The Stacks project

Proof. We claim $H^ p(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$-module and $p > 0$. Since $X$ is quasi-compact by Properties, Definition 28.26.1 the claim finishes the proof by Lemma 30.3.1. The scheme $X$ is separated by Properties, Lemma 28.26.8. Say $X$ is covered by $e + 1$ affine opens. Then $H^ p(X, \mathcal{F}) = 0$ for $p > e$, see Lemma 30.4.2. Thus we may use descending induction on $p$ to prove the claim. Writing $\mathcal{F}$ as a filtered colimit of finite type quasi-coherent modules (Properties, Lemma 28.22.3) and using Cohomology, Lemma 20.19.1 we may assume $\mathcal{F}$ is of finite type. Then we can choose $n > n_0$ such that $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ is globally generated, see Properties, Proposition 28.26.13. This means there is a short exact sequence

for some set $I$ (in fact we can choose $I$ finite). By induction hypothesis we have $H^{p + 1}(X, \mathcal{F}') = 0$ and by assumption (combined with the already used commutation of cohomology with colimits) we have $H^ p(X, \bigoplus _{i \in I} \mathcal{L}^{\otimes -n}) = 0$. From the long exact cohomology sequence we conclude that $H^ p(X, \mathcal{F}) = 0$ as desired. $\square$