The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 60.14.1. 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 58.3.3). 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 60.4.1) 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 36.38.10. 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 36.38.8 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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