Lemma 69.20.1 (08AQ)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.20: Higher direct images of coherent sheaves
Lemma 69.20.1 (cite)

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

\[ \xymatrix{ X \ar[r]_ i \ar[rd]_ f & \mathbf{P}^ n_ Y \ar[d] \\ & Y } \]

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 $p > 0$.

Proof. Checking whether $R^ pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes d})$ is zero can be done étale locally on $Y$, see Equation (69.3.0.1). Hence we may assume $Y$ is the spectrum of a Noetherian ring. In this case $X$ is a scheme and the result follows from Cohomology of Schemes, Lemma 30.16.2. $\square$

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