hcju/stacksqa huggingface dataset:
Note that $f^{-1}B, f^{-1}U, f^{-1}R \in \Sh((\Sch'/S)_{fppf})$ are algebraic spaces by Spaces, Lemma \ref{spaces-lemma-change-big-site} and hence $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$ is a groupoid in algebraic spaces over $f^{-1}B$. Thus the statement makes sense. \medskip\noindent The category $u_p[U/_{\!p}R]$ is the localization of the category $u_{pp}[U/_{\!p}R]$ at right multiplicative system $I$ of morphisms. An object of $u_{pp}[U/_{\!p}R]$ is a triple $$ (T', \phi : T' \to T, x) $$ where $T' \in \Ob((\Sch'/S)_{fppf})$, $T \in \Ob((\Sch/S)_{fppf})$, $\phi$ is a morphism of schemes over $S$, and $x : T \to U$ is a morphism of sheaves on $(\Sch/S)_{fppf}$. Note that the morphism of schemes $\phi : T' \to T$ is the same thing as a morphism $\phi : T' \to u(T)$, and since $u(T)$ represents $f^{-1}T$ it is the same thing as a morphism $T' \to f^{-1}T$. Moreover, as $f^{-1}$ on algebraic spaces is fully faithful, see Spaces, Lemma \ref{spaces-lemma-fully-faithful}, we may think of $x$ as a morphism $x : f^{-1}T \to f^{-1}U$ as well. From now on we will make such identifications without further mention. A morphism $$ (a, a', \alpha) : (T'_1, \phi_1 : T'_1 \to T_1, x_1) \longrightarrow (T'_2, \phi_2 : T'_2 \to T_2, x_2) $$ of $u_{pp}[U/_{\!p}R]$ is a commutative diagram $$ \xymatrix{ & & U \\ T'_1 \ar[d]_{a'} \ar[r]_{\phi_1} & T_1 \ar[d]_a \ar[ru]^{x_1} \ar[r]_\alpha & R \ar[d]^t \ar[u]_s \\ T'_2 \ar[r]^{\phi_2} & T_2 \ar[r]^{x_2} & U } $$ and such a morphism is an element of $I$ if and only if $T'_1 = T'_2$ and $a' = \text{id}$. We define a functor $$ u_{pp}[U/_{\!p}R] \longrightarrow [f^{-1}U/_{\!p}f^{-1}R] $$ by the rules $$ (T', \phi : T' \to T, x) \longmapsto (x \circ \phi : T' \to f^{-1}U) $$ on objects and $$ (a, a', \alpha) \longmapsto (\alpha \circ \phi_1 : T'_1 \to f^{-1}R) $$ on morphisms as above. It is clear that elements of $I$ are transformed into isomorphisms as $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$ is a groupoid in algebraic spaces over $f^{-1}B$. Hence this functor factors in a canonical way through a functor $$ u_p[U/_{\!p}R] \longrightarrow [f^{-1}U/_{\!p}f^{-1}R] $$ Applying stackification we obtain a functor of stacks $$ f^{-1}[U/R] \longrightarrow [f^{-1}U/f^{-1}R] $$ over $(\Sch'/S)_{fppf}$, as by Stacks, Lemma \ref{stacks-lemma-technical-up} the stack $f^{-1}[U/R]$ is the stackification of $u_p[U/_{\!p}R]$. \medskip\noindent At this point we have a morphism of stacks, and to verify that it is an equivalence it suffices to show that it is fully faithful and that objects are locally in the essential image, see Stacks, Lemmas \ref{stacks-lemma-characterize-ff} and \ref{stacks-lemma-characterize-essentially-surjective-when-ff}. The statement on objects holds as $f^{-1}R$ admits a surjective \'etale morphism $f^{-1}W \to f^{-1}R$ for some object $W$ of $(\Sch/S)_{fppf}$. To show that the functor is ``full'', it suffices to show that morphisms are locally in the image of the functor which holds as $f^{-1}U$ admits a surjective \'etale morphism $f^{-1}W \to f^{-1}U$ for some object $W$ of $(\Sch/S)_{fppf}$. We omit the proof that the functor is faithful.
The group algebraic space $G \to U$ is the base change of $R \to U \times_B U$ by the diagonal morphism $U \to U \times_B U$, see Lemma \ref{lemma-groupoid-stabilizer}. Hence if $j$ is separated (resp.\ locally separated, resp.\ quasi-separated), then $G \to U$ is separated (resp.\ locally separated, resp.\ quasi-separated). See Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-separated}. Thus (a) $\Rightarrow$ (b) in (1), (2), and (3). \medskip\noindent Conversely, if $G \to U$ is separated (resp.\ locally separated, resp.\ quasi-separated), then the morphism $e : U \to G$, as a section of the structure morphism $G \to U$ is a closed immersion (resp.\ an immersion, resp.\ quasi-compact), see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion}. Thus (b) $\Rightarrow$ (c) in (1), (2), and (3). \medskip\noindent If $e$ is a closed immersion (resp.\ an immersion, resp.\ quasi-compact) then by the result of Lemma \ref{lemma-diagram-diagonal} (and Spaces, Lemma \ref{spaces-lemma-base-change-immersions}, and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-quasi-compact}) we see that $\Delta_{R/U \times_B U}$ is a closed immersion (resp.\ an immersion, resp.\ quasi-compact). Thus (c) $\Rightarrow$ (a) in (1), (2), and (3).
As $f$ is finite it is representable. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Then $U = V \times_Y X$ is a scheme with a surjective \'etale morphism towards $X$ and a finite morphism $\psi : U \to V$ (the base change of $f$). Since $\psi_*(\mathcal{F}|_U) = f_*\mathcal{F}|_V$ the result of the lemma follows immediately from the schemes version which is Descent, Lemma \ref{descent-lemma-finite-over-finite-module}.
As $f$ is finite it is representable. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Then $U = V \times_Y X$ is a scheme with a surjective \'etale morphism towards $X$ and a finite morphism $\psi : U \to V$ (the base change of $f$). Since $\psi_*(\mathcal{F}|_U) = f_*\mathcal{F}|_V$ the result of the lemma follows immediately from the schemes version which is Descent, Lemma \ref{descent-lemma-finite-finitely-presented-module}.
By Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-refine-fpqc-schemes} we may assume each $T_i$ is a scheme. Choose a scheme $U$ and a surjective \'etale morphism $U \to T$. Then $\{T_i \times_T U \to U\}$ is an fpqc covering of $U$ and $T_i \times_T U$ is a scheme for each $i$. Hence we see that the collection of opens $W_i \times_T U$ comes from a unique open subscheme $W' \subset U$ by Descent, Lemma \ref{descent-lemma-open-fpqc-covering}. As $U \to X$ is open we can define $W \subset X$ the Zariski open which is the image of $W'$, see Properties of Spaces, Section \ref{spaces-properties-section-points}. We omit the verification that this works, i.e., that $W_i$ is the inverse image of $W$ for each $i$.
We know this is true if $\{T_i \to T\}$ is an fpqc covering of schemes, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. This is the key fact and we encourage the reader to skip the rest of the proof which is formal. Choose a scheme $U$ and a surjective \'etale morphism $U \to T$. Let $U_i$ be a scheme and let $U_i \to T_i \times_T U$ be a surjective \'etale morphism. Then $\{U_i \to U\}$ is an fpqc covering. This follows from Topologies on Spaces, Lemmas \ref{spaces-topologies-lemma-fpqc} and \ref{spaces-topologies-lemma-recognize-fpqc-covering}. By the above we have the result for $\{U_i \to U\}$. \medskip\noindent What this means is the following: Suppose that $b_i : T_i \to B$ is a family of morphisms with $b_i \circ \text{pr}_0 = b_j \circ \text{pr}_1$ as morphisms $T_i \times_T T_j \to B$. Then we let $a_i : U_i \to B$ be the composition of $U_i \to T_i$ with $b_i$. By what was said above we find a unique morphism $a : U \to B$ such that $a_i$ is the composition of $a$ with $U_i \to U$. The uniqueness guarantees that $a \circ \text{pr}_0 = a \circ \text{pr}_1$ as morphisms $U \times_T U \to B$. Then since $T = U/(U \times_T U)$ as a sheaf, we find that $a$ comes from a unique morphism $b : T \to B$. Chasing diagrams we find that $b$ is the morphism we are looking for.
Choose a scheme $W$ and a surjective \'etale morphism $W \to Z$. Choose a scheme $V$ and a surjective \'etale morphism $V \to W \times_Z Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. By assumption we can find an fppf covering $\{V_i \to V\}$ and lifts $V_i \to X$ of the morphism $V_i \to Y$. Since $U \to X$ is surjective \'etale we see that over the members of the fppf covering $\{V_i \times_X U \to V\}$ we have lifts into $U$. Hence $U \to V$ induces a surjection of sheaves on $(\Sch/S)_{fppf}$. By our definition of what it means to have property $P$ for a morphism of algebraic spaces (see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-flat}, Definition \ref{spaces-morphisms-definition-locally-finite-type}, and Definition \ref{spaces-morphisms-definition-locally-finite-presentation}) we see that $U \to W$ has $P$ and we have to show $V \to W$ has $P$. Thus we reduce the question to the case of morphisms of schemes which is treated in Descent, Lemma \ref{descent-lemma-curiosity}.
Since $\{X \to Y\}$ is an fppf covering, it induces a surjection of fppf sheaves (Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fppf-covering-surjective}) and the lemma is a special case of Lemma \ref{lemma-curiosity}. On the other hand, an easier argument is to deduce it from the analogue for schemes. Namely, the problem is \'etale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-finite-type-local} and \ref{spaces-morphisms-lemma-finite-presentation-local}). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}), we can choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma \ref{descent-lemma-flat-finitely-presented-permanence}.
We deduce this from the analogue for schemes. Namely, the problem is \'etale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-syntomic-local}, \ref{spaces-morphisms-lemma-smooth-local}, and \ref{spaces-morphisms-lemma-etale-local}). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}), we can choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma \ref{descent-lemma-syntomic-smooth-etale-permanence}.
We deduce this from the analogue for schemes. Namely, the problem is \'etale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-smooth-local} and \ref{spaces-morphisms-lemma-etale-local}). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}), we can choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma \ref{descent-lemma-smooth-permanence}.
We deduce this from the analogue for schemes. Namely, the problem is \'etale local on $B$ and $Y$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-syntomic-local}). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}), we can choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma \ref{descent-lemma-syntomic-permanence}.
Consider the map of \'etale local rings $\mathcal{O}_{Y, f(\overline{x})} \to \mathcal{O}_{X, \overline{x}}$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings} this is flat. Hence if $\mathcal{O}_{X, \overline{x}}$ has a unique minimal prime, so does $\mathcal{O}_{Y, f(\overline{x})}$ (by going down, see Algebra, Lemma \ref{algebra-lemma-flat-going-down}).
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to X \times_Y V$. As $f$ is surjective and flat, the morphism of schemes $U \to V$ is surjective and flat. In this way we reduce the problem to the case of schemes (as reducedness of $X$ and $Y$ is defined in terms of reducedness of $U$ and $V$, see Properties of Spaces, Section \ref{spaces-properties-section-types-properties}). The case of schemes is Descent, Lemma \ref{descent-lemma-descend-reduced}.
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to X \times_Y V$. As $f$ is surjective, flat, and locally of finite presentation the morphism of schemes $U \to V$ is surjective, flat, and locally of finite presentation. In this way we reduce the problem to the case of schemes (as being locally Noetherian for $X$ and $Y$ is defined in terms of being locally Noetherian of $U$ and $V$, see Properties of Spaces, Section \ref{spaces-properties-section-types-properties}). In the case of schemes the result follows from Descent, Lemma \ref{descent-lemma-Noetherian-local-fppf}.
By Lemma \ref{lemma-descend-locally-Noetherian} we know that $Y$ is locally Noetherian. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. It suffices to prove that the local rings of $V$ are all regular local rings, see Properties, Lemma \ref{properties-lemma-characterize-regular}. Choose a scheme $U$ and a surjective \'etale morphism $U \to X \times_Y V$. As $f$ is surjective and flat the morphism of schemes $U \to V$ is surjective and flat. By assumption $U$ is a regular scheme in particular all of its local rings are regular (by the lemma above). Hence the lemma follows from Algebra, Lemma \ref{algebra-lemma-flat-under-regular}.
Choose a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$ and $V$ are schemes, the vertical arrows are surjective and \'etale, and $U \to X \times_Y V$ is surjective \'etale. Observe that $X$ is a reduced algebraic space if and only if $U$ is a reduced scheme by our definition of reduced algebraic spaces in Properties of Spaces, Section \ref{spaces-properties-section-types-properties}. Similarly for $Y$ and $V$.  The morphism $U \to V$ is a smooth morphism of schemes, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-smooth-local}. Since being reduced is local for the smooth topology for schemes (Descent, Lemma \ref{descent-lemma-reduced-local-smooth}) we see that $U$ is reduced if $V$ is reduced. On the other hand, if $X \to Y$ is surjective, then $U \to V$ is surjective and in this case if $U$ is reduced, then $V$ is reduced.
Consider the union $W_{set} \subset |Y|$ of the images $g(|Y'|) \subset |Y|$ of morphisms $g : Y' \to Y$ with the properties: \begin{enumerate} \item $g$ is flat and locally of finite presentation, syntomic, smooth, or \'etale, and \item the base change $Y' \times_{g, Y} X \to Y'$ has property $\mathcal{P}$. \end{enumerate} Since such a morphism $g$ is open (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}) we see that $W_{set}$ is an open subset of $|Y|$. Denote $W \subset Y$ the open subspace whose underlying set of points is $W_{set}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-open-subspaces}. Since $\mathcal{P}$ is local in the $\tau$ topology the restriction $X_W \to W$ has property $\mathcal{P}$ because we are given a covering $\{Y' \to W\}$ of $W$ such that the pullbacks have $\mathcal{P}$. This proves the existence and proves that $W(f)$ has property (1). To see property (2) note that $W(f') \supset g^{-1}(W(f))$ because $\mathcal{P}$ is stable under base change by flat and locally of finite presentation, syntomic, smooth, or \'etale morphisms, see Lemma \ref{lemma-pullback-property-local-target}. On the other hand, if $Y'' \subset Y'$ is an open such that $X_{Y''} \to Y''$ has property $\mathcal{P}$, then $Y'' \to Y$ factors through $W$ by construction, i.e., $Y'' \subset g^{-1}(W(f))$. This proves (2). Assertion (3) follows from (2) because each morphism $Y_i \to Y$ is flat and locally of finite presentation, syntomic, smooth, or \'etale by our definition of a $\tau$-covering.
If $\mathcal{P}$ has property (2), then it is automatically stable under any base change. Hence the direct implication in Definition \ref{definition-property-morphisms-local}. \medskip\noindent Let $\{Y_i \to Y\}_{i \in I}$ be an fpqc covering of algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume each base change $f_i : Y_i \times_Y X \to Y_i$ has property $\mathcal{P}$. Our goal is to show that $f$ has $\mathcal{P}$. Let $Z$ be an affine scheme, and let $Z \to Y$ be a morphism. By (2) it suffices to show that the morphism of algebraic spaces $Z \times_Y X \to Z$ has $\mathcal{P}$. Since $\{Y_i \to Y\}_{i \in I}$ is an fpqc covering we know there exists a standard fpqc covering $\{Z_j \to Z\}_{j = 1, \ldots , n}$ and morphisms $Z_j \to Y_{i_j}$ over $Y$ for suitable indices $i_j \in I$. Since $f_{i_j}$ has $\mathcal{P}$ we see that $$ Z_j \times_Y X = Z_j \times_{Y_{i_j}} (Y_{i_j} \times_Y X) \longrightarrow Z_j $$ has $\mathcal{P}$ as a base change of $f_{i_j}$ (see first remark of the proof). Set $Z' = \coprod_{j = 1, \ldots, n} Z_j$, so that $Z' \to Z$ is a flat and surjective morphism of affine schemes over $S$. By (1) we conclude that $Z' \times_Y X \to Z'$ has property $\mathcal{P}$. Since this is the base change of the morphism $Z \times_Y X \to Z$ by the morphism $Z' \to Z$ we conclude that $Z \times_Y X \to Z$ has property $\mathcal{P}$ as desired.
We will use Lemma \ref{lemma-descending-properties-morphisms} to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-local}. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times_Z X \to Z'$ is quasi-compact. We have to show that $f$ is quasi-compact. To see this, using Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-local} again, it is enough to show that for every affine scheme $Y$ and morphism $Y \to Z$ the fibre product $Y \times_Z X$ is quasi-compact. Here is a picture: \begin{equation} \label{equation-cube} \vcenter{ \xymatrix{ Y \times_Z Z' \times_Z X \ar[dd] \ar[rr] \ar[rd] & & Z' \times_Z X \ar'[d][dd]^{f'} \ar[rd] \\ & Y \times_Z X \ar[dd] \ar[rr] & & X \ar[dd]^f \\ Y \times_Z Z' \ar'[r][rr] \ar[rd] & & Z' \ar[rd] \\ & Y \ar[rr] & & Z } } \end{equation} Note that all squares are cartesian and the bottom square consists of affine schemes. The assumption that $f'$ is quasi-compact combined with the fact that $Y \times_Z Z'$ is affine implies that $Y \times_Z Z' \times_Z X$ is quasi-compact. Since $$ Y \times_Z Z' \times_Z X \longrightarrow Y \times_Z X $$ is surjective as a base change of $Z' \to Z$ we conclude that $Y \times_Z X$ is quasi-compact, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-surjection-from-quasi-compact}. This finishes the proof.
A base change of a quasi-separated morphism is quasi-separated, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-separated}. Hence the direct implication in Definition \ref{definition-property-morphisms-local}. \medskip\noindent Let $\{Y_i \to Y\}_{i \in I}$ be an fpqc covering of algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume each base change $X_i := Y_i \times_Y X \to Y_i$ is quasi-separated. This means that each of the morphisms $$ \Delta_i : X_i \longrightarrow X_i \times_{Y_i} X_i = Y_i \times_Y (X \times_Y X) $$ is quasi-compact. The base change of a fpqc covering is an fpqc covering, see Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fpqc} hence $\{Y_i \times_Y (X \times_Y X) \to X \times_Y X\}$ is an fpqc covering of algebraic spaces. Moreover, each $\Delta_i$ is the base change of the morphism $\Delta : X \to X \times_Y X$. Hence it follows from Lemma \ref{lemma-descending-property-quasi-compact} that $\Delta$ is quasi-compact, i.e., $f$ is quasi-separated.
We will use Lemma \ref{lemma-descending-properties-morphisms} to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-local}. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times_Z X \to Z'$ is universally closed. We have to show that $f$ is universally closed. To see this, using Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-local} again, it is enough to show that for every affine scheme $Y$ and morphism $Y \to Z$ the map $|Y \times_Z X| \to |Y|$ is closed. Consider the cube (\ref{equation-cube}). The assumption that $f'$ is universally closed implies that $|Y \times_Z Z' \times_Z X| \to |Y \times_Z Z'|$ is closed. As $Y \times_Z Z' \to Y$ is quasi-compact, surjective, and flat as a base change of $Z' \to Z$ we see the map $|Y \times_Z Z'| \to |Y|$ is submersive, see Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}. Moreover the map $$ |Y \times_Z Z' \times_Z X| \longrightarrow |Y \times_Z Z'| \times_{|Y|} |Y \times_Z X| $$ is surjective, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. It follows by elementary topology that $|Y \times_Z X| \to |Y|$ is closed.
We will use Lemma \ref{lemma-descending-properties-morphisms} to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-local}. Let $Z' \to Z$ be a flat surjective morphism of affine schemes over $S$ and let $f : X \to Z$ be a morphism from an algebraic space to $Z$. Assume that the base change $f' : X' \to Z'$ is universally injective. Let $K$ be a field, and let $a, b : \Spec(K) \to X$ be two morphisms such that $f \circ a = f \circ b$. As $Z' \to Z$ is surjective there exists a field extension $K \subset K'$ and a morphism $\Spec(K') \to Z'$ such that the following solid diagram commutes $$ \xymatrix{ \Spec(K') \ar[rrd] \ar@{-->}[rd]_{a', b'} \ar[dd] \\  & X' \ar[r] \ar[d] & Z' \ar[d] \\ \Spec(K) \ar[r]^{a, b} & X \ar[r] & Z } $$ As the square is cartesian we get the two dotted arrows $a'$, $b'$ making the diagram commute. Since $X' \to Z'$ is universally injective we get $a' = b'$. This forces $a = b$ as $\{\Spec(K') \to \Spec(K)\}$ is an fpqc covering, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. Hence $f$ is universally injective as desired.
This can be proved in exactly the same manner as Lemma \ref{lemma-descending-property-universally-closed}. Alternatively, one can use that a map of topological spaces is a homeomorphism if and only if it is injective, surjective, and open. Thus a universal homeomorphism is the same thing as a surjective, universally injective, and universally open morphism. See Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-surjective} and Morphisms of Spaces, Definitions \ref{spaces-morphisms-definition-universally-injective}, \ref{spaces-morphisms-definition-surjective}, \ref{spaces-morphisms-definition-open}, \ref{spaces-morphisms-definition-universal-homeomorphism}. Thus the lemma follows from Lemmas \ref{lemma-descending-property-surjective}, \ref{lemma-descending-property-universally-injective}, and \ref{lemma-descending-property-universally-open}.
We will use Lemma \ref{lemma-descending-properties-morphisms} to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-type-local}. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times_Z X \to Z'$ is locally of finite type. We have to show that $f$ is locally of finite type. Let $U$ be a scheme and let $U \to X$ be surjective and \'etale. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-type-local} again, it is enough to show that $U \to Z$ is locally of finite type. Since $f'$ is locally of finite type, and since $Z' \times_Z U$ is a scheme \'etale over $Z' \times_Z X$ we conclude (by the same lemma again) that $Z' \times_Z U \to Z'$ is locally of finite type. As $\{Z' \to Z\}$ is an fpqc covering we conclude that $U \to Z$ is locally of finite type by Descent, Lemma \ref{descent-lemma-descending-property-locally-finite-type} as desired.