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Tag 0BB6

Lemma 59.13.1. 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 étale morphisms with $U$ and $V$ affine schemes. We have to show that $W = U \times_X V$ is quasi-compact (Properties of Spaces, Lemma 57.3.3). 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 59.4.1) 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. $\square$

The code snippet corresponding to this tag is a part of the file decent-spaces.tex and is located in lines 2729–2732 (see updates for more information).

\begin{lemma}
\label{lemma-locally-Noetherian-decent-quasi-separated}
Any locally Noetherian decent algebraic space is quasi-separated.
\end{lemma}

\begin{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 \'etale morphisms with $U$ and $V$
affine schemes. We have to show that $W = U \times_X V$ is quasi-compact
(Properties of Spaces, Lemma
\ref{spaces-properties-lemma-characterize-quasi-separated}).
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 $\Spec(k) \to X$.
Then $U_k$ and $V_k$ are finite disjoint unions of spectra of
finite separable extensions of $k$ (Remark \ref{remark-recall})
and we see that $W_k = U_k \times_{\Spec(k)} V_k$ 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 \ref{more-morphisms-lemma-stratify-flat-fp-lqf}.
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
\ref{more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded}
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.
\end{proof}

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