The Stacks project

Proposition 69.7.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and separated. Let $U$ be an affine scheme, and let $f : U \to X$ be a surjective ├ętale morphism. Let $d$ be an upper bound for the size of the fibres of $|U| \to |X|$. Then for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^ q(X, \mathcal{F}) = 0$ for $q \geq d$.

Proof. We will use the spectral sequence of Lemma 69.6.6. The lemma applies since $f$ is separated as $U$ is separated, see Morphisms of Spaces, Lemma 67.4.10. Since $X$ is separated the scheme $U \times _ X \ldots \times _ X U$ is a closed subscheme of $U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \ldots \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U$ hence is affine. Thus $W_ p$ is affine. Hence $U_ p = W_ p/S_{p + 1}$ is an affine scheme by Groupoids, Proposition 39.23.9. The discussion in Section 69.3 shows that cohomology of quasi-coherent sheaves on $W_ p$ (as an algebraic space) agrees with the cohomology of the corresponding quasi-coherent sheaf on the underlying affine scheme, hence vanishes in positive degrees by Cohomology of Schemes, Lemma 30.2.2. By Lemma 69.7.1 the sheaves $\mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)$ are quasi-coherent. Hence $H^ q(W_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is zero when $q > 0$. By our definition of the integer $d$ we see that $W_ p = \emptyset $ for $p \geq d$. Hence also $H^0(W_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is zero when $p \geq d$. This proves the proposition. $\square$

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