Lemma 69.7.3. 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
an affine scheme U,
a surjective étale morphism f : U \to X,
an integer d bounding the degrees of the fibres of U \to X,
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 69.6.6, and
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 66.6.3. By Decent Spaces, Lemma 68.5.1 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 65.14.5. By Properties of Spaces, Lemma 66.6.3 we can find the morphisms V_ p \to W_ p. By Decent Spaces, Lemma 68.5.1 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 69.6.6. By definition of the integer d we see that U_ p = 0 for p \geq d. By Proposition 69.7.2 and Lemma 69.7.1 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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