The Stacks project

Proof. The vanishing in (1) is Limits, Lemma 32.19.2. 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 37.72.2, and since $H^1(X, -)$ commutes with direct sums (Cohomology, Lemma 20.19.1) we conclude. $\square$