The Stacks project

Lemma 68.14.1. Any locally Noetherian decent algebraic space is quasi-separated.

Proof. Namely, let $X$ be an algebraic space (over some base scheme, for example over $\mathbf{Z}$) which is decent and locally Noetherian. Let $U \to X$ and $V \to X$ be ├ętale morphisms with $U$ and $V$ affine schemes. We have to show that $W = U \times _ X V$ is quasi-compact (Properties of Spaces, Lemma 66.3.3). Since $X$ is locally Noetherian, the schemes $U$, $V$ are Noetherian and $W$ is locally Noetherian. Since $X$ is decent, the fibres of the morphism $W \to U$ are finite. Namely, we can represent any $x \in |X|$ by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$. Then $U_ k$ and $V_ k$ are finite disjoint unions of spectra of finite separable extensions of $k$ (Remark 68.4.1) and we see that $W_ k = U_ k \times _{\mathop{\mathrm{Spec}}(k)} V_ k$ is finite. Let $n$ be the maximum degree of a fibre of $W \to U$ at a generic point of an irreducible component of $U$. Consider the stratification

\[ U = U_0 \supset U_1 \supset U_2 \supset \ldots \]

associated to $W \to U$ in More on Morphisms, Lemma 37.45.5. By our choice of $n$ above we conclude that $U_{n + 1}$ is empty. Hence we see that the fibres of $W \to U$ are universally bounded. Then we can apply More on Morphisms, Lemma 37.45.3 to find a stratification

\[ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_ n = U \]

by closed subsets such that with $S_ r = Z_ r \setminus Z_{r - 1}$ the morphism $W \times _ U S_ r \to S_ r$ is finite locally free. Since $U$ is Noetherian, the schemes $S_ r$ are Noetherian, whence the schemes $W \times _ U S_ r$ are Noetherian, whence $W = \coprod W \times _ U S_ r$ is quasi-compact as desired. $\square$

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