$ by Lemma 37.69.2, and since $H^1(X, -)$ commutes with direct sums (Cohomology, Lemma 20.19.1) we conclude. $\square$


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