The Stacks project

This result is almost identical to [Proposition 5.7.8, GruRay].

Lemma 68.8.6. Let $X$ be a quasi-compact and quasi-separated algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. There exist an integer $n$ and open subspaces

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

with the following property: setting $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_ p$ and a surjective ├ętale morphism $f_ p : V_ p \to U_ p$ such that $f_ p^{-1}(T_ p) \to T_ p$ is an isomorphism.

Proof. The proof of this lemma is identical to the proof of Lemma 68.8.3. Observe that a quasi-separated space is reasonable, see Lemma 68.5.1 and Definition 68.6.1. Hence we find that $U_{n + 1} = \emptyset $ as in Lemma 68.8.4. At the end of the argument we add that since $X$ is quasi-separated the schemes $U \times _ X \ldots \times _ X U$ are all quasi-compact. Hence the schemes $W_ p$ are quasi-compact. Hence the quotients $V_ p = W_ p/S_ p$ by the symmetric group $S_ p$ are quasi-compact schemes. $\square$

Comments (1)

Comment #1615 by on

The notation is potentially confusing here: it refers to the symmetric group on elements, as is explained in tag 68.8.3. I'm not saying you should change, but if people are confused when looking at this on the website this clears up things.

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Your email address will not be published.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 07ST. Beware of the difference between the letter 'O' and the digit '0'.