hcju/stacksqa huggingface dataset:
The assumptions imply that $Y$ is Noetherian (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}). Then $Y$ is quasi-separated, hence has a dense open subscheme (Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}). Thus we may assume that $Y$ is a Noetherian scheme. By removing intersections of irreducible components of $Y$ (use Topology, Lemma \ref{topology-lemma-Noetherian} and Properties, Lemma \ref{properties-lemma-Noetherian-topology}) we may assume that $Y$ is a disjoint union of irreducible Noetherian schemes. Since there is an immersion $$ \mathbf{A}^n_X \amalg \mathbf{A}^m_X \longrightarrow \mathbf{A}^{\max(n, m) + 1}_X $$ (details omitted) we see that it suffices to prove the result in case $Y$ is irreducible. \medskip\noindent Assume $Y$ is an irreducible scheme. Let $T \subset |X|$ be the closure of the image of $f : Y \to X$. Note that since $|Y|$ and $|X|$ are sober topological spaces (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated-sober}) $T$ is irreducible with a unique generic point $\xi$ which is the image of the generic point $\eta$ of $Y$. Let $\mathcal{I} \subset X$ be a quasi-coherent sheaf of ideals cutting out the reduced induced space structure on $T$ (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}). Since $\mathcal{O}_{Y, \eta}$ is an Artinian local ring we see that for some $n > 0$ we have $f^{-1}\mathcal{I}^n \mathcal{O}_{Y, \eta} = 0$. As $f^{-1}\mathcal{I}\mathcal{O}_Y$ is a finite type quasi-coherent ideal we conclude that $f^{-1}\mathcal{I}^n\mathcal{O}_V = 0$ for some nonempty open $V \subset Y$. Let $Z \subset X$ be the closed subspace cut out by $\mathcal{I}^n$. By construction $V \to Y \to X$ factors through $Z$. Because $\mathbf{A}^n_Z \to \mathbf{A}^n_X$ is an immersion, we may replace $X$ by $Z$ and $Y$ by $V$. Hence we reach the situation where $Y$ and $X$ are irreducible and $Y \to X$ maps the generic point of $Y$ onto the generic point of $X$. \medskip\noindent Assume $Y$ and $X$ are irreducible, $Y$ is a scheme, and $Y \to X$ maps the generic point of $Y$ onto the generic point of $X$. By Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme} $X$ has a dense open subscheme $U \subset X$. Choose a nonempty affine open $V \subset Y$ whose image in $X$ is contained in $U$. By Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S} we may factor $V \to U$ as $V \to \mathbf{A}^n_U \to U$. Composing with $\mathbf{A}^n_U \to \mathbf{A}^n_X$ we obtain the desired immersion.
Let $W$ be an affine scheme and let $\varphi : W \to X$ be a surjective \'etale morphism, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. If (1) holds, then $\varphi^{-1}(U) \to W$ is quasi-compact, hence $\varphi^{-1}(U)$ is quasi-compact, hence $U$ is quasi-compact (as $|\varphi^{-1}(U)| \to |U|$ is surjective). If (2) holds, then $\varphi^{-1}(U)$ is quasi-compact because $\varphi$ is quasi-compact since $X$ is quasi-separated (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}). Hence $\varphi^{-1}(U) \to W$ is a quasi-compact morphism of schemes by Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}. It follows that $U \to X$ is quasi-compact by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-local}. Thus (1) and (2) are equivalent. \medskip\noindent Assume (1) and (2). By Properties of Spaces, Lemma \ref{spaces-properties-lemma-reduced-closed-subspace} there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subspace structure on $|X| \setminus |U|$. Note that $\mathcal{J}|_U = \mathcal{O}_U$ which is an $\mathcal{O}_U$-modules of finite type. As $U$ is quasi-compact it follows from Lemma \ref{lemma-directed-colimit-finite-type} that there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_U = \mathcal{J}|_U$. Then $|X| \setminus |U| = |V(\mathcal{I})|$ and we obtain (3). Conversely, if $\mathcal{I}$ is as in (3), then $\varphi^{-1}(U) \subset W$ is a quasi-compact open by the lemma for schemes (Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}) applied to $\varphi^{-1}\mathcal{I}$ on $W$. Thus (2) holds.
Omitted. Hint: The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}) so that Lemma \ref{lemma-sections-annihilated-by-ideal} applies to $\mathcal{I}$ and $f_*\mathcal{F}$.
Omitted. Hints: $|X| \setminus |f|^{-1}T$ is the support of the open subspace $U = f^{-1}V \subset X$. Since $V \to Y$ is quasi-compact, so is $U \to X$ (by base change). The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent. Hence Lemma \ref{lemma-sections-supported-on-closed-subset} applies to $T$ and $f_*\mathcal{F}$ as well as to $|f|^{-1}T$ and $\mathcal{F}$. The equality of the given quasi-coherent modules is immediate from the definitions.
We may and do view $f : X \to Y$ as a morphism of algebraic space over $\Spec(\mathbf{Z})$ (see Spaces, Definition \ref{spaces-definition-base-change}). Note that a finite morphism is affine and universally closed, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-universally-closed}. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-universally-closed-separated} we see that $Y$ is a separated algebraic space. As $f$ is surjective and $X$ is quasi-compact we see that $Y$ is quasi-compact. \medskip\noindent By Lemma \ref{lemma-finite-in-finite-and-finite-presentation} we can write $X = \lim X_a$ with each $X_a \to Y$ finite and of finite presentation. By Lemma \ref{lemma-limit-is-affine} we see that $X_a$ is affine for $a$ large enough. Hence we may and do assume that $f : X \to Y$ is finite, surjective, and of finite presentation. \medskip\noindent By Proposition \ref{proposition-approximate} we may write $Y = \lim Y_i$ as a directed limit of algebraic spaces of finite presentation over $\mathbf{Z}$. By Lemma \ref{lemma-descend-finite-presentation} we can find $0 \in I$ and a morphism $X_0 \to Y_0$ of finite presentation such that $X_i = X_0 \times_{Y_0} Y_i$ for $i \geq 0$ and such that $X = \lim_i X_i$. By Lemma \ref{lemma-descend-finite} we see that $X_i \to Y_i$ is finite for $i$ large enough. By Lemma \ref{lemma-descend-surjective} we see that $X_i \to Y_i$ is surjective for $i$ large enough. By Lemma \ref{lemma-limit-is-affine} we see that $X_i$ is affine for $i$ large enough. Hence for $i$ large enough we can apply Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian} to conclude that $Y_i$ is affine. This implies that $Y$ is affine and we conclude.
Let $U' \subset X_{red}$ be an open affine subscheme. Let $U \subset X$ be the open subspace corresponding to the open $|U'| \subset |X_{red}| = |X|$. Then $U' \to U$ is surjective and integral. Hence $U$ is affine by Proposition \ref{proposition-affine}. Thus every point is contained in an open subscheme of $X$, i.e., $X$ is a scheme.
An integral morphism is representable by definition, hence if $Y$ is a scheme, so is $X$. Conversely, assume that $X$ is a scheme. Let $U \subset X$ be an affine open. An integral morphism is closed and $|f|$ is bijective, hence $|f|(|U|) \subset |Y|$ is open as the complement of $|f|(|X| \setminus |U|)$. Let $V \subset Y$ be the open subspace with $|V| = |f|(|U|)$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-open-subspaces}. Then $U \to V$ is integral and surjective, hence $V$ is an affine scheme by Proposition \ref{proposition-affine}. This concludes the proof.
Assumptions (1) and (2) imply that $B_{red} = B'_{red}$. Set $X' = X \times_B B'$. Then $X' \to X$ is closed immersion and $X'_{red} = X_{red}$. Let $U \to B$ be an \'etale morphism with $U$ affine. Then $X' \times_B U \to X \times_B U$ is a closed immersion of algebraic spaces inducing an isomorphism on underlying reduced spaces. Since $X' \times_B U$ is a scheme (as $B' \to B$ and $X' \to B'$ are representable) so is $X \times_B U$ by Lemma \ref{lemma-reduction-scheme}. Hence $X \to B$ is representable too. Thus we reduce to the case of schemes, see Morphisms, Lemma \ref{morphisms-lemma-check-closed-infinitesimally}.
We prove this by Zorn's lemma. Let $\mathcal{Z}$ be the set of closed subspaces $Z$ which are not schemes ordered by inclusion. By assumption $\mathcal{Z}$ contains $X$, hence is nonempty. If $Z_\alpha$ is a totally ordered subset of $\mathcal{Z}$, then $Z = \bigcap Z_\alpha$ is in $\mathcal{Z}$. Namely, $$ Z = \lim Z_\alpha $$ and the transition morphisms are affine. Thus we may apply Lemma \ref{lemma-limit-is-scheme} to see that if $Z$ were a scheme, then so would one of the $Z_\alpha$. (This works even if $Z = \emptyset$, but note that by Lemma \ref{lemma-limit-nonempty} this cannot happen.) Thus $\mathcal{Z}$ has minimal elements by Zorn's lemma.
We see that $X$ is reduced by Lemma \ref{lemma-reduction-scheme}. Choose closed subsets $T_1 \subset |X|$ and $T_2 \subset |X|$ such that $|X| = T_1 \cup T_2$. If $T_1$ and $T_2$ are proper closed subsets, then the corresponding reduced induced closed subspaces $Z_1, Z_2 \subset X$ (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}) are schemes and so is $Z = Z_1 \times_X Z_2 = Z_1 \cap Z_2$ as a closed subscheme of either $Z_1$ or $Z_2$. Observe that the coproduct $Z_1 \amalg_Z Z_2$ exists in the category of schemes, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-closed-immersions}. One way to proceed, is to show that $Z_1 \amalg_Z Z_2$ is isomorphic to $X$, but we cannot use this here as the material on pushouts of algebraic spaces comes later in the theory. Instead we will use Lemma \ref{lemma-affine} to find an affine neighbourhood of every point. Namely, let $x \in |X|$. If $x \not \in Z_1$, then $x$ has a neighbourhood which is a scheme, namely, $X \setminus Z_1$. Similarly if $x \not \in Z_2$. If $x \in Z = Z_1 \cap Z_2$, then we choose an affine open $U \subset Z_1 \amalg_Z Z_2$ containing $z$. Then $U_1 = Z_1 \cap U$ and $U_2 = Z_2 \cap U$ are affine opens whose intersections with $Z$ agree. Since $|Z_1| = T_1$ and $|Z_2| = T_2$ are closed subsets of $|X|$ which intersect in $|Z|$, we find an open $W \subset |X|$ with $W \cap T_1 = |U_1|$ and $W \cap T_2 = |U_2|$. Let $W$ denote the corresponding open subspace of $X$. Then $x \in |W|$ and the morphism $U_1 \amalg U_2 \to W$ is a surjective finite morphism whose source is an affine scheme. Thus $W$ is an affine scheme by Lemma \ref{lemma-affine}.
Let $x \in |X|$. It suffices to find an open neighbourhood $U$ of $s = f(x)$ such that $X \times_S U$ is a scheme. As $X \times_S \Spec(\mathcal{O}_{S, s})$ is a scheme, then, since $\mathcal{O}_{S, s} = \colim \mathcal{O}_S(U)$ where the colimit is over affine open neighbourhoods of $s$ in $S$ we see that $$ X \times_S \Spec(\mathcal{O}_{S, s}) = \lim X \times_S U $$ By Lemma \ref{lemma-limit-is-scheme} we see that $X \times_S U$ is a scheme for some $U$.
We prove this by Zorn's lemma. Let $\mathcal{I}$ be the set of ideals $I$ such that $X_{A/I}$ is not a scheme. By assumption $\mathcal{I}$ contains $(0)$. If $I_\alpha$ is a chain of ideals in $\mathcal{I}$, then $I = \bigcup I_\alpha$ is in $\mathcal{I}$. Namely, $A/I = \colim A/I_\alpha$, hence $$ X_{A/I} = \lim X_{A/I_\alpha} $$ Thus we may apply Lemma \ref{lemma-limit-is-scheme} to see that if $X_{A/I}$ were a scheme, then so would be one of the $X_{A/I_\alpha}$. Thus $\mathcal{I}$ has maximal elements by Zorn's lemma.
First, since $S = U \cup W$ is a Zariski covering, we see that the category of sheaves on $(\Sch/S)_{fppf}$ is equivalent to the category of triples $(\mathcal{F}_U, \mathcal{F}_W, \varphi)$ where $\mathcal{F}_U$ is a sheaf on $(\Sch/U)_{fppf}$, $\mathcal{F}_W$ is a sheaf on $(\Sch/W)_{fppf}$, and $$ \varphi : \mathcal{F}_U|_{(\Sch/U \cap W)_{fppf}} \longrightarrow \mathcal{F}_W|_{(\Sch/U \cap W)_{fppf}} $$ is an isomorphism. See Sites, Lemma \ref{sites-lemma-mapping-property-glue} (note that no other gluing data are necessary because $U \times_S U = U$, $W \times_S W = W$ and that the cocycle condition is automatic for the same reason). Now, if the sheaf $\mathcal{F}$ on $(\Sch/S)_{fppf}$ maps to $(\mathcal{F}_U, \mathcal{F}_W, \varphi)$ via this equivalence, then $\mathcal{F}$ is an algebraic space if and only if $\mathcal{F}_U$ and $\mathcal{F}_W$ are algebraic spaces. This follows immediately from Algebraic Spaces, Lemma \ref{spaces-lemma-glueing-algebraic-spaces} as $\mathcal{F}_U \to \mathcal{F}$ and $\mathcal{F}_W \to \mathcal{F}$ are representable by open immersions and cover $\mathcal{F}$. Finally, in this case the algebraic space $\mathcal{F}$ is of finite presentation over $S$ if and only if $\mathcal{F}_U$ is of finite presentation over $U$ and $\mathcal{F}_W$ is of finite presentation over $W$ by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-quasi-compact-local}, \ref{spaces-morphisms-lemma-separated-local}, and \ref{spaces-morphisms-lemma-finite-presentation-local}.
Let $W \subset S$ be an open neighbourhood of $s$. The functor $$ FP_S \to FP_U \times_{FP_{W \setminus \{s\}}} FP_W $$ is an equivalence of categories by Lemma \ref{lemma-relative-glueing}. We have $\mathcal{O}_{S, s} = \colim \mathcal{O}_W(W)$ where $W$ runs over the affine open neighbourhoods of $s$. Hence $\Spec(\mathcal{O}_{S, s}) = \lim W$ where $W$ runs over the affine open neighbourhoods of $s$. Thus the category of algebraic spaces of finite presentation over $\Spec(\mathcal{O}_{S, s})$ is the limit of the category of algebraic spaces of finite presentation over $W$ where $W$ runs over the affine open neighbourhoods of $s$, see Lemma \ref{lemma-descend-finite-presentation}. For every affine open $s \in W$ we see that $U \cap W$ is quasi-compact as $U \to S$ is quasi-compact. Hence $V = \lim W \cap U = \lim W \setminus \{s\}$ is a limit of quasi-compact and quasi-separated schemes (see Limits, Lemma \ref{limits-lemma-directed-inverse-system-has-limit}). Thus also the category of algebraic spaces of finite presentation over $V$ is the limit of the categories of algebraic spaces of finite presentation over $W \cap U$ where $W$ runs over the affine open neighbourhoods of $s$. The lemma follows formally from a combination of these results.
Let $W \subset S$ be an open neighbourhood of $s$. By Lemma \ref{lemma-relative-glueing} the functor $$ FP_{U \cup W} \longrightarrow FP_U \times_{FP_{U \cap W}} FP_W $$ is an equivalence of categories. We have $\mathcal{O}_{S, s} = \colim \mathcal{O}_W(W)$ where $W$ runs over the affine open neighbourhoods of $s$. Hence $\Spec(\mathcal{O}_{S, s}) = \lim W$ where $W$ runs over the affine open neighbourhoods of $s$. Thus the category of algebraic spaces of finite presentation over $\Spec(\mathcal{O}_{S, s})$ is the limit of the category of algebraic spaces of finite presentation over $W$ where $W$ runs over the affine open neighbourhoods of $s$, see Lemma \ref{lemma-descend-finite-presentation}. For every affine open $s \in W$ we see that $U \cap W$ is quasi-compact as $U \to S$ is quasi-compact. Hence $V = \lim W \cap U$ is a limit of quasi-compact and quasi-separated schemes (see Limits, Lemma \ref{limits-lemma-directed-inverse-system-has-limit}). Thus also the category of algebraic spaces of finite presentation over $V$ is the limit of the categories of algebraic spaces of finite presentation over $W \cap U$ where $W$ runs over the affine open neighbourhoods of $s$. The lemma follows formally from a combination of these results.
Since $V \to W$ is separated we see that $V \times_W V = \Delta(V) \amalg U$ for some open and closed subspace $U$ of $V \times_W V$. By the assumption that $f^{-1}T \to T$ is an isomorphism we see that $U \times_W T = \emptyset$, i.e., the two projections $U \to V$ maps into $V^0$. \medskip\noindent Given $h : Y \to V$ in the right hand category, consider the contravariant functor $X$ on $(\Sch/S)_{fppf}$ defined by the rule $$ X(T) = \{(w, y) \mid w :  T \to W,\ y : T \times_{w, W} V \to Y\text{ morphism over }V\} $$ Denote $g : X \to W$ the map sending $(w, y) \in X(T)$ to $w \in W(T)$. Since $h^{-1}V^0 \to V^0$ is an isomorphism, we see that if $w : T \to W$ maps into $W^0$, then there is a unique choice for $h$. In other words $X \times_{g, W} W^0 = W^0$. On the other hand, consider a $T$-valued point $(w, y, v)$ of $X \times_{g, W, f} V$. Then $w = f \circ v$ and $$ y : T \times_{f \circ v, W} V \longrightarrow V $$ is a morphism over $V$. Consider the morphism $$ T \times_{f \circ v, W} V \xrightarrow{(v, \text{id}_V)} V \times_W V = V \amalg U $$ The inverse image of $V$ is $T$ embedded via $(\text{id}_T, v) : T \to T \times_{f \circ v, W} V$. The composition $y' = y \circ (\text{id}_T, v) : T \to Y$ is a morphism with $v = h \circ y'$ which determines $y$ because the restriction of $y$ to the other part is uniquely determined as $U$ maps into $V^0$ by the second projection. It follows that $X \times_{g, W, f} V \to Y$, $(w, y, v) \mapsto y'$ is an isomorphism. \medskip\noindent Thus if we can show that $X$ is an algebraic space, then we are done. Since $V \to W$ is separated and \'etale it is representable by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable} (and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-locally-quasi-finite}). Of course $W^0 \to W$ is representable and \'etale as it is an open immersion. Thus $$ W^0 \amalg Y = X \times_{g, W} W^0 \amalg X \times_{g, W, f} V = X \times_{g, W} (W^0 \amalg V) \longrightarrow X $$ is representable, surjective, and \'etale by Spaces, Lemmas \ref{spaces-lemma-base-change-representable-transformations} and \ref{spaces-lemma-base-change-representable-transformations-property}. Thus $X$ is an algebraic space by Spaces, Lemma \ref{spaces-lemma-etale-locally-representable-by-space-gives-space}.
If $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, finite, so is $h$ as a base change of $g$ by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-quasi-compact}, \ref{spaces-morphisms-lemma-base-change-separated}, \ref{spaces-morphisms-lemma-base-change-finite-presentation}, \ref{spaces-morphisms-lemma-base-change-finite-type}, \ref{spaces-morphisms-lemma-base-change-proper}, \ref{spaces-morphisms-lemma-base-change-integral}. Conversely, let $P$ be a property of morphisms of algebraic spaces which is \'etale local on the base and which holds for the identity morphism of any algebraic space. Since $\{W^0 \to W, V \to W\}$ is an \'etale covering, to prove that $g$ has $P$ it suffices to show that $h$ has $P$. Thus we conclude using Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-quasi-compact-local}, \ref{spaces-morphisms-lemma-separated-local}, \ref{spaces-morphisms-lemma-finite-presentation-local}, \ref{spaces-morphisms-lemma-finite-type-local}, \ref{spaces-morphisms-lemma-proper-local}, \ref{spaces-morphisms-lemma-integral-local}.
Let $a : (W, w) \to (X, x)$ be an elementary \'etale neighbourhood of $x$ with $W$ affine as in Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}. Since $x$ is a closed point of $X$ and $w$ is the unique point of $W$ lying over $x$, we see that $w$ is a closed point of $W$. Since $a$ is \'etale and identifies residue fields at $x$ and $w$, it follows that $a$ induces an isomorphism $a^{-1}x \to x$ (as closed subspaces of $X$ and $W$). Thus we may apply Lemma \ref{lemma-excision-modifications} and \ref{lemma-excision-modifications-properties} to reduce the problem to the case where $X$ is an affine scheme. \medskip\noindent Assume $X$ is an affine scheme. Recall that $\mathcal{O}_{X, x}^h$ is the colimit of $\Gamma(U, \mathcal{O}_U)$ over affine elementary \'etale neighbourhoods $(U, u) \to (X, x)$. Recall that the category of these neighbourhoods is cofiltered, see Decent Spaces, Lemma \ref{decent-spaces-lemma-elementary-etale-neighbourhoods} or More on Morphisms, Lemma \ref{more-morphisms-lemma-elementary-etale-neighbourhoods}. Then $\Spec(\mathcal{O}_{X, x}^h) = \lim U$ and $V = \lim U \setminus \{u\}$ (Lemma \ref{lemma-directed-inverse-system-has-limit}) where the limits are taken over the same category. Thus by Lemma \ref{lemma-descend-finite-presentation} The category on the right is the colimit of the categories for the pairs $(U, u)$. And by the material in the first paragraph, each of these categories is equivalent to the category for the pair $(X, x)$. This finishes the proof.
We will deduce this from the corresponding result for morphisms of schemes. Let $y \in |Y|$ be the image of $z$. First we choose an affine \'etale neighbourhood $(U, u) \to (Y, y)$ and then we choose an affine \'etale neighbourhood $(V, v) \to (Z, z)$ such that the morphism $V \to Y$ factors through $U$. Then we may replace \begin{enumerate} \item $X \to Y$ by $X \times_Y U \to U$, \item $Z \to Y$ by $V \to U$, \item $z$ by $v$, and \item $T$ by its inverse image in $|(X \times_Y U) \times_U V| = |X \times_Y V|$. \end{enumerate} In fact, below we will show that after replacing $V$ by an affine open neighbourhood of $v$ there will be a morphism $a : V \to Z'$ for some $Z' \to U$ of finite presentation and a closed subset $T'$ of $|(X \times_Y U) \times_U Z'| = |X \times_Y Z'|$ such that $T$ maps into $T'$ and $a(v) \not \in \Im(T' \to |Z'|)$. Thus we may and do assume that $Z$ and $Y$ are affine schemes with the proviso that we need to find a solution where $V$ is an open neighbourhood of $z$. \medskip\noindent Since $f$ is quasi-compact and $Y$ is affine, the algebraic space $X$ is quasi-compact. Choose an affine scheme $W$ and a surjective \'etale morphism $W \to X$. Let $T_W \subset |W \times_Y Z|$ be the inverse image of $T$. Then $z$ is not in the image of $T_W$. By the schemes case (Limits, Lemma \ref{limits-lemma-separate}) we can find an open neighbourhood $V \subset Z$ of $z$ a commutative diagram of schemes $$ \xymatrix{ V \ar[d] \ar[r]_a & Z' \ar[d]^b \\ Z \ar[r]^g & Y, } $$ and a closed subset $T' \subset |W \times_Y Z'|$ such that \begin{enumerate} \item the morphism $b : Z' \to Y$ is locally of finite presentation, \item with $z' = a(z)$ we have $z' \not \in \Im(T' \to Z')$, and \item $T_W \cap |W \times_Y V|$ maps into $T'$ via $|W \times_Y V| \to |W \times_Y Z'|$. \end{enumerate} The commutative diagram $$ \xymatrix{ W \times_Y V \ar[r]_b \ar[d]_c & W \times_Y Z' \ar[d]_q \\ X \times_Y V \ar[r]^a & X \times_Y Z' } $$ is cartesian. The vertical maps are surjective \'etale hence surjective and open. Also $T_1 = T_W \cap |W \times_Y V|$ is the inverse image of $T_2 = T \cap |X \times_Y V|$ by $c$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} we get $b(T_1) = q^{-1}(a(T_2))$. By Topology, Lemma \ref{topology-lemma-open-morphism-quotient-topology} we get $$ q^{-1}(\overline{a(T_1)}) = \overline{q^{-1}(a(T_1))} = \overline{b(T_2)} \subset T' $$ As $q$ is surjective the image of $\overline{a(T_1)} \to |Z'|$ does not contain $z'$ since the same is true for $T'$. This concludes the proof.