The Stacks project

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 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

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

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. $\square$

Proof. Assume $|X|$ is a singleton. It follows immediately from Theorem 68.10.2 that $X$ is a scheme, but we can also argue directly as follows. Choose an affine scheme $U$ and a surjective étale morphism $U \to X$. Set $R = U \times _ X U$. Then $U$ and $R$ have finitely many points by Lemma 68.4.5 (and the definition of a decent space). All of these points are closed in $U$ and $R$ by Lemma 68.12.1. It follows that $U$ and $R$ are affine schemes. We may shrink $U$ to a singleton space. Then $U$ is the spectrum of a henselian local ring, see Algebra, Lemma 10.153.10. The projections $R \to U$ are étale, hence finite étale because $U$ is the spectrum of a $0$-dimensional henselian local ring, see Algebra, Lemma 10.153.3. It follows that $X$ is a scheme by Groupoids, Proposition 39.23.9.

Part (2) follows from (1) and the fact that a reduced singleton scheme is the spectrum of a field. $\square$

Proof. Suppose that $x \leadsto x'$ for some $x' \in |X|$. Choose an étale morphism $U \to X$ where $U$ is a scheme and a point $u' \in U'$ mapping to $x'$. Choose a specialization $u \leadsto u'$ in $U$ with $u$ mapping to $x$ in $X$, see Lemma 68.12.2. Then $u$ is the image of a point $w$ of the scheme $W = \mathop{\mathrm{Spec}}(k) \times _ X U$. Since the projection $W \to \mathop{\mathrm{Spec}}(k)$ is étale we see that $\kappa (w) \supset k$ is finite. Hence $\kappa (w) \supset \kappa (s)$ is algebraic. Hence $\kappa (u) \supset \kappa (s)$ is algebraic. Thus $u$ is a closed point of $U$ by Morphisms, Lemma 29.20.2. Thus $u = u'$, whence $x = x'$. $\square$

Proof. By Lemma 68.14.4 the image point $x \in |X|$ is closed. Let $Z \subset X$ be the reduced closed subspace with $|Z| = \{ x\} $ (Properties of Spaces, Lemma 66.12.3). Note that $Z$ is a decent algebraic space by Lemma 68.6.5. By Lemma 68.14.2 we see that $Z = \mathop{\mathrm{Spec}}(k')$ for some field $k'$. Of course $k \supset k' \supset \kappa (s)$. Then $\mathop{\mathrm{Spec}}(k) \to Z$ is a finite morphism of schemes and $Z \to X$ is a finite morphism as it is a closed immersion. Hence $\mathop{\mathrm{Spec}}(k) \to X$ is finite (Morphisms of Spaces, Lemma 67.45.4). If $k = \kappa (s)$, then $\mathop{\mathrm{Spec}}(k) = Z$ and $\mathop{\mathrm{Spec}}(k) \to X$ is a closed immersion. $\square$

Proof. We know that $x$ can be represented by a quasi-compact monomorphism $i : \mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field (Definition 68.6.1). Let $U \to X$ be an étale morphism where $U$ is an affine scheme. As $x$ is closed and $X$ decent, the fibre $F$ of $|U| \to |X|$ over $x$ consists of closed points (Lemma 68.12.1). As $i$ is a monomorphism, so is $U_ k = U \times _ X \mathop{\mathrm{Spec}}(k) \to U$. In particular, the map $|U_ k| \to F$ is injective. Since $U_ k$ is quasi-compact and étale over a field, we see that $U_ k$ is a finite disjoint union of spectra of fields (Remark 68.4.1). Say $U_ k = \mathop{\mathrm{Spec}}(k_1) \amalg \ldots \amalg \mathop{\mathrm{Spec}}(k_ r)$. Since $\mathop{\mathrm{Spec}}(k_ i) \to U$ is a monomorphism, we see that its image $u_ i$ has residue field $\kappa (u_ i) = k_ i$. Since $u_ i \in F$ is a closed point we conclude the morphism $\mathop{\mathrm{Spec}}(k_ i) \to U$ is a closed immersion. As the $u_ i$ are pairwise distinct, $U_ k \to U$ is a closed immersion. Hence $i$ is a closed immersion (Morphisms of Spaces, Lemma 67.12.1). This finishes the proof. $\square$