The Stacks project

67.21 Ample invertible sheaves and cohomology

Here is a criterion for ampleness on proper algebraic spaces over affine bases in terms of vanishing of cohomology after twisting.

Lemma 67.21.1. Let $R$ be a Noetherian ring. Let $X$ be a proper algebraic space over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The following are equivalent

  1. $X$ is a scheme and $\mathcal{L}$ is ample on $X$,

  2. for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$, and

  3. for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$.

Proof. The implication (1) $\Rightarrow $ (2) follows from Cohomology of Schemes, Lemma 30.17.1. The implication (2) $\Rightarrow $ (3) is trivial. The implication (3) $\Rightarrow $ (1) is Lemma 67.16.9. $\square$

Lemma 67.21.2. Let $R$ be a Noetherian ring. Let $f : Y \to X$ be a morphism of algebraic spaces proper over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $f$ is finite and surjective. The following are equivalent

  1. $X$ is a scheme and $\mathcal{L}$ is ample, and

  2. $Y$ is a scheme and $f^*\mathcal{L}$ is ample.

Proof. Assume (1). Then $Y$ is a scheme as a finite morphism is representable (by schemes), see Morphisms of Spaces, Lemma 65.45.3. 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 67.21.1. We are going to apply Lemma 67.14.5. 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 67.17.1 and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma 67.14.5 which will finish the proof. Property (3)(a) we have seen in Lemma 67.17.1. 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 67.17.2 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 67.12.9 and 67.12.8. As $Y$ is a scheme and $\mathcal{L}$ is ample we see from Lemma 67.21.1 that there exists an $n_0$ such that

\[ H^ p(Y, i'_*\mathcal{I}' \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n}) = 0 \]

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

\begin{align*} H^ p(X, i_*\mathcal{G}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) & = H^ p(Z, \mathcal{G'} \otimes _{\mathcal{O}_ Z} i^*\mathcal{L}^{\otimes n}) \\ & = H^ p(Z, f'_*\mathcal{I}' \otimes _{\mathcal{O}_ Z} i^*\mathcal{L}^{\otimes n})) \\ & = H^ p(Z, f'_*(\mathcal{I}' \otimes _{\mathcal{O}_{Z'}} (f')^*i^*\mathcal{L}^{\otimes n})) \\ & = H^ p(Z, f'_*(\mathcal{I}' \otimes _{\mathcal{O}_{Z'}} (i')^*f^*\mathcal{L}^{\otimes n})) \\ & = H^ p(Z', \mathcal{I}' \otimes _{\mathcal{O}_{Z'}} (i')^*f^*\mathcal{L}^{\otimes n})) \\ & = H^ p(Y, i'_*\mathcal{I}' \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*}

Here we have used the projection formula and the Leray spectral sequence (see Cohomology on Sites, Sections 21.48 and 21.14) and Lemma 67.4.1. This verifies property (3)(b) of Lemma 67.14.5 as desired. $\square$


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