Lemma 68.7.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Then we can choose

1. an affine scheme $U$,

2. a surjective étale morphism $f : U \to X$,

3. an integer $d$ bounding the degrees of the fibres of $U \to X$,

4. for every $p = 0, 1, \ldots , d$ a surjective étale morphism $V_ p \to U_ p$ from an affine scheme $V_ p$ where $U_ p$ is as in Lemma 68.6.6, and

5. an integer $d_ p$ bounding the degree of the fibres of $V_ p \to U_ p$.

Moreover, whenever we have (1) – (5), then for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^ q(X, \mathcal{F}) = 0$ for $q \geq \max (d_ p + p)$.

Proof. Since $X$ is quasi-compact we can find a surjective étale morphism $U \to X$ with $U$ affine, see Properties of Spaces, Lemma 65.6.3. By Decent Spaces, Lemma 67.5.1 the fibres of $f$ are universally bounded, hence we can find $d$. We have $U_ p = W_ p/S_{p + 1}$ and $W_ p \subset U \times _ X \ldots \times _ X U$ is open and closed. Since $X$ is quasi-separated the schemes $W_ p$ are quasi-compact, hence $U_ p$ is quasi-compact. Since $U$ is separated, the schemes $W_ p$ are separated, hence $U_ p$ is separated by (the absolute version of) Spaces, Lemma 64.14.5. By Properties of Spaces, Lemma 65.6.3 we can find the morphisms $V_ p \to W_ p$. By Decent Spaces, Lemma 67.5.1 we can find the integers $d_ p$.

At this point the proof uses the spectral sequence

$E_1^{p, q} = H^ q(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \Rightarrow H^{p + q}(X, \mathcal{F})$

see Lemma 68.6.6. By definition of the integer $d$ we see that $U_ p = 0$ for $p \geq d$. By Proposition 68.7.2 and Lemma 68.7.1 we see that $H^ q(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is zero for $q \geq d_ p$ for $p = 0, \ldots , d$. Whence the lemma. $\square$

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