Proposition 66.7.2. 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 étale 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 66.6.6. The lemma applies since $f$ is separated as $U$ is separated, see Morphisms of Spaces, Lemma 64.4.10. 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 66.3 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 66.7.1 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)