The Stacks project

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

Lemma 66.8.6. 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 66.8.3. Observe that a quasi-separated space is reasonable, see Lemma 66.5.1 and Definition 66.6.1. Hence we find that $U_{n + 1} = \emptyset $ as in Lemma 66.8.4. 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 66.8.3. I'm not saying you should change, but if people are confused when looking at this on the website this clears up things.


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