The Stacks project

Proof. Assume (1). Then $Y$ is a scheme as a finite morphism is representable (by schemes), see Morphisms of Spaces, Lemma 66.45.3. 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 68.21.1. We are going to apply Lemma 68.14.5. 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 68.17.1 and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma 68.14.5 which will finish the proof. Property (3)(a) we have seen in Lemma 68.17.1. 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 68.17.2 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 68.12.9 and 68.12.8. As $Y$ is a scheme and $\mathcal{L}$ is ample we see from Lemma 68.21.1 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 68.4.1. This verifies property (3)(b) of Lemma 68.14.5 as desired. $\square$