The Stacks project

Lemma 66.4.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent:

  1. for every affine scheme $U$, any étale morphism $\varphi : U \to X$ setting $R = U \times _ X U$ the fibres of both

    \[ |U| \longrightarrow |X| \quad \text{and}\quad |R| \longrightarrow |X| \]

    over $x$ are finite,

  2. there exist schemes $U_ i$ and étale morphisms $U_ i \to X$ such that $\coprod U_ i \to X$ is surjective and for each $i$, setting $R_ i = U_ i \times _ X U_ i$ the fibres of both

    \[ |U_ i| \longrightarrow |X| \quad \text{and}\quad |R_ i| \longrightarrow |X| \]

    over $x$ are finite,

  3. there exists a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$, and for any affine scheme $U$ and étale morphism $U \to X$ the fibre product $\mathop{\mathrm{Spec}}(k) \times _ X U$ is a finite scheme over $k$,

  4. there exists a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$, and

  5. there exists a quasi-compact morphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$.

Proof. The equivalence of (1) and (3) follows on applying Lemma 66.4.4 to every étale morphism $U \to X$ with $U$ affine. It is clear that (3) implies (2). Assume $U_ i \to X$ and $R_ i$ are as in (2). We conclude from Lemma 66.4.2 that for any affine scheme $U$ and étale morphism $U \to X$ the fibre of $|U| \to |X|$ over $x$ is finite. Say this fibre is $\{ u_1, \ldots , u_ n\} $. Then, as Lemma 66.4.3 (1) applies to $U_ i \to X$ for some $i$ such that $x$ is in the image of $|U_ i| \to |X|$, we see that the fibre of $|R = U \times _ X U| \to |U| \times _{|X|} |U|$ is finite over $(u_ a, u_ b)$, $a, b \in \{ 1, \ldots , n\} $. Hence the fibre of $|R| \to |X|$ over $x$ is finite. In this way we see that (1) holds. At this point we know that (1), (2), and (3) are equivalent.

If (4) holds, then for any affine scheme $U$ and étale morphism $U \to X$ the scheme $\mathop{\mathrm{Spec}}(k) \times _ X U$ is on the one hand étale over $k$ (hence a disjoint union of spectra of finite separable extensions of $k$ by Remark 66.4.1) and on the other hand quasi-compact over $U$ (hence quasi-compact). Thus we see that (3) holds. Conversely, if $U_ i \to X$ is as in (2) and $\mathop{\mathrm{Spec}}(k) \to X$ is a monomorphism as in (3), then

\[ \coprod \mathop{\mathrm{Spec}}(k) \times _ X U_ i \longrightarrow \coprod U_ i \]

is quasi-compact (because over each $U_ i$ we see that $\mathop{\mathrm{Spec}}(k) \times _ X U_ i$ is a finite disjoint union spectra of fields). Thus $\mathop{\mathrm{Spec}}(k) \to X$ is quasi-compact by Morphisms of Spaces, Lemma 65.8.8.

It is immediate that (4) implies (5). Conversely, let $\mathop{\mathrm{Spec}}(k) \to X$ be a quasi-compact morphism in the equivalence class of $x$. Let $U \to X$ be an étale morphism with $U$ affine. Consider the fibre product

\[ \xymatrix{ F \ar[r] \ar[d] & U \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & X } \]

Then $F \to U$ is quasi-compact, hence $F$ is quasi-compact. On the other hand, $F \to \mathop{\mathrm{Spec}}(k)$ is étale, hence $F$ is a finite disjoint union of spectra of finite separable extensions of $k$ (Remark 66.4.1). Since the image of $|F| \to |U|$ is the fibre of $|U| \to |X|$ over $x$ (Properties of Spaces, Lemma 64.4.3), we conclude that the fibre of $|U| \to |X|$ over $x$ is finite. The scheme $F \times _{\mathop{\mathrm{Spec}}(k)} F$ is also a finite union of spectra of fields because it is also quasi-compact and étale over $\mathop{\mathrm{Spec}}(k)$. There is a monomorphism $F \times _ X F \to F \times _{\mathop{\mathrm{Spec}}(k)} F$, hence $F \times _ X F$ is a finite disjoint union of spectra of fields (Schemes, Lemma 26.23.11). Thus the image of $F \times _ X F \to U \times _ X U = R$ is finite. Since this image is the fibre of $|R| \to |X|$ over $x$ by Properties of Spaces, Lemma 64.4.3 we conclude that (1) holds. $\square$


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