hcju/stacksqa huggingface dataset:
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. Thus the lemma follows from Lemmas \ref{lemma-descending-property-surjective}, \ref{lemma-descending-property-universally-injective}, and \ref{lemma-descending-property-universally-open}.
Being locally of finite type is preserved under base change, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-type}. Being locally of finite type is Zariski local on the base, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-type-characterize}. Finally, let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is locally of finite type. Let $U \subset X$ be an affine open. Then $U' = S' \times_S U$ is affine and of finite type over $S'$. Write $S = \Spec(R)$, $S' = \Spec(R')$, $U = \Spec(A)$, and $U' = \Spec(A')$. We know that $R \to R'$ is faithfully flat, $A' = R' \otimes_R A$ and $R' \to A'$ is of finite type. We have to show that $R \to A$ is of finite type. This is the result of Algebra, Lemma \ref{algebra-lemma-finite-type-descends}. It follows that $f$ is locally of finite type. Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
Being locally of finite presentation is preserved under base change, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-presentation}. Being locally of finite type is Zariski local on the base, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}. Finally, let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is locally of finite presentation. Let $U \subset X$ be an affine open. Then $U' = S' \times_S U$ is affine and of finite type over $S'$. Write $S = \Spec(R)$, $S' = \Spec(R')$, $U = \Spec(A)$, and $U' = \Spec(A')$. We know that $R \to R'$ is faithfully flat, $A' = R' \otimes_R A$ and $R' \to A'$ is of finite presentation. We have to show that $R \to A$ is of finite presentation. This is the result of Algebra, Lemma \ref{algebra-lemma-finite-presentation-descends}. It follows that $f$ is locally of finite presentation. Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
Being flat is preserved under arbitrary base change, see Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}. Being flat is Zariski local on the base by definition. Finally, let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is flat. Let $U \subset X$ be an affine open. Then $U' = S' \times_S U$ is affine. Write $S = \Spec(R)$, $S' = \Spec(R')$, $U = \Spec(A)$, and $U' = \Spec(A')$. We know that $R \to R'$ is faithfully flat, $A' = R' \otimes_R A$ and $R' \to A'$ is flat. Goal: Show that $R \to A$ is flat. This follows immediately from Algebra, Lemma \ref{algebra-lemma-flatness-descends}. Hence $f$ is flat. Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
The property of being an open immersion is stable under base change, see Schemes, Lemma \ref{schemes-lemma-base-change-immersion}. The property of being an open immersion is Zariski local on the base (this is obvious). \medskip\noindent Let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is an open immersion. We claim that $f$ is an open immersion. Then $f'$ is universally open, and universally injective. Hence we conclude that $f$ is universally open by Lemma \ref{lemma-descending-property-universally-open}, and universally injective by Lemma \ref{lemma-descending-property-universally-injective}. In particular $f(X) \subset S$ is open. If for every affine open $U \subset f(X)$ we can prove that $f^{-1}(U) \to U$ is an isomorphism, then $f$ is an open immersion and we're done. If $U' \subset S'$ denotes the inverse image of $U$, then $U' \to U$ is a faithfully flat morphism of affines and $(f')^{-1}(U') \to U'$ is an isomorphism (as $f'(X')$ contains $U'$ by our choice of $U$). Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Let $S' \to S$ be a flat surjective morphism of affine schemes, let $f : X \to S$ be a morphism, and assume that the base change $f' : X' \to S'$ is an isomorphism. We have to show that $f$ is an isomorphism also. It is clear that $f$ is surjective, universally injective, and universally open (see arguments above for the last two). Hence $f$ is bijective, i.e., $f$ is a homeomorphism. Thus $f$ is affine by Morphisms, Lemma \ref{morphisms-lemma-homeomorphism-affine}. Since $$ \mathcal{O}(S') \to \mathcal{O}(X') = \mathcal{O}(S') \otimes_{\mathcal{O}(S)} \mathcal{O}(X) $$ is an isomorphism and since $\mathcal{O}(S) \to \mathcal{O}(S')$ is faithfully flat this implies that $\mathcal{O}(S) \to \mathcal{O}(X)$ is an isomorphism. Thus $f$ is an isomorphism. This finishes the proof of the claim above. Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
A base change of an affine morphism is affine, see Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}. Being affine is Zariski local on the base, see Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. Finally, let $g : S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is affine. In other words, $X'$ is affine, say $X' = \Spec(A')$. Also write $S = \Spec(R)$ and $S' = \Spec(R')$. We have to show that $X$ is affine. \medskip\noindent By Lemmas \ref{lemma-descending-property-quasi-compact} and \ref{lemma-descending-property-separated} we see that $X \to S$ is separated and quasi-compact. Thus $f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Hence $f_*\mathcal{O}_X = \widetilde{A}$ for some $R$-algebra $A$. In fact $A = \Gamma(X, \mathcal{O}_X)$ of course. Also, by flat base change (see for example Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) we have $g^*f_*\mathcal{O}_X = f'_*\mathcal{O}_{X'}$. In other words, we have $A' = R' \otimes_R A$. Consider the canonical morphism $$ X \longrightarrow \Spec(A) $$ over $S$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. By the above the base change of this morphism to $S'$ is an isomorphism. Hence it is an isomorphism by Lemma \ref{lemma-descending-property-isomorphism}. Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
Let $f : X \to Y$ be a morphism of schemes. Let $\{Y_i \to Y\}$ be an fpqc covering. Assume that each $f_i : Y_i \times_Y X \to Y_i$ is a closed immersion. This implies that each $f_i$ is affine, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-affine}. By Lemma \ref{lemma-descending-property-affine} we conclude that $f$ is affine. It remains to show that $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is surjective. For every $y \in Y$ there exists an $i$ and a point $y_i \in Y_i$ mapping to $y$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} the sheaf $f_{i, *}(\mathcal{O}_{Y_i \times_Y X})$ is the pullback of $f_*\mathcal{O}_X$. By assumption it is a quotient of $\mathcal{O}_{Y_i}$. Hence we see that $$ \Big( \mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y \Big) \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{Y_i, y_i} $$ is surjective. Since $\mathcal{O}_{Y_i, y_i}$ is faithfully flat over $\mathcal{O}_{Y, y}$ this implies the surjectivity of $\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y$ as desired.
Let $f : X \to Y$ be a morphism of schemes. Let $\{g_i : Y_i \to Y\}$ be an fpqc covering. Assume that each $f_i : Y_i \times_Y X \to Y_i$ is quasi-affine. This implies that each $f_i$ is quasi-compact and separated. By Lemmas \ref{lemma-descending-property-quasi-compact} and \ref{lemma-descending-property-separated} this implies that $f$ is quasi-compact and separated. Consider the sheaf of $\mathcal{O}_Y$-algebras $\mathcal{A} = f_*\mathcal{O}_X$. By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} it is a quasi-coherent $\mathcal{O}_Y$-algebra. Consider the canonical morphism $$ j : X \longrightarrow \underline{\Spec}_Y(\mathcal{A}) $$ see Constructions, Lemma \ref{constructions-lemma-canonical-morphism}. By flat base change (see for example Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) we have $g_i^*f_*\mathcal{O}_X = f_{i, *}\mathcal{O}_{X'}$ where $g_i : Y_i \to Y$ are the given flat maps. Hence the base change $j_i$ of $j$ by $g_i$ is the canonical morphism of Constructions, Lemma \ref{constructions-lemma-canonical-morphism} for the morphism $f_i$. By assumption and Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine} all of these morphisms $j_i$ are quasi-compact open immersions. Hence, by Lemmas \ref{lemma-descending-property-quasi-compact} and \ref{lemma-descending-property-open-immersion} we see that $j$ is a quasi-compact open immersion. Hence by Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine} again we conclude that $f$ is quasi-affine.
Let $f : X \to Y$ be a morphism of schemes. Let $\{Y_i \to Y\}$ be an fpqc covering. Write $X_i = Y_i \times_Y X$ and $f_i : X_i \to Y_i$ the base change of $f$. Also denote $q_i : Y_i \to Y$ the given flat morphisms. Assume each $f_i$ is a quasi-compact immersion. By Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms} each $f_i$ is separated. By Lemmas \ref{lemma-descending-property-quasi-compact} and \ref{lemma-descending-property-separated} this implies that $f$ is quasi-compact and separated. Let $X \to Z \to Y$ be the factorization of $f$ through its scheme theoretic image. By Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image} the closed subscheme $Z \subset Y$ is cut out by the quasi-coherent sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$ as $f$ is quasi-compact. By flat base change (see for example Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}; here we use $f$ is separated) we see $f_{i, *}\mathcal{O}_{X_i}$ is the pullback $q_i^*f_*\mathcal{O}_X$. Hence $Y_i \times_Y Z$ is cut out by the quasi-coherent sheaf of ideals $q_i^*\mathcal{I} = \Ker(\mathcal{O}_{Y_i} \to f_{i, *}\mathcal{O}_{X_i})$. By Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-immersion} the morphisms $X_i \to Y_i \times_Y Z$ are open immersions. Hence by Lemma \ref{lemma-descending-property-open-immersion} we see that $X \to Z$ is an open immersion and hence $f$ is a immersion as desired (we already saw it was quasi-compact).
Let $f : X \to S$ be a morphism of schemes, and let $\{S_i \to S\}$ be an fpqc covering such that each base change $f_i : X_i \to S_i$ is locally quasi-finite. We have already seen (Lemma \ref{lemma-descending-property-locally-finite-type}) that ``locally of finite type'' is fpqc local on the base, and hence we see that $f$ is locally of finite type. Then it follows from Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-finite} that $f$ is locally quasi-finite. The quasi-finite case follows as we have already seen that ``quasi-compact'' is fpqc local on the base (Lemma \ref{lemma-descending-property-quasi-compact}).
A morphism is syntomic if and only if it is locally of finite presentation, flat, and has locally complete intersections as fibres. We have seen already that being flat and locally of finite presentation are fpqc local on the base (Lemmas \ref{lemma-descending-property-flat}, and \ref{lemma-descending-property-locally-finite-presentation}). Hence the result follows for syntomic from Morphisms, Lemma \ref{morphisms-lemma-set-points-where-fibres-lci}.
A morphism is smooth if and only if it is locally of finite presentation, flat, and has smooth fibres. We have seen already that being flat and locally of finite presentation are fpqc local on the base (Lemmas \ref{lemma-descending-property-flat}, and \ref{lemma-descending-property-locally-finite-presentation}). Hence the result follows for smooth from Morphisms, Lemma \ref{morphisms-lemma-set-points-where-fibres-smooth}.
A morphism is unramified (resp.\ G-unramified) if and only if it is locally of finite type (resp.\ finite presentation) and its diagonal morphism is an open immersion (see Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}). We have seen already that being locally of finite type (resp.\ locally of finite presentation) and an open immersion is fpqc local on the base (Lemmas \ref{lemma-descending-property-locally-finite-presentation}, \ref{lemma-descending-property-locally-finite-type}, and \ref{lemma-descending-property-open-immersion}). Hence the result follows formally.
A morphism is \'etale if and only if it flat and G-unramified. See Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale}. We have seen already that being flat and G-unramified are fpqc local on the base (Lemmas \ref{lemma-descending-property-flat}, and \ref{lemma-descending-property-unramified}). Hence the result follows.
Being finite locally free is equivalent to being finite, flat and locally of finite presentation (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}). Hence this follows from Lemmas \ref{lemma-descending-property-finite}, \ref{lemma-descending-property-flat}, and \ref{lemma-descending-property-locally-finite-presentation}. If $f : Z \to U$ is finite locally free, and $\{U_i \to U\}$ is a surjective family of morphisms such that each pullback $Z \times_U U_i \to U_i$ has degree $d$, then $Z \to U$ has degree $d$, for example because we can read off the degree in a point $u \in U$ from the fibre $(f_*\mathcal{O}_Z)_u \otimes_{\mathcal{O}_{U, u}} \kappa(u)$.
Let $f : X \to S$ be a morphism of schemes. Let $\{S_i \to S\}$ be an fpqc covering, and assume each of the base changes $f_i : X_i \to S_i$ of $f$ is a monomorphism. Let $a, b : T \to X$ be two morphisms such that $f \circ a = f \circ b$. We have to show that $a = b$. Since $f_i$ is a monomorphism we see that $a_i = b_i$, where $a_i, b_i : S_i \times_S T \to X_i$ are the base changes. In particular the compositions $S_i \times_S T \to T \to X$ are equal. Since $\coprod S_i \times_S T \to T$ is an epimorphism (see e.g.\ Lemma \ref{lemma-fpqc-universal-effective-epimorphisms}) we conclude $a = b$.
We will use the criterion of Lemma \ref{lemma-descending-properties-morphisms} to prove this. By Divisors, Definition \ref{divisors-definition-regular-immersion} being a Koszul-regular (resp.\ $H_1$-regular, quasi-regular) immersion is Zariski local on the base. By Divisors, Lemma \ref{divisors-lemma-flat-base-change-regular-immersion} being a Koszul-regular (resp.\ $H_1$-regular, quasi-regular) immersion is preserved under flat base change. The final hypothesis (3) of Lemma \ref{lemma-descending-properties-morphisms} translates into the following algebra statement: Let $A \to B$ be a faithfully flat ring map. Let $I \subset A$ be an ideal. If $IB$ is locally on $\Spec(B)$ generated by a Koszul-regular (resp.\ $H_1$-regular, quasi-regular) sequence in $B$, then $I \subset A$ is locally on $\Spec(A)$ generated by a Koszul-regular (resp.\ $H_1$-regular, quasi-regular) sequence in $A$. This is More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-regular-ideal}.
The property of being an immersion is stable under base change, see Schemes, Lemma \ref{schemes-lemma-base-change-immersion}. The property of being an immersion is Zariski local on the base. Finally, let $\pi : S' \to S$ be a surjective morphism of affine schemes, which is flat and locally of finite presentation. Note that $\pi : S' \to S$ is open by Morphisms, Lemma \ref{morphisms-lemma-fppf-open}. Let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is an immersion. In particular we see that $f'(X') = \pi^{-1}(f(X))$ is locally closed. Hence by Topology, Lemma \ref{topology-lemma-open-morphism-quotient-topology} we see that $f(X) \subset S$ is locally closed. Let $Z \subset S$ be the closed subset $Z = \overline{f(X)} \setminus f(X)$. By Topology, Lemma \ref{topology-lemma-open-morphism-quotient-topology} again we see that $f'(X')$ is closed in $S' \setminus Z'$. Hence we may apply Lemma \ref{lemma-descending-property-closed-immersion} to the fpqc covering $\{S' \setminus Z' \to S \setminus Z\}$ and conclude that $f : X \to S \setminus Z$ is a closed immersion. In other words, $f$ is an immersion. Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
As $f$ is a flat, quasi-compact, surjective morphism we see $\{X \to Y\}$ is an fpqc covering of $Y$. The diagonal $\Delta : X \to X \times_Y X$ is an isomorphism. This implies that the base change of $f$ by $f$ is an isomorphism. Hence we see $f$ is an isomorphism by Lemma \ref{lemma-descending-property-isomorphism}.
[First proof] Let $f : X \to Y$ be an \'etale morphism which is universally injective. Then $f$ is open (Morphisms, Lemma \ref{morphisms-lemma-etale-open}) hence we can replace $Y$ by $f(X)$ and we may assume that $f$ is surjective. Then $f$ is bijective and open hence a homeomorphism. Hence $f$ is quasi-compact. Thus by Lemma \ref{lemma-flat-surjective-quasi-compact-monomorphism-isomorphism} it suffices to show that $f$ is a monomorphism. As $X \to Y$ is \'etale the morphism $\Delta_{X/Y} : X \to X \times_Y X$ is an open immersion by Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism} (and Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale}). As $f$ is universally injective $\Delta_{X/Y}$ is also surjective, see Morphisms, Lemma \ref{morphisms-lemma-universally-injective}. Hence $\Delta_{X/Y}$ is an isomorphism, i.e., $X \to Y$ is a monomorphism.
We have to show that $\Delta : X \to X \times_Y X$ is surjective, see Morphisms, Lemma \ref{morphisms-lemma-universally-injective}. As $X \to Y$ is separated, the image of $\Delta$ is closed. Thus if $\Delta$ is not surjective, we can find a generic point $\eta \in X \times_S X$ of an irreducible component of $X \times_S X$ which is not in the image of $\Delta$. The projection $\text{pr}_1 : X \times_Y X \to X$ is flat as a base change of the flat morphism $X \to Y$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}. Hence generalizations lift along $\text{pr}_1$, see Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}. We conclude that $\xi = \text{pr}_1(\eta) \in X^0$. However, assumptions (2) and (3) guarantee that the scheme $(X \times_Y X)_{f(\xi)}$ has at most one point for every $\xi \in X^0$. In other words, we have $\Delta(\xi) = \eta$ a contradiction.
Endow $Z$ with the reduced induced closed subscheme structure, see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}. For every $i$ the base change $Y_i \times_Y Z$ is a closed subscheme of $Y_i \times_Y X$ whose underlying closed subset is $Z_i$. By definition (via Cohomology of Schemes, Lemma \ref{coherent-lemma-closed-proper-over-base}) we conclude that the projections $Y_i \times_Y Z \to Y_i$ are proper morphisms. Hence $Z \to Y$ is a proper morphism by Lemma \ref{lemma-descending-property-proper}. Thus $Z$ is proper over $Y$ by definition.
The implication (1) $\Rightarrow$ (2) follows from Morphisms, Lemma \ref{morphisms-lemma-ample-base-change}. Assume $\mathcal{L}_i$ is ample on $X_i/S_i$ for every $i \in I$. By Morphisms, Definition \ref{morphisms-definition-relatively-ample} this implies that $X_i \to S_i$ is quasi-compact and by Morphisms, Lemma \ref{morphisms-lemma-relatively-ample-separated} this implies $X_i \to S$ is separated. Hence $f$ is quasi-compact and separated by Lemmas \ref{lemma-descending-property-quasi-compact} and \ref{lemma-descending-property-separated}. \medskip\noindent This means that $\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}$ is a quasi-coherent graded $\mathcal{O}_S$-algebra (Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}). Moreover, the formation of $\mathcal{A}$ commutes with flat base change by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}. In particular, if we set $\mathcal{A}_i = \bigoplus_{d \geq 0} f_{i, *}\mathcal{L}_i^{\otimes d}$ then we have $\mathcal{A}_i = g_i^*\mathcal{A}$. It follows that the natural maps $\psi_d : f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}$ of $\mathcal{O}_X$ pullback to give the natural maps $\psi_{i, d} : f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}$ of $\mathcal{O}_{X_i}$-modules. Since $\mathcal{L}_i$ is ample on $X_i/S_i$ we see that for any point $x_i \in X_i$, there exists a $d \geq 1$ such that $f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}$ is surjective on stalks at $x_i$. This follows either directly from the definition of a relatively ample module or from Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}. If $x \in X$, then we can choose an $i$ and an $x_i \in X_i$ mapping to $x$. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X_i, x_i}$ is flat hence faithfully flat, we conclude that for every $x \in X$ there exists a $d \geq 1$ such that $f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}$ is surjective on stalks at $x$. This implies that the open subset $U(\psi) \subset X$ of Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj} corresponding to the map $\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$ of graded $\mathcal{O}_X$-algebras is equal to $X$. Consider the corresponding morphism $$ r_{\mathcal{L}, \psi} : X \longrightarrow \underline{\text{Proj}}_S(\mathcal{A}) $$ It is clear from the above that the base change of $r_{\mathcal{L}, \psi}$ to $S_i$ is the morphism $r_{\mathcal{L}_i, \psi_i}$ which is an open immersion by Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}. Hence $r_{\mathcal{L}, \psi}$ is an open immersion by Lemma \ref{lemma-descending-property-open-immersion} and we conclude $\mathcal{L}$ is ample on $X/S$ by Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}.
Consider the union $W$ of the images $g(X') \subset X$ of morphisms $g : X' \to X$ with the properties: \begin{enumerate} \item $g$ is flat and locally of finite presentation, syntomic, smooth, or \'etale, and \item the composition $X' \to X \to Y$ has property $\mathcal{P}$. \end{enumerate} Since such a morphism $g$ is open (see Morphisms, Lemma \ref{morphisms-lemma-fppf-open}) we see that $W \subset X$ is an open subset of $X$. Since $\mathcal{P}$ is local in the $\tau$ topology the restriction $f|_W : W \to Y$ has property $\mathcal{P}$ because we are given a $\tau$ covering $\{X' \to W\}$ of $W$ such that the pullbacks have $\mathcal{P}$. This proves the existence of $W(f)$. The compatibility stated in the last sentence follows immediately from the construction of $W(f)$.
This follows almost immediately from the definition of a $\tau$-covering, see Topologies, Definition \ref{topologies-definition-fpqc-covering} \ref{topologies-definition-fppf-covering} \ref{topologies-definition-etale-covering} \ref{topologies-definition-smooth-covering}, or \ref{topologies-definition-syntomic-covering} and Topologies, Lemma \ref{topologies-lemma-fpqc-affine}, \ref{topologies-lemma-fppf-affine}, \ref{topologies-lemma-etale-affine}, \ref{topologies-lemma-smooth-affine}, or \ref{topologies-lemma-syntomic-affine}. Details omitted. (Hint: Use locality on the source and target to reduce the verification of property $\mathcal{P}$ to the case of a morphism between affines. Then apply (1) and (4).)