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 étale 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 Pieter Belmans on