The Stacks project

Lemma 30.16.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that

  1. $S$ is Noetherian,

  2. $f$ is proper,

  3. $\mathcal{F}$ is coherent, and

  4. $\mathcal{L}$ is relatively ample on $X/S$.

Then there exists an $n_0$ such that for all $n \geq n_0$ we have

\[ R^ pf_*\left(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}\right) = 0 \]

for all $p > 0$.

Proof. Choose a finite affine open covering $S = \bigcup V_ j$ and set $X_ j = f^{-1}(V_ j)$. Clearly, if we solve the question for each of the finitely many systems $(X_ j \to V_ j, \mathcal{L}|_{X_ j}, \mathcal{F}|_{V_ j})$ then the result follows. Thus we may assume $S$ is affine. In this case the vanishing of $R^ pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes n})$ is equivalent to the vanishing of $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$, see Lemma 30.4.6. Thus the required vanishing follows from Lemma 30.16.1 (which applies because $\mathcal{L}$ is ample on $X$ by Morphisms, Lemma 29.39.4). $\square$

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