Lemma 69.20.1. Let $S$ be a scheme. Consider a commutative diagram

of algebraic spaces over $S$. Assume $i$ is a closed immersion and $Y$ Noetherian. Set $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ Y}(1)$. Let $\mathcal{F}$ be a coherent module on $X$. Then there exists an integer $d_0$ such that for all $d \geq d_0$ we have $R^ pf_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$.

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