The Stacks project

Proof. Assume (1). Then $Y$ is a scheme as a finite morphism is representable (by schemes), see Morphisms of Spaces, Lemma 67.45.3. Hence (2) follows from Cohomology of Schemes, Lemma 30.17.2.

Assume (2). Let $P$ be the following property on coherent $\mathcal{O}_ X$-modules $\mathcal{F}$: there exists an $n_0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$. We will prove that $P$ holds for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$, which implies $\mathcal{L}$ is ample by Lemma 69.21.1. We are going to apply Lemma 69.14.5. Thus we have to verify (1), (2) and (3) of that lemma for $P$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves and the fact that tensoring with an invertible sheaf is an exact functor. Property (2) follows since $H^ p(X, -)$ is an additive functor.

To see (3) let $i : Z \to X$ be a reduced closed subspace with $|Z|$ irreducible. Let $i' : Z' \to Y$ and $f' : Z' \to Z$ be as in Lemma 69.17.1 and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma 69.14.5 which will finish the proof. Property (3)(a) we have seen in Lemma 69.17.1. To see (3)(b) let $\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$. Denote $\mathcal{I}' \subset \mathcal{O}_{Z'}$ the quasi-coherent ideal $(f')^{-1}\mathcal{I} \mathcal{O}_{Z'}$, i.e., the image of $(f')^*\mathcal{I} \to \mathcal{O}_{Z'}$. By Lemma 69.17.2 we have $f_*\mathcal{I}' = \mathcal{I} \mathcal{G}$. We claim the common value $\mathcal{G}' = \mathcal{I} \mathcal{G} = f'_*\mathcal{I}'$ satisfies the condition expressed in (3)(b). First, it is clear that the support of $\mathcal{G}/\mathcal{G}'$ is contained in the support of $\mathcal{O}_ Z/\mathcal{I}$ which is a proper subspace of $|Z|$ as $\mathcal{I}$ is a nonzero ideal sheaf on the reduced and irreducible algebraic space $Z$. Recall that $f'_*$, $i_*$, and $i'_*$ transform coherent modules into coherent modules, see Lemmas 69.12.9 and 69.12.8. As $Y$ is a scheme and $\mathcal{L}$ is ample we see from Lemma 69.21.1 that there exists an $n_0$ such that

for $n \geq n_0$ and $p > 0$. Now we get

Here we have used the projection formula and the Leray spectral sequence (see Cohomology on Sites, Sections 21.50 and 21.14) and Lemma 69.4.1. This verifies property (3)(b) of Lemma 69.14.5 as desired. $\square$