hcju/stacksqa huggingface dataset:
Assume $x$ is a finite type point. Choose an affine scheme $U$, a closed point $u \in U$, and an \'etale morphism $\varphi : U \to X$ with $\varphi(u) = x$, see Lemma \ref{lemma-identify-finite-type-points}. Set $u = \Spec(\kappa(u))$ as usual. The projection morphisms $u \times_X u \to u$ are the compositions $$ u \times_X u \to u \times_X U \to u \times_X X = u $$ where the first arrow is a closed immersion (a base change of $u \to U$) and the second arrow is \'etale (a base change of the \'etale morphism $U \to X$). Hence $u \times_X U$ is a disjoint union of spectra of finite separable extensions of $k$ (see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}) and therefore the closed subscheme $u \times_X u$ is a disjoint union of finite separable extension of $k$, i.e., $u \times_X u \to u$ is \'etale. By Spaces, Theorem \ref{spaces-theorem-presentation} we see that $Z = u/u \times_X u$ is an algebraic space. By construction the diagram $$ \xymatrix{ u \ar[d] \ar[r] & U \ar[d] \\ Z \ar[r] & X } $$ is commutative with \'etale vertical arrows. Hence $Z \to X$ is locally of finite type (see Lemma \ref{lemma-finite-type-local}). By construction the morphism $Z \to X$ is a monomorphism and the image of $z$ is $x$. Thus (3) holds. \medskip\noindent It is clear that (3) implies (2). If (2) holds then $x$ is a finite type point of $X$ by Lemma \ref{lemma-finite-type-points-morphism} (and Lemma \ref{lemma-enough-finite-type-points} to see that $Z_{\text{ft-pts}}$ is nonempty, i.e., the unique point of $Z$ is a finite type point of $Z$).
Let $V$ be a scheme and let $V \to Y$ be a surjective \'etale morphism. Let $U$ be a scheme and let $U \to X \times_Y V$ be a surjective \'etale morphism. If $Y$ is Nagata, then $V$ is a Nagata scheme. If $X \to Y$ is locally of finite type, then $U \to V$ is locally of finite type. Hence $V$ is a Nagata scheme by Morphisms, Lemma \ref{morphisms-lemma-finite-type-nagata}. Then $X$ is Nagata by definition.
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to Y' \times_Y V$. Set $U' = V' \times_V U$ so that $U' \to X'$ is a surjective \'etale morphism as well. Picture $$ \vcenter{ \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ V' \ar[r] & V } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } } $$ Choose $u \in |U|$ with image $x \in |X|$. The property of being "locally quasi-finite" is \'etale local on the source-and-target, see Descent, Remark \ref{descent-remark-list-local-source-target}. Hence Lemmas \ref{lemma-local-source-target-at-point} and \ref{lemma-local-source-target-global-implies-local} apply and we see that $f : X \to Y$ is quasi-finite at $x$ if and only if $U \to V$ is quasi-finite at $u$. Similarly for $f' : X' \to Y'$ and the morphism $U' \to V'$. Hence parts (1), (2), and (3) reduce to Morphisms, Lemmas \ref{morphisms-lemma-base-change-quasi-finite} and \ref{morphisms-lemma-quasi-finite-points-open}. Part (4) follows from (3) and Lemma \ref{lemma-base-change-quasi-compact}.
Assume $f$ is locally quasi-finite. Let $\Spec(k) \to Y$ be as in (2). Choose a surjective \'etale morphism $U \to X$ where $U$ is a scheme. Then $U_k = \Spec(k) \times_Y U \to X_k$ is an \'etale morphism of algebraic spaces by Properties of Spaces, Lemma \ref{spaces-properties-lemma-base-change-etale}. By Lemma \ref{lemma-base-change-quasi-finite} we see that $X_k \to \Spec(k)$ is locally quasi-finite. By definition this means that $U_k \to \Spec(k)$ is locally quasi-finite. Hence $|U_k|$ is discrete by Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-fibres}. Since $|U_k| \to |X_k|$ is surjective and open we conclude that $|X_k|$ is discrete. \medskip\noindent Conversely, assume (2). Choose a surjective \'etale morphism $V \to Y$ where $V$ is a scheme. Choose a surjective \'etale morphism $U \to V \times_Y X$ where $U$ is a scheme. Note that $U \to V$ is locally of finite type as $f$ is locally of finite type. Picture $$ \xymatrix{ U \ar[r] \ar[rd] & X \times_Y V \ar[d] \ar[r] & V \ar[d] \\ & X \ar[r] & Y } $$ If $f$ is not locally quasi-finite then $U \to V$ is not locally quasi-finite. Hence there exists a specialization $u \leadsto u'$ for some $u, u' \in U$ lying over the same point $v \in V$, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}. We claim that $u, u'$ do not have the same image in $X_v = \Spec(\kappa(v)) \times_Y X$ which will contradict the assumption that $|X_v|$ is discrete as desired. Let $d = \text{trdeg}_{\kappa(v)}(\kappa(u))$ and $d' = \text{trdeg}_{\kappa(v)}(\kappa(u'))$. Then we see that $d > d'$ by Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-specialization}. Note that $U_v$ (the fibre of $U \to V$ over $v$) is the fibre product of $U$ and $X_v$ over $X \times_Y V$, hence $U_v \to X_v$ is \'etale (as a base change of the \'etale morphism $U \to X \times_Y V$). If $u, u' \in U_v$ map to the same element of $|X_v|$ then there exists a point $r \in R_v = U_v \times_{X_v} U_v$ with $t(r) = u$ and $s(r) = u'$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. Note that $s, t : R_v \to U_v$ are \'etale morphisms of schemes over $\kappa(v)$, hence $\kappa(u) \subset \kappa(r) \supset \kappa(u')$ are finite separable extensions of fields over $\kappa(v)$ (see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}). We conclude that the transcendence degrees are equal. This contradiction finishes the proof.
Choose a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } $$ with vertical arrows \'etale and surjective. (See Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}.) Apply Morphisms, Lemma \ref{morphisms-lemma-permanence-quasi-finite} to the top row.
Choose a field $k$ and a morphism $\Spec(k) \to Y$ in the equivalence class determined by $y$. The fibre $X_k = \Spec(k) \times_Y X$ is an algebraic space of finite type over a field, in particular quasi-compact. The map $|X_k| \to |X|$ surjects onto the fibre of $|X| \to |Y|$ over $y$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}). Moreover, the set of points where $X_k \to \Spec(k)$ is quasi-finite maps onto the set of points lying over $y$ where $f$ is quasi-finite by Lemma \ref{lemma-base-change-quasi-finite-locus}. Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X_k$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U \to \Spec(k)$ is a morphism of finite type and there are at most a finite number of points where this morphism is quasi-finite, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-a-finite-number-of-points}. Since $X_k \to \Spec(k)$ is quasi-finite at a point $x'$ if and only if it is the image of a point of $U$ where $U \to \Spec(k)$ is quasi-finite, we conclude.
A monomorphism is separated, see Lemma \ref{lemma-monomorphism-separated}. By Lemma \ref{lemma-quasi-finite-local} it suffices to prove the lemma after performing a base change by $Z \to Y$ with $Z$ affine. Hence we may assume that $Y$ is an affine scheme. Choose an affine scheme $U$ and an \'etale morphism $U \to X$. Since $X \to Y$ is locally of finite type the morphism of affine schemes $U \to Y$ is of finite type. Since $X \to Y$ is a monomorphism we have $U \times_X U = U \times_Y U$. In particular the maps $U \times_Y U \to U$ are \'etale. Let $y \in Y$. Then either $U_y$ is empty, or $\Spec(\kappa(u)) \times_{\Spec(\kappa(y))} U_y$ is isomorphic to the fibre of $U \times_Y U \to U$ over $u$ for some $u \in U$ lying over $y$. This implies that the fibres of $U \to Y$ are finite discrete sets (as $U \times_Y U \to U$ is an \'etale morphism of affine schemes, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}). Hence $U \to Y$ is quasi-finite, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}. As $U \to X$ was an arbitrary \'etale morphism with $U$ affine this implies that $X \to Y$ is locally quasi-finite.
Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite presentation. This means there exists a diagram as in Lemma \ref{lemma-local-source-target} with $h$ locally of finite presentation and surjective vertical arrow $a$. By Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-finite-type} $h$ is locally of finite type. Hence $X \to Y$ is locally of finite type by definition. If $f$ is of finite presentation then it is quasi-compact and it follows that $f$ is of finite type.
Assume $f$ is of finite presentation and $Y$ Noetherian. By Lemmas \ref{lemma-finite-presentation-finite-type} and \ref{lemma-locally-finite-type-locally-noetherian} we see that $X$ is locally Noetherian. As $f$ is quasi-compact and $Y$ is quasi-compact we see that $X$ is quasi-compact. As $f$ is of finite presentation it is quasi-separated (see Definition \ref{definition-locally-finite-presentation}) and as $Y$ is Noetherian it is quasi-separated (see Properties of Spaces, Definition \ref{spaces-properties-definition-noetherian}). Hence $X$ is quasi-separated by Lemma \ref{lemma-separated-over-separated}. Hence we have checked all three conditions of Properties of Spaces, Definition \ref{spaces-properties-definition-noetherian} and we win.
Assume $f : X \to Y$ locally of finite type and $Y$ locally Noetherian. This means there exists a diagram as in Lemma \ref{lemma-local-source-target} with $h$ locally of finite type and surjective vertical arrow $a$. By Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation} $h$ is locally of finite presentation. Hence $X \to Y$ is locally of finite presentation by definition. This proves (1). If $f$ is of finite type and quasi-separated then it is also quasi-compact and quasi-separated and (2) follows immediately.
Choose a scheme $W$ and a surjective \'etale morphism $W \to Z$. Then choose a scheme $V$ and a surjective \'etale morphism $V \to W \times_Z Y$. Finally choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. By definition $U$ is locally of finite presentation over $W$ and $V$ is locally of finite type over $W$. By Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-permanence} the morphism $U \to V$ is locally of finite presentation. Hence $f$ is locally of finite presentation.
Note that $\Delta$ is a morphism over $X$ (via the second projection $X \times_Y X \to X$). Assume $f$ is locally of finite type. Note that $X$ is of finite presentation over $X$ and $X \times_Y X$ is of finite type over $X$ (by Lemma \ref{lemma-base-change-finite-type}). Thus the first statement holds by Lemma \ref{lemma-finite-presentation-permanence}. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is separated (Lemma \ref{lemma-properties-diagonal}).
Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. By Lemma \ref{lemma-finite-presentation-local} we see that $i' : Z \times_X U \to U$ is of finite presentation if and only if $i$ is. By Properties of Spaces, Section \ref{spaces-properties-section-properties-modules} we see that $\mathcal{I}$ is of finite type if and only if $\mathcal{I}|_U = \Ker(\mathcal{O}_U \to i'_*\mathcal{O}_{Z \times_X U})$ is. Hence the result follows from the case of schemes, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-finite-presentation}.
Choose a scheme $V$ and a surjective \'etale morphism $\varphi : V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Then $U \to X$ is surjective \'etale and the inverse image of $f^{-1}(E)$ in $U$ is the inverse image of $\varphi^{-1}(E)$ by $U \to V$. Thus the lemma follows from the case of schemes for $U \to V$ (Morphisms, Lemma \ref{morphisms-lemma-inverse-image-constructible}) and the definition (Properties of Spaces, Definition \ref{spaces-properties-definition-locally-constructible}).
Choose a commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes, the vertical arrows are \'etale, and $a$ is surjective. By definition $f$ is flat if and only if $h$ is flat (Definition \ref{definition-P}). By definition $f$ is flat at $x \in |X|$ if and only if $h$ is flat at some (equivalently any) $u \in U$ which maps to $x$ (Definition \ref{definition-P-at-point}). Thus the lemma follows from the fact that a morphism of schemes is flat if and only if it is flat at all points of the source (Morphisms, Definition \ref{morphisms-definition-flat}).
Let $f : X \to Y$ be a flat morphism locally of finite presentation of algebraic spaces over $S$. Choose a diagram $$ \xymatrix{ U \ar[r]_\alpha \ar[d] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. By Lemmas \ref{lemma-flat-local} and \ref{lemma-finite-presentation-local} the morphism $\alpha$ is flat and locally of finite presentation. Hence by Morphisms, Lemma \ref{morphisms-lemma-fppf-open} we see that $\alpha$ is universally open. Hence $X \to Y$ is universally open according to Lemma \ref{lemma-universally-open-local}.
Choose affine schemes $V_i$ and \'etale morphisms $V_i \to Y$ such that $V = \coprod V_i \to Y$ is surjective, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. For each $i$ the algebraic space $V_i \times_Y X$ is quasi-compact. Hence we can find an affine scheme $U_i$ and a surjective \'etale morphism $U_i \to V_i \times_Y X$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. Then the composition $U_i \to V_i \times_Y X \to V_i$ is a surjective, flat morphism of affines. Of course then $U = \coprod U_i \to X$ is surjective and \'etale and $U = V \times_Y X$. Moreover, the morphism $U \to V$ is the disjoint union of the morphisms $U_i \to V_i$. Hence $U \to V$ is surjective, quasi-compact and flat. Consider the diagram $$ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y } $$ By definition of the topology on $|Y|$ the set $T$ is closed (resp.\ open) if and only if $g^{-1}(T) \subset |V|$ is closed (resp.\ open). The same holds for $f^{-1}(T)$ and its inverse image in $|U|$. Since $U \to V$ is quasi-compact, surjective,  and flat we win by Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}.
Choose a commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes, $a, b$ are \'etale, and $u \in U$ mapping to $x$. We can find a geometric point $\overline{u} : \Spec(k) \to U$ lying over $u$ with $\overline{x} = a \circ \overline{u}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-geometric-lift-to-usual}. Set $\overline{v} = h \circ \overline{u}$ with image $v \in V$. We know that $$ \mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh} \quad\text{and}\quad \mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh} $$ see Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring}. We obtain a commutative diagram $$ \xymatrix{ \mathcal{O}_{U, u} \ar[r] & \mathcal{O}_{X, \overline{x}} \\ \mathcal{O}_{V, v} \ar[u] \ar[r] & \mathcal{O}_{Y, \overline{y}} \ar[u] } $$ of local rings with flat horizontal arrows. We have to show that the left vertical arrow is flat if and only if the right vertical arrow is. Algebra, Lemma \ref{algebra-lemma-flatness-descends-more-general} tells us $\mathcal{O}_{U, u}$ is flat over $\mathcal{O}_{V, v}$ if and only if $\mathcal{O}_{X, \overline{x}}$ is flat over $\mathcal{O}_{V, v}$. Hence the result follows from More on Flatness, Lemma \ref{flat-lemma-flat-up-down-henselization}.
Flatness of $(f_{small}, f^\sharp)$ is defined in terms of flatness of $\mathcal{O}_X$ as a $f_{small}^{-1}\mathcal{O}_Y$-module. This can be checked at stalks, see Modules on Sites, Lemma \ref{sites-modules-lemma-check-flat-stalks} and Properties of Spaces, Theorem \ref{spaces-properties-theorem-exactness-stalks}. But we've already seen that flatness of $f$ can be checked on stalks, see Lemma \ref{lemma-flat-at-point-etale-local-rings}.
Let $Y' \to Y$ be a surjective \'etale morphism such that $Y'$ is a disjoint union of affine schemes (Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}). Let $X' \to X \times_Y Y'$ be a surjective \'etale morphism such that $X'$ is a disjoint union of affine schemes. By Lemma \ref{lemma-flat-local} the morphism $X' \to Y'$ is flat. Set $V' = V \times_Y Y'$. By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} the inverse image of $Z$ in $Y'$ is the scheme theoretic image of $V' \to Y'$ and the inverse image of $Z'$ in $X'$ is the scheme theoretic image of $V' \times_{Y'} X' \to X'$. Since $X' \to X$ is surjective \'etale, it suffices to prove the result in the case of the morphisms $X' \to Y'$ and $V' \to Y'$. Thus we may assume $X$ and $Y$ are affine schemes. In this case $V$ is a quasi-compact algebraic space. Choose an affine scheme $W$ and a surjective \'etale morphism $W \to V$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). It is clear that the scheme theoretic image of $V \to Y$ agrees with the scheme theoretic image of $W \to Y$ and similarly for $V \times_Y X \to Y$ and $W \times_Y X \to X$. Thus we reduce to the case of schemes which is Morphisms, Lemma \ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}.
During this proof we fix a geometric proof $\overline{x} : \Spec(k) \to X$ over $x$ and we denote $\overline{y} = f \circ \overline{x}$ its image in $Y$. Given a diagram as in (1) we can find a geometric point $\overline{u} : \Spec(k) \to U$ lying over $u$ with $\overline{x} = a \circ \overline{u}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-geometric-lift-to-usual}. Set $\overline{v} = h \circ \overline{u}$ with image $v \in V$. We know that $$ \mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh} \quad\text{and}\quad \mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh} $$ see Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring}. We obtain a commutative diagram $$ \xymatrix{ \mathcal{O}_{U, u} \ar[r] & \mathcal{O}_{X, \overline{x}} \\ \mathcal{O}_{V, v} \ar[u] \ar[r] & \mathcal{O}_{Y, \overline{y}} \ar[u] } $$ of local rings. Finally, we have $$ \mathcal{F}_{\overline{x}} = (\varphi^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} $$ by Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-quasi-coherent}. Thus Algebra, Lemma \ref{algebra-lemma-flatness-descends-more-general} tells us $(\varphi^*\mathcal{F})_u$ is flat over $\mathcal{O}_{V, v}$ if and only if $\mathcal{F}_{\overline{x}}$ is flat over $\mathcal{O}_{V, v}$. Hence the result follows from More on Flatness, Lemma \ref{flat-lemma-flat-up-down-henselization}.
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to V \times_Y Y'$. Then $U' = V' \times_V U$ is a scheme endowed with a surjective \'etale morphism $U' = V' \times_V U \to Y' \times_Y X = X'$. Pick $u' \in U'$ mapping to $x' \in |X'|$. Then we can check flatness of $\mathcal{F}'$ at $x'$ over $Y'$ in terms of flatness of $\mathcal{F}'|_{U'}$ at $u'$ over $V'$. Hence the lemma follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-locus-base-change}.
Pick $\overline{x}$ and $\overline{y}$ as in part (3) and denote $\overline{z}$ the induced geometric point of $Z$. Via the characterization of flatness in Lemmas \ref{lemma-flat-at-point} and \ref{lemma-flat-at-point-etale-local-rings} the lemma reduces to a purely algebraic question on the local ring map $\mathcal{O}_{Z, \overline{z}} \to \mathcal{O}_{Y, \overline{y}}$ and the module $\mathcal{F}_{\overline{x}}$. Part (1) follows from Algebra, Lemma \ref{algebra-lemma-composition-flat}. We remark that condition (2)(b) guarantees that $\mathcal{F}_{\overline{x}}/ \mathfrak m_{\overline{y}} \mathcal{F}_{\overline{x}}$ is nonzero. Hence (2)(a) $+$ (2)(b) imply that $\mathcal{F}_{\overline{x}}$ is a faithfully flat $\mathcal{O}_{Y, \overline{y}}$-module, see Algebra, Lemma \ref{algebra-lemma-ff}. Thus (2) is a special case of (3). Finally, (3) follows from Algebra, Lemma \ref{algebra-lemma-flat-permanence}.
Pick a geometric point $\overline{x}$ of $X$ and denote $\overline{y}$ the image in $Y$ and $\overline{z}$ the image in $Z$. Via the characterization of flatness in Lemmas \ref{lemma-flat-at-point} and \ref{lemma-flat-at-point-etale-local-rings} and the description of the stalk of $f^*\mathcal{G}$ at $\overline{x}$ of Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback-quasi-coherent} the lemma reduces to a purely algebraic question on the local ring maps $\mathcal{O}_{Z, \overline{z}} \to \mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ and the module $\mathcal{G}_{\overline{y}}$. This algebraic statement is Algebra, Lemma \ref{algebra-lemma-flatness-descends-more-general}.
Choose a surjective \'etale morphism $\varphi : V \to Y$ where $V$ is a scheme. Choose a surjective \'etale morphism $U \to V \times_Y X$ where $U$ is a scheme. Then it suffices to prove the lemma for $U \to V$ and the quasi-coherent $\mathcal{O}_V$-module $\varphi^*\mathcal{F}$. Hence this lemma follows from the case of schemes, see Morphisms, Lemma \ref{morphisms-lemma-pf-flat-module-open}.
We can choose commutative diagrams $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } \quad\quad \xymatrix{ u \ar[d] \ar[r] & v \ar[d] \ar[r] & w \ar[d] \\ x \ar[r] & y \ar[r] & z } $$ where $U, V, W$ are schemes and the vertical arrows are \'etale. By definition the morphism $U \to V$ is locally quasi-finite which implies that $\kappa(v) \subset \kappa(u)$ is finite, see Morphisms, Lemma \ref{morphisms-lemma-residue-field-quasi-finite}. Hence the result is clear.
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to X \times_Y V$. By Lemma \ref{lemma-finite-type-points-surjective-morphism} we can find a finite type point $u \in U$ mapping to $x$. After shrinking $U$ we may assume $u \in U$ is closed (Morphisms, Lemma \ref{morphisms-lemma-identify-finite-type-points}). Let $v \in V$ be the image of $u$. By Morphisms, Lemma \ref{morphisms-lemma-jacobson-finite-type-points} the extension $\kappa(u)/\kappa(v)$ is finite. This finishes the proof.
Choose a diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } $$ with $U, V, W$ schemes and vertical arrows \'etale and surjective. (See Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}.) Choose $u \in U$ mapping to $x$. Set $v, w$ equal to the images of $u$ in $V, W$. Apply Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point-additive} to the top row and the points $u, v, w$. Details omitted.
Choose a surjective \'etale morphism $V \to Y$ with $V$ a scheme. Choose a surjective \'etale morphism $U \to V \times_Y X$ with $U$ a scheme. Choose a surjective \'etale morphism $V' \to V \times_Y Y'$ with $V'$ a scheme. Set $U' = V' \times_V U$. Then the induced morphism $U' \to X'$ is also surjective and \'etale (argument omitted). Choose $u' \in U'$ mapping to $x'$. At this point parts (1) and (2) follow by applying Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change} to the diagram of schemes involving $U', U, V', V$ and the point $u'$. To prove (3) first choose $v \in V$ mapping to $y$. Then using Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} we can choose $v' \in V'$ mapping to $y'$ and $v$ and $u \in U$ mapping to $x$ and $v$. Finally, according to Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change} we can choose $u' \in U'$ mapping to $v'$ and $u$ such that the integer is zero. Then taking $x' \in |X'|$ the image of $u'$ works.