The Stacks project

Lemma 30.17.2. Let $R$ be a Noetherian ring. Let $f : Y \to X$ be a morphism of schemes proper over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $f$ is finite and surjective. Then $\mathcal{L}$ is ample if and only if $f^*\mathcal{L}$ is ample.

Proof. The pullback of an ample invertible sheaf by a quasi-affine morphism is ample, see Morphisms, Lemma 29.37.7. This proves one of the implications as a finite morphism is affine by definition.

Assume that $f^*\mathcal{L}$ is ample. 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 30.17.1. We are going to apply Lemma 30.12.8. 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 $Z \subset X$ be an integral closed subscheme with generic point $\xi $. Let $\mathcal{F}$ be a coherent sheaf on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi $ is annihilated by $\mathfrak m_\xi $, see Lemma 30.13.1. We claim that taking $\mathcal{G} = f_*\mathcal{F}$ works. We only have to verify part (3)(c) of Lemma 30.12.8. Hence assume that $\mathcal{J} \subset \mathcal{O}_ X$ is a quasi-coherent sheaf of ideals such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi }$. A finite morphism is affine hence by Lemma 30.13.2 we see that $\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})$. Also, as pointed out in the proof of Lemma 30.13.2 the sheaf $f^{-1}\mathcal{J}\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module. As $\mathcal{L}$ is ample we see from Lemma 30.17.1 that there exists an $n_0$ such that

\[ H^ p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n}) = 0, \]

for $n \geq n_0$ and $p > 0$. Since $f$ is finite, hence affine, we see that

\begin{align*} H^ p(X, \mathcal{J}\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) & = H^ p(X, f_*(f^{-1}\mathcal{J}\mathcal{F}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \\ & = H^ p(X, f_*(f^{-1}\mathcal{J}\mathcal{F} \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n})) \\ & = H^ p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*}

Here we have used the projection formula (Cohomology, Lemma 20.52.2) and Lemma 30.2.4. Hence the quasi-coherent subsheaf $\mathcal{G}' = \mathcal{J}\mathcal{G}$ satisfies $P$. This verifies property (3)(c) of Lemma 30.12.8 as desired. $\square$

Comments (2)

Comment #2725 by Ariyan Javanpeykar on

This is closely related to EGA II, Corollaire 6.6.3. But some of the conditions are unclear. Johan, can you verify?

Comment #2851 by on

Yeah, it is related but it is different enough that it wouldn't be the correct reference.

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