hcju/stacksqa huggingface dataset:
Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which satisfies conditions (1), (2) and (3) of the lemma. By Lemma \ref{lemma-precompose-property-local-source} we see that $\mathcal{P}$ is stable under precomposing with smooth morphisms. By Lemma \ref{lemma-pullback-property-local-target} we see that $\mathcal{P}$ is stable under smooth base change. Hence it suffices to prove part (3) of Definition \ref{definition-local-source-target} holds. \medskip\noindent More precisely, suppose that $f : X \to Y$ is a morphism of algebraic spaces over $S$ which satisfies Definition \ref{definition-local-source-target} part (3)(b). In other words, for every $x \in X$ there exists a smooth morphism $a_x : U_x \to X$, a point $u_x \in |U_x|$ mapping to $x$, a smooth morphism $b_x : V_x \to Y$, and a morphism $h_x : U_x \to V_x$ such that $f \circ a_x = b_x \circ h_x$ and $h_x$ has $\mathcal{P}$. The proof of the lemma is complete once we show that $f$ has $\mathcal{P}$. Set $U = \coprod U_x$, $a = \coprod a_x$, $V = \coprod V_x$, $b = \coprod b_x$, and $h = \coprod h_x$. We obtain a commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $a$, $b$ smooth, $a$ surjective. Note that $h$ has $\mathcal{P}$ as each $h_x$ does and $\mathcal{P}$ is smooth local on the target. Because $a$ is surjective and $\mathcal{P}$ is smooth local on the source, it suffices to prove that $b \circ h$ has $\mathcal{P}$. This follows as we assumed that $\mathcal{P}$ is stable under postcomposing with a smooth morphism and as $b$ is smooth.
We write everything out completely. \medskip\noindent Proof of (1). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{X_i \to X\}_{i \in I}$ be an \'etale covering of $X$. If each composition $h_i : X_i \to Y$ has $\mathcal{P}$, then for each $|x| \in X$ we can find an $i \in I$ and a point $x_i \in |X_i|$ mapping to $x$. Then $(X_i, x_i) \to (X, x)$ is an \'etale morphism of pairs, and $\text{id}_Y : Y \to Y$ is a smooth morphism, and $h_i$ is as in part (3) of Definition \ref{definition-etale-smooth-local-source-target}. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$ then each $X_i \to Y$ has $\mathcal{P}$ by Definition \ref{definition-etale-smooth-local-source-target} part (1). \medskip\noindent Proof of (2). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{Y_i \to Y\}_{i \in I}$ be a smooth covering of $Y$. Write $X_i = Y_i \times_Y X$ and $h_i : X_i \to Y_i$ for the base change of $f$. If each  $h_i : X_i \to Y_i$ has $\mathcal{P}$, then for each $x \in |X|$ we pick an $i \in I$ and a point $x_i \in |X_i|$ mapping to $x$. Then $X_i \to X \times_Y Y_i$ is an \'etale morphism (because it is an isomorphism), $Y_i \to Y$ is smooth, and $h_i$ is as in part (3) of Definition \ref{definition-local-source-target}. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$, then each $X_i \to Y_i$ has $\mathcal{P}$ by Definition \ref{definition-local-source-target} part (2). \medskip\noindent Proof of (3). Assume $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is \'etale. The morphism $X \to Y \times_Z X$ is \'etale as a morphism between algebraic spaces \'etale over $X$ ( Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-permanence}). Also $Y \to Z$ is \'etale hence a smooth morphism. Thus the diagram $$ \xymatrix{ X \ar[d] \ar[r]_f & Y \ar[d] \\ X \ar[r]^{g \circ f} & Z } $$ works for every $x \in |X|$ in part (3) of Definition \ref{definition-local-source-target} and we conclude that $g \circ f$ has $\mathcal{P}$. \medskip\noindent Proof of (4). Let $f : X \to Y$ be a morphism and $g : Y \to Z$ \'etale such that $g \circ f$ has $\mathcal{P}$. Then by Definition \ref{definition-etale-smooth-local-source-target} part (2) we see that $\text{pr}_Y : Y \times_Z X \to Y$ has $\mathcal{P}$. But the morphism $(f, 1) : X \to Y \times_Z X$ is \'etale as a section to the \'etale projection $\text{pr}_X : Y \times_Z X \to X$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-permanence}. Hence $f = \text{pr}_Y \circ (f, 1)$ has $\mathcal{P}$ by Definition \ref{definition-etale-smooth-local-source-target} part (1).
The equivalence of (a) and (b) is part of Definition \ref{definition-etale-smooth-local-source-target}. The equivalence of (a) and (e) is Lemma \ref{lemma-etale-smooth-local-source-target-implies} part (2). The equivalence of (a) and (f) is Lemma \ref{lemma-etale-smooth-local-source-target-implies} part (1). As (a) is now equivalent to (e) and (f) it follows that (a) equivalent to (g). \medskip\noindent It is clear that (c) implies (b). If (b) holds, then for any $x \in |X|$ we can choose a smooth morphism a smooth morphism $b_x : V_x \to Y$, an \'etale morphism $U_x \to V_x \times_Y X$, and $u_x \in |U_x|$ mapping to $x$ such that $U_x \to V_x$ has $\mathcal{P}$. Then $h = \coprod h_x : \coprod U_x \to \coprod V_x$ with $a = \coprod a_x$ and $b = \coprod b_x$ is a diagram as in (c). (Note that $h$ has property $\mathcal{P}$ as $\{V_x \to \coprod V_x\}$ is a smooth covering and $\mathcal{P}$ is smooth local on the target.) Thus (b) is equivalent to (c). \medskip\noindent Now we know that (a), (b), (c), (e), (f), and (g) are equivalent. Suppose (a) holds. Let $U, V, a, b, h$ be as in (d). Then $X \times_Y V \to V$ has $\mathcal{P}$ as $\mathcal{P}$ is stable under smooth base change, whence $U \to V$ has $\mathcal{P}$ as $\mathcal{P}$ is stable under precomposing with \'etale morphisms. Conversely, if (d) holds, then setting $U = X$ and $V = Y$ we see that $f$ has $\mathcal{P}$.
Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which satisfies conditions (1), (2) and (3) of the lemma. By Lemma \ref{lemma-precompose-property-local-source} we see that $\mathcal{P}$ is stable under precomposing with \'etale morphisms. By Lemma \ref{lemma-pullback-property-local-target} we see that $\mathcal{P}$ is stable under smooth base change. Hence it suffices to prove part (3) of Definition \ref{definition-local-source-target} holds. \medskip\noindent More precisely, suppose that $f : X \to Y$ is a morphism of algebraic spaces over $S$ which satisfies Definition \ref{definition-local-source-target} part (3)(b). In other words, for every $x \in X$ there exists a smooth morphism $b_x : V_x \to Y$, an \'etale morphism $U_x \to V_x \times_Y X$, and a point $u_x \in |U_x|$ mapping to $x$ such that $h_x : U_x \to V_x$ has $\mathcal{P}$. The proof of the lemma is complete once we show that $f$ has $\mathcal{P}$. \medskip\noindent Let $a_x : U_x \to X$ be the composition $U_x \to V_x \times_Y X \to X$. Set $U = \coprod U_x$, $a = \coprod a_x$, $V = \coprod V_x$, $b = \coprod b_x$, and $h = \coprod h_x$. We obtain a commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $b$ smooth, $U \to V \times_Y X$ \'etale, $a$ surjective. Note that $h$ has $\mathcal{P}$ as each $h_x$ does and $\mathcal{P}$ is smooth local on the target. In the next paragraph we prove that we may assume $U, V, X, Y$ are schemes; we encourage the reader to skip it. \medskip\noindent Let $X, Y, U, V, a, b, f, h$ be as in the previous paragraph. We have to show $f$ has $\mathcal{P}$. Let $X' \to X$ be a surjective \'etale morphism with $X_i$ a scheme. Set $U' = X' \times_X U$. Then $U' \to X'$ is surjective and $U' \to X' \times_Y V$ is \'etale. Since $\mathcal{P}$ is \'etale local on the source, we see that $U' \to V$ has $\mathcal{P}$ and that it suffices to show that $X' \to Y$ has $\mathcal{P}$. In other words, we may assume that $X$ is a scheme. Next, choose a surjective \'etale morphism $Y' \to Y$ with $Y'$ a scheme. Set $V' = V \times_Y Y'$, $X' = X \times_Y Y'$, and $U' = U \times_Y Y'$. Then $U' \to X'$ is surjective and $U' \to X' \times_{Y'} V'$ is \'etale. Since $\mathcal{P}$ is smooth local on the target, we see that $U' \to V'$ has $\mathcal{P}$ and that it suffices to prove $X' \to Y'$ has $\mathcal{P}$. Thus we may assume both $X$ and $Y$ are schemes. Choose a surjective \'etale morphism $V' \to V$ with $V'$ a scheme. Set $U' = U \times_V V'$. Then $U' \to X$ is surjective and $U' \to X \times_Y V'$ is \'etale. Since $\mathcal{P}$ is smooth local on the source, we see that $U' \to V'$ has $\mathcal{P}$. Thus we may replace $U, V$ by $U', V'$ and assume $X, Y, V$ are schemes. Finally, we replace $U$ by a scheme surjective \'etale over $U$ and we see that we may assume $U, V, X, Y$ are all schemes. \medskip\noindent If $U, V, X, Y$ are schemes, then $f$ has $\mathcal{P}$ by Descent, Lemma \ref{descent-lemma-etale-tau-local-source-target}.
We omit the proof of (1), but we remark that the morphism $\varphi'$ is the morphism $(f \times f)^*\varphi$ in the notation introduced in Remark \ref{remark-easier}. For (2) we indicate which morphism $f_0^*V \to f_1^*V$ gives the functorial isomorphism. Namely, since $f_0$ and $f_1$ both fit into the commutative diagram we see there is a unique morphism $r : Y' \to Y \times_X Y$ with $f_i = \text{pr}_i \circ r$. Then we take \begin{eqnarray*} f_0^*V & = & Y' \times_{f_0, Y} V \\ & = & Y' \times_{\text{pr}_0 \circ r, Y} V \\ & = & Y' \times_{r, Y \times_X Y} (Y \times_X Y) \times_{\text{pr}_0, Y} V \\ & \xrightarrow{\varphi} & Y' \times_{r, Y \times_X Y} (Y \times_X Y) \times_{\text{pr}_1, Y} V \\ & = & Y' \times_{\text{pr}_1 \circ r, Y} V \\ & = & Y' \times_{f_1, Y} V \\ & = & f_1^*V \end{eqnarray*} We omit the verification that this works.
This is more or less a formal consequence of the corresponding result for schemes, see Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. Here is a strategy for a proof: \begin{enumerate} \item The fact that $\{X_i \to X\}$ is a refinement of the trivial covering $\{X \to X\}$ gives, via Lemma \ref{lemma-map-families}, a functor $\QCoh(\mathcal{O}_X) \to DD(\{X_i \to X\})$ from the category of quasi-coherent $\mathcal{O}_X$-modules to the category of descent data for the given family. \item In order to prove the proposition we will construct a quasi-inverse functor $back : DD(\{X_i \to X\}) \to \QCoh(\mathcal{O}_X)$. \item Applying again Lemma \ref{lemma-map-families} we see that there is a functor $DD(\{X_i \to X\}) \to DD(\{T_j \to X\})$ if $\{T_j \to X\}$ is a refinement of the given family. Hence in order to construct the functor $back$ we may assume that each $X_i$ is a scheme, see Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-refine-fpqc-schemes}. This reduces us to the case where all the $X_i$ are schemes. \item A quasi-coherent sheaf on $X$ is by definition a quasi-coherent $\mathcal{O}_X$-module on $X_\etale$. Now for any $U \in \Ob(X_\etale)$ we get an fppf covering $\{U_i \times_X X_i \to U\}$ by schemes and a morphism $g : \{U_i \times_X X_i \to U\} \to \{X_i \to X\}$ of coverings lying over $U \to X$. Given a descent datum $\xi = (\mathcal{F}_i, \varphi_{ij})$ we obtain a quasi-coherent $\mathcal{O}_U$-module $\mathcal{F}_{\xi, U}$ corresponding to the pullback $g^*\xi$ of Lemma \ref{lemma-map-families} to the covering of $U$ and using effectivity for fppf covering of schemes, see Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. \item Check that $\xi \mapsto \mathcal{F}_{\xi, U}$ is functorial in $\xi$. Omitted. \item Check that $\xi \mapsto \mathcal{F}_{\xi, U}$ is compatible with morphisms $U \to U'$ of the site $X_\etale$, so that the system of sheaves $\mathcal{F}_{\xi, U}$ corresponds to a quasi-coherent $\mathcal{F}_\xi$ on $X_\etale$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent-small-etale}. Details omitted. \item Check that $back : \xi \mapsto \mathcal{F}_\xi$ is quasi-inverse to the functor constructed in (1). Omitted. \end{enumerate} This finishes the proof.
Assume $X$ is a decent algebraic space for which the theorem is false. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} there exists a largest open subspace $X' \subset X$ which is a scheme. Since $X'$ is not dense in $X$, there exists an open subspace $X'' \subset X$ such that $|X''| \cap |X'| = \emptyset$. Replacing $X$ by $X''$ we get a nonempty decent algebraic space $X$ which does not contain {\it any} open subspace which is a scheme. \medskip\noindent Choose a nonempty affine scheme $U$ and an \'etale morphism $U \to X$. We may and do replace $X$ by the open subscheme corresponding to the image of $|U| \to |X|$. Consider the sequence of open subspaces $$ X = X_0 \supset X_1 \supset X_2 \ldots $$ constructed in Lemma \ref{lemma-stratify-flat-fp-lqf} for the morphism $U \to X$. Note that $X_0 = X_1$ as $U \to X$ is surjective. Let $U = U_0 = U_1 \supset U_2 \ldots$ be the induced sequence of open subschemes of $U$. \medskip\noindent Choose a nonempty open affine $V_1 \subset U_1$ (for example $V_1 = U_1$). By induction we will construct a sequence of nonempty affine opens $V_1 \supset V_2 \supset \ldots$ with $V_n \subset U_n$. Namely, having constructed $V_1, \ldots, V_{n - 1}$ we can always choose $V_n$ unless $V_{n - 1} \cap U_n = \emptyset$. But if $V_{n - 1} \cap U_n = \emptyset$, then the open subspace $X' \subset X$ with $|X'| = \Im(|V_{n - 1}| \to |X|)$ is contained in $|X| \setminus |X_n|$. Hence $V_{n - 1} \to X'$ is an \'etale morphism whose fibres have degree bounded by $n - 1$. In other words, $X'$ is reasonable (by definition), hence $X'$ contains a nonempty open subscheme by Proposition \ref{proposition-reasonable-open-dense-scheme}. This is a contradiction which shows that we can pick $V_n$. \medskip\noindent By Limits, Lemma \ref{limits-lemma-limit-nonempty} the limit $V_\infty = \lim V_n$ is a nonempty scheme. Pick a morphism $\Spec(k) \to V_\infty$. The composition $\Spec(k) \to V_\infty \to U \to X$ has image contained in all $X_d$ by construction. In other words, the fibred $U \times_X \Spec(k)$ has infinite degree which contradicts the definition of a decent space. This contradiction finishes the proof of the theorem.
Let $n$ be an integer which works for $V \to U$, and let $m$ be an integer which works for $U \to X$ in Definition \ref{definition-universally-bounded}. Let $\Spec(k) \to X$ be a morphism, where $k$ is a field. Consider the morphisms $$ \Spec(k) \times_X V \longrightarrow \Spec(k) \times_X U \longrightarrow \Spec(k). $$ By assumption the scheme $\Spec(k) \times_X U$ is finite of degree at most $m$ over $k$, and $n$ is an integer which bounds the degree of the fibres of the first morphism. Hence by Morphisms, Lemma \ref{morphisms-lemma-composition-universally-bounded} we conclude that $\Spec(k) \times_X V$ is finite over $k$ of degree at most $nm$.
Let $n \geq 0$ be an integer bounding the degrees of the fibre products $\Spec(k) \times_Y (U \times_X Y)$ as in Definition \ref{definition-universally-bounded} for the morphism $f'$. We claim that $n$ works for $f$ also. Namely, suppose that $x : \Spec(k) \to X$ is a morphism from the spectrum of a field. Then either $\Spec(k) \times_X U$ is empty (and there is nothing to prove), or $x$ is in the image of $|f|$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} and the assumption of the lemma we see that this means there exists a field extension $k \subset k'$ and a commutative diagram $$ \xymatrix{ \Spec(k') \ar[r] \ar[d] & Y \ar[d] \\ \Spec(k) \ar[r] & X } $$ Hence we see that $$ \Spec(k') \times_Y (U \times_X Y) = \Spec(k') \times_{\Spec(k)} (\Spec(k) \times_X U) $$ Since the scheme $\Spec(k') \times_Y (U \times_X Y)$ is assumed finite of degree $\leq n$ over $k'$ it follows that also $\Spec(k) \times_X U$ is finite of degree $\leq n$ over $k$ as desired. (Some details omitted.)
Assume $g$ has universally bounded fibres, and $f$ is surjective and flat. Say $n \geq 0$ is an integer which bounds the degrees of the schemes $\Spec(k) \times_X U$ as in Definition \ref{definition-universally-bounded}. We claim $n$ also works for $h$. Let $\Spec(k) \to X$ be a morphism from the spectrum of a field to $X$. Consider the morphism of schemes $$ \Spec(k) \times_X V \longrightarrow \Spec(k) \times_X U $$ It is flat and surjective. By assumption the scheme on the left is finite of degree $\leq n$ over $\Spec(k)$. It follows from Morphisms, Lemma \ref{morphisms-lemma-universally-bounded-permanence} that the degree of the scheme on the right is also bounded by $n$ as desired.
The integer $n$ of Definition \ref{definition-universally-bounded} works. Namely, pick $x \in |X|$. Represent $x$ by a morphism $x : \Spec(k) \to X$. Then we get a commutative diagram $$ \xymatrix{ \Spec(k) \times_X U \ar[r] \ar[d] & U \ar[d] \\ \Spec(k) \ar[r]^x & X } $$ which shows (via Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}) that the inverse image of $x$ in $|U|$ is the image of the top horizontal arrow. Since $\Spec(k) \times_X U$ is finite of degree $\leq n$ over $k$ it has at most $n$ points.
The implication (2) $\Rightarrow$ (1) is trivial. Let $\varphi_i : U_i \to X$ be a family of \'etale morphisms as in (1). Let $\varphi : U \to X$ be an \'etale morphism from an affine scheme towards $X$. Consider the fibre product diagrams $$ \xymatrix{ U \times_X U_i \ar[r]_-{p_i} \ar[d]_{q_i} & U_i \ar[d]^{\varphi_i} \\ U \ar[r]^\varphi & X } \quad \quad \xymatrix{ \coprod U \times_X U_i \ar[r]_-{\coprod p_i} \ar[d]_{\coprod q_i} & \coprod U_i \ar[d]^{\coprod \varphi_i} \\ U \ar[r]^\varphi & X } $$ Since $q_i$ is \'etale it is open (see Remark \ref{remark-recall}). Moreover, the morphism $\coprod q_i$ is surjective. Hence there exist finitely many indices $i_1, \ldots, i_n$ and a quasi-compact opens $W_{i_j} \subset U \times_X U_{i_j}$ which surject onto $U$. The morphism $p_i$ is \'etale, hence locally quasi-finite (see remark on \'etale morphisms above). Thus we may apply Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded} to see the fibres of $p_{i_j}|_{W_{i_j}} : W_{i_j} \to U_i$ are finite. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} and the assumption on $\varphi_i$ we conclude that the fibre of $\varphi$ over $x$ is finite. In other words (2) holds.
Assume (1). This clearly implies the first condition of Lemma \ref{lemma-R-finite-above-x} and hence we obtain a monomorphism $\Spec(k) \to X$ in the class of $x$. Taking the fibre product we see that $\Spec(k) \times_X U \to \Spec(k)$ is a scheme \'etale over $\Spec(k)$ with finitely many points, hence a finite nonempty scheme over $k$, i.e., (2) holds. \medskip\noindent Assume (2). By assumption $x$ is in the image of $|U| \to |X|$. The finiteness of the fibre of $|U| \to |X|$ over $x$ is clear since this fibre is equal to $|\Spec(k) \times_X U|$ by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. The finiteness of the fibre of $|R| \to |X|$ above $x$ is also clear since it is equal to the set underlying the scheme $$ (\Spec(k) \times_X U) \times_{\Spec(k)} (\Spec(k) \times_X U) $$ which is finite over $k$. Thus (1) holds.
The equivalence of (1) and (3) follows on applying Lemma \ref{lemma-weak-UR-finite-above-x} to every \'etale 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 \ref{lemma-U-finite-above-x} that for any affine scheme $U$ and \'etale 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 \ref{lemma-R-finite-above-x} (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. \medskip\noindent If (4) holds, then for any affine scheme $U$ and \'etale morphism $U \to X$ the scheme $\Spec(k) \times_X U$ is on the one hand \'etale over $k$ (hence a disjoint union of spectra of finite separable extensions of $k$ by Remark \ref{remark-recall}) 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 $\Spec(k) \to X$ is a monomorphism as in (3), then $$ \coprod \Spec(k) \times_X U_i \longrightarrow \coprod U_i $$ is quasi-compact (because over each $U_i$ we see that $\Spec(k) \times_X U_i$ is a finite disjoint union spectra of fields). Thus $\Spec(k) \to X$ is quasi-compact by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-local}. \medskip\noindent It is immediate that (4) implies (5). Conversely, let $\Spec(k) \to X$ be a quasi-compact morphism in the equivalence class of $x$. Let $U \to X$ be an \'etale morphism with $U$ affine. Consider the fibre product $$ \xymatrix{ F \ar[r] \ar[d] & U \ar[d] \\ \Spec(k) \ar[r] & X } $$ Then $F \to U$ is quasi-compact, hence $F$ is quasi-compact. On the other hand, $F \to \Spec(k)$ is \'etale, hence $F$ is a finite disjoint union of spectra of finite separable extensions of $k$ (Remark \ref{remark-recall}). Since the image of $|F| \to |U|$ is the fibre of $|U| \to |X|$ over $x$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}), we conclude that the fibre of $|U| \to |X|$ over $x$ is finite. The scheme $F \times_{\Spec(k)} F$ is also a finite union of spectra of fields because it is also quasi-compact and \'etale over $\Spec(k)$. There is a monomorphism $F \times_X F \to F \times_{\Spec(k)} F$, hence $F \times_X F$ is a finite disjoint union of spectra of fields (Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field}). 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 \ref{spaces-properties-lemma-points-cartesian} we conclude that (1) holds.
The implication (2) $\Rightarrow$ (1) is trivial. Assume (1). Let $(\varphi_i : U_i \to X)_{i \in I}$ be a collection of \'etale morphisms from schemes towards $X$, covering $X$, such that each $\varphi_i$ has universally bounded fibres. Let $\psi : U \to X$ be an \'etale morphism from an affine scheme towards $X$. For each $i$ consider the fibre product diagram $$ \xymatrix{ U \times_X U_i \ar[r]_{p_i} \ar[d]_{q_i} & U_i \ar[d]^{\varphi_i} \\ U \ar[r]^\psi & X } $$ Since $q_i$ is \'etale it is open (see Remark \ref{remark-recall}). Moreover, we have $U = \bigcup \Im(q_i)$, since the family $(\varphi_i)_{i \in I}$ is surjective. Since $U$ is affine, hence quasi-compact we can finite finitely many $i_1, \ldots, i_n \in I$ and quasi-compact opens $W_j \subset U \times_X U_{i_j}$ such that $U = \bigcup p_{i_j}(W_j)$. The morphism $p_{i_j}$ is \'etale, hence locally quasi-finite (see remark on \'etale morphisms above). Thus we may apply Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded} to see the fibres of $p_{i_j}|_{W_j} : W_j \to U_{i_j}$ are universally bounded. Hence by Lemma \ref{lemma-composition-universally-bounded} we see that the fibres of $W_j \to X$ are universally bounded. Thus also $\coprod_{j = 1, \ldots, n} W_j \to X$ has universally bounded fibres. Since $\coprod_{j = 1, \ldots, n} W_j \to X$ factors through the surjective \'etale map $\coprod q_{i_j}|_{W_j} : \coprod_{j = 1, \ldots, n} W_j \to U$ we see that the fibres of $U \to X$ are universally bounded by Lemma \ref{lemma-universally-bounded-permanence}. In other words (2) holds.