The Stacks project

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

  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 69.6.6, 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 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. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 072C. Beware of the difference between the letter 'O' and the digit '0'.