hcju/stacksqa huggingface dataset:
It is clear that (1) implies (2) and that (2) implies (3) by taking $V$ to be a disjoint union of affines \'etale over $Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. Assume $V \to Y$ is as in (3). Then for every affine open $W$ of $V$ we see that $W \times_Y X$ is an affine open of $V \times_Y X$. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} we conclude that $V \times_Y X$ is a scheme. Moreover the morphism $V \times_Y X \to V$ is affine. This means we can apply Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} because the class of integral (resp.\ finite) morphisms satisfies all the required properties (see Morphisms, Lemmas \ref{morphisms-lemma-base-change-finite} and Descent, Lemmas \ref{descent-lemma-descending-property-integral}, \ref{descent-lemma-descending-property-finite}, and \ref{descent-lemma-affine}). The conclusion of applying this lemma is that $f$ is representable and integral (resp.\ finite), i.e., (1) holds. \medskip\noindent The equivalence of (1) and (4) follows from the fact that being integral (resp.\ finite) is Zariski local on the target (the reference above shows that being integral or finite is in fact fpqc local on the target).
Let $f : X \to Y$ be a morphism of algebraic spaces. By Definition \ref{definition-integral} and Lemmas \ref{lemma-quasi-compact-local} and \ref{lemma-quasi-finite-local} both properties may be checked after base change to an affine over $Y$, i.e., we may assume $Y$ affine. If $f$ is finite then $X$ is a scheme. Hence the result follows from the corresponding result for schemes, see Morphisms, Lemma \ref{morphisms-lemma-finite-quasi-finite}.
Assume $g \circ f$ is finite (resp.\ integral) and $g$ separated. The base change $X \times_Z Y \to Y$ is finite (resp.\ integral) by Lemma \ref{lemma-base-change-integral}. The morphism $X \to X \times_Z Y$ is a closed immersion as $Y \to Z$ is separated, see Lemma \ref{lemma-section-immersion}. A closed immersion is finite (resp.\ integral), see Lemma \ref{lemma-closed-immersion-finite}. The composition of finite (resp.\ integral) morphisms is finite (resp.\ integral), see Lemma \ref{lemma-composition-integral}. Thus we win.
Assume $f$ is finite locally free (as defined in Section \ref{section-representable}). This means that for every morphism $V \to Y$ whose source is a scheme the base change $f' : V \times_Y X \to V$ is a finite locally free morphism of schemes. This in turn means (by the definition of a finite locally free morphism of schemes) that $f'_*\mathcal{O}_{V \times_Y X}$ is a finite locally free $\mathcal{O}_V$-module. We may choose $V \to Y$ to be surjective and \'etale. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules} we conclude the restriction of $f_*\mathcal{O}_X$ to $V$ is finite locally free. Hence by Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object} applied to the sheaf $f_*\mathcal{O}_X$ on $Y_{spaces, \etale}$ we conclude that $f_*\mathcal{O}_X$ is finite locally free. \medskip\noindent Conversely, assume $f$ is affine and that $f_*\mathcal{O}_X$ is a finite locally free $\mathcal{O}_Y$-module. Let $V$ be a scheme, and let $V \to Y$ be a surjective \'etale morphism. Again by Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules} we see that $f'_*\mathcal{O}_{V \times_Y X}$ is finite locally free. Hence $f' : V \times_Y X \to V$ is finite locally free (as it is also affine). By Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} we conclude that $f$ is finite locally free (use Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-locally-free} Descent, Lemmas \ref{descent-lemma-descending-property-finite-locally-free} and \ref{descent-lemma-affine}). Thus we win.
It is clear that (1) implies (2) and that (2) implies (3) by taking $V$ to be a disjoint union of affines \'etale over $Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. Assume $V \to Y$ is as in (3). Then for every affine open $W$ of $V$ we see that $W \times_Y X$ is an affine open of $V \times_Y X$. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} we conclude that $V \times_Y X$ is a scheme. Moreover the morphism $V \times_Y X \to V$ is affine. This means we can apply Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} because the class of finite locally free morphisms satisfies all the required properties (see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-locally-free} Descent, Lemmas \ref{descent-lemma-descending-property-finite-locally-free} and \ref{descent-lemma-affine}). The conclusion of applying this lemma is that $f$ is representable and finite locally free, i.e., (1) holds. \medskip\noindent The equivalence of (1) and (4) follows from the fact that being finite locally free is Zariski local on the target (the reference above shows that being finite locally free is in fact fpqc local on the target).
In each of the three cases the morphism is representable and you can check the property after base change by a surjective \'etale morphism $V \to Y$, see Lemmas \ref{lemma-integral-local}, \ref{lemma-finite-locally-free-local}, \ref{lemma-flat-local}, and \ref{lemma-finite-presentation-local}. If $Y$ is locally Noetherian, then $V$ is locally Noetherian. Hence the result follows from the corresponding result in the schemes case, see Morphisms, Lemma \ref{morphisms-lemma-finite-flat}.
Let $(V, g)$ and $(V', g')$ be representatives of $\varphi$. Then $g, g'$ agree on a dense open subspace $W \subset V \cap V'$. On the other hand, the equalizer $E$ of $g|_{V \cap V'}$ and $g'|_{V \cap V'}$ is a closed subspace of $V \cap V'$ because it is the base change of $\Delta : Y \to Y \times_S Y$ by the morphism $V \cap V' \to Y \times_S Y$ given by $g|_{V \cap V'}$ and $g'|_{V \cap V'}$. Now $W \subset E$ implies that $|E| = |V \cap V'|$. As $V \cap V'$ is reduced we conclude $E = V \cap V'$ scheme theoretically, i.e., $g|_{V \cap V'} = g'|_{V \cap V'}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-map-into-reduction}. It follows that we can glue the representatives $g : V \to Y$ of $\varphi$ to a morphism $f : U \to Y$ because $\coprod V \to U$ is a surjection of fppf sheaves and $\coprod_{V, V'} V \cap V' = (\coprod V) \times_U (\coprod V)$.
Assume $X$ and $Y$ are birational. Let $f : U \to Y$ and $g : V \to X$ define inverse dominant rational maps from $X$ to $Y$ and from $Y$ to $X$. After shrinking $U$ we may assume $f : U \to Y$ factors through $V$. As $g \circ f$ is the identity as a dominant rational map, we see that the composition $U \to V \to X$ is the identity on a dense open of $U$. Thus after replacing $U$ by a smaller open we may assume that $U \to V \to X$ is the inclusion of $U$ into $X$. By symmetry we find there exists an open subspace $V' \subset V$ such that $g|_{V'} : V' \to X$ factors through $U \subset X$ and such that $V' \to U \to Y$ is the identity. The inverse image of $|V'|$ by $|U| \to |V|$ is an open of $|U|$ and hence equal to $|U'|$ for some open subspace $U' \subset U$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-open-subspaces}. Then $U' \subset U \to V$ factors as $U' \to V'$. Similarly $V' \to U$ factors as $V' \to U'$. The reader finds that $U' \to V'$ and $V' \to U'$ are mutually inverse morphisms of algebraic spaces over $S$ and the proof is complete.
By Properties of Spaces, Lemma \ref{spaces-properties-lemma-alternative} it suffices to prove that the rule given above defines a quasi-coherent module on $X_{affine, \etale}$. To see this it suffices to show the following: Let $U_1 \to U_2$ be a morphism of affine objects of $X_\etale$. Say $U_i = \Spec(R_i)$. Say $\mathcal{A}|_{(U_1)_\etale}$ is the quasi-coherent sheaf associated to the $R_2$-algebra $A$. Let $A' \subset A$ be the integral closure of $R_2$ in $A$. Then $A' \otimes_{R_2} R_1$ is the integral closure of $R_1$ in $A \otimes_{R_2} R_1$. This is Algebra, Lemma \ref{algebra-lemma-integral-closure-commutes-etale}.
Let $\mathcal{O}' \subset f_*\mathcal{O}_Y$ be the integral closure of $\mathcal{O}_X$ as in Definition \ref{definition-normalization-X-in-Y}. The morphism $\nu$ is integral by construction, which proves (1). Assume given a factorization $f = \pi \circ g$ with $\pi : Z \to X$ integral as in (2). By Definition \ref{definition-integral} $\pi$ is affine, and hence $Z$ is the relative spectrum of a quasi-coherent sheaf of $\mathcal{O}_X$-algebras $\mathcal{B}$. The morphism $g : X \to Z$ corresponds to a map of $\mathcal{O}_X$-algebras $\chi : \mathcal{B} \to f_*\mathcal{O}_Y$. Since $\mathcal{B}(U)$ is integral over $\mathcal{O}_X(U)$ for every affine $U$ \'etale over $X$ (by Definition \ref{definition-integral}) we see from Lemma \ref{lemma-integral-closure} that $\chi(\mathcal{B}) \subset \mathcal{O}'$. By the functoriality of the relative spectrum Lemma \ref{lemma-affine-equivalence-algebras} this provides us with a unique morphism $h : X' \to Z$. We omit the verification that the diagram commutes. \medskip\noindent It is clear that (1) and (2) characterize the factorization $f = \nu \circ f'$ since it characterizes it as an initial object in a category. The morphism $h$ in (2) is integral by Lemma \ref{lemma-finite-permanence}. Given a factorization $g = \pi' \circ g'$ with $\pi' : Z' \to Z$ integral, we get a factorization $f = (\pi \circ \pi') \circ g'$ and we get a morphism $h' : X' \to Z'$. Uniqueness implies that $\pi' \circ h' = h$. Hence the characterization (1), (2) applies to the morphism $h : X' \to Z$ which gives the last statement of the lemma.
By Lemma \ref{lemma-properties-normalization} and the definitions, we may assume that $X = \Spec(A)$ is affine. Then $X' = \Spec(A')$ where $A'$ is the integral closure of $A$ in $\Gamma(Y, \mathcal{O}_Y)$ and $x'$ corresponds to a minimal prime of $A'$. Choose a surjective \'etale morphism $V \to Y$ where $V = \Spec(B)$ is affine. Then $A' \to B$ is injective, hence every minimal prime of $A'$ is the image of a minimal prime of $B$, see Algebra, Lemma \ref{algebra-lemma-injective-minimal-primes-in-image}. The lemma follows.
The question is \'etale local on $Y$, hence we may reduce to the case where $Y = \Spec(R)$ is affine. Let $h \in \Gamma(X, \mathcal{O}_X)$. We have to show that $h$ satisfies a monic equation over $R$. Think of $h$ as a morphism as in the following commutative diagram $$ \xymatrix{ X \ar[rr]_h \ar[rd]_f & & \mathbf{A}^1_Y \ar[ld] \\ & Y & } $$ Let $Z \subset \mathbf{A}^1_Y$ be the scheme theoretic image of $h$, see Definition \ref{definition-scheme-theoretic-image}. The morphism $h$ is quasi-compact as $f$ is quasi-compact and $\mathbf{A}^1_Y \to Y$ is separated, see Lemma \ref{lemma-quasi-compact-permanence}. By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} the morphism $X \to Z$ has dense image on underlying topological spaces. By Lemma \ref{lemma-universally-closed-permanence} the morphism $X \to Z$ is closed. Hence $h(X) = Z$ (set theoretically). Thus we can use Lemma \ref{lemma-image-proper-is-proper} to conclude that $Z \to Y$ is universally closed (and even proper). Since $Z \subset \mathbf{A}^1_Y$, we see that $Z \to Y$ is affine and proper, hence integral by Lemma \ref{lemma-integral-universally-closed}. Writing $\mathbf{A}^1_Y = \Spec(R[T])$ we conclude that the ideal $I \subset R[T]$ of $Z$ contains a monic polynomial $P(T) \in R[T]$. Hence $P(h) = 0$ and we win.
The question is \'etale local on $Y$, see Lemma \ref{lemma-properties-normalization}. Thus we may assume $Y = \Spec(R)$ is affine. Then $R$ is a Noetherian Nagata ring and we have to show that the integral closure of $R$ in $\Gamma(X, \mathcal{O}_X)$ is finite over $R$. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $\Gamma(X, \mathcal{O}_X) \subset \Gamma(U, \mathcal{O}_X)$. Since $R$ is Noetherian it suffices to show that the integral closure of $R$ in $\Gamma(U, \mathcal{O}_U)$ is finite over $R$. As $U \to Y$ is of finite type this follows from Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization-finite}.
The equivalence of (1) and (2) and the final statement follow from Descent, Lemma \ref{descent-lemma-locally-finite-nr-irred-local-fppf} and Properties of Spaces, Lemma \ref{spaces-properties-lemma-type-property}. It is clear that (3) implies (1) and (2). Conversely, assume (2) and let $Y \to X$ be an \'etale morphism of algebraic spaces with $Y$ quasi-compact. Then we can choose an affine scheme $V$ and a surjective \'etale morphism $V \to Y$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Since $V$ has finitely many irreducible components by (2) and since $|V| \to |Y|$ is surjective and continuous, we conclude that $|Y|$ has finitely many irreducible components.
Let $U \to X$ be a surjective \'etale morphism where $U$ is a scheme. Set $R = U \times_X U$ with projections $s, t : R \to U$ and $j = (t, s) : R \to U \times_S U$ so that $X = U/R$, see Spaces, Lemma \ref{spaces-lemma-space-presentation}. The assumption on $X$ means that the normalization $U^\nu$ of $U$ is defined, see Morphisms, Definition \ref{morphisms-definition-normalization}. By More on Morphisms, Lemma \ref{more-morphisms-lemma-normalization-and-smooth} taking normalization commutes with \'etale morphisms of schemes. Thus we see that the normalization $R^\nu$ of $R$ is isomorphic to both $R \times_{s, U} U^\nu$ and $U^\nu \times_{U, t} R$. Thus we obtain two \'etale morphisms $s^\nu : R^\nu \to U^\nu$ and $t^\nu : R^\nu \to U^\nu$ of schemes. The induced morphism $j^\nu : R^\nu \to U^\nu \times_S U^\nu$ is a monomorphism as $R^\nu$ is a subscheme of the restriction of $R$ to $U^\nu$. A formal computation with fibre products shows that $R^\nu \times_{s^\nu, U^\nu, t^\nu} R^\nu$ is the normalization of $R \times_{s, U, t} R$. Hence the (\'etale) morphism $c : R \times_{s, U, t} R \to R$ extends to $c^\nu$ as well. Combined we see that we obtain an \'etale equivalence relation. Setting $X^\nu = U^\nu/R^\nu$ (Spaces, Theorem \ref{spaces-theorem-presentation}) we see that we have $U^\nu = X^\nu \times_X U$ by Groupoids, Lemma \ref{groupoids-lemma-criterion-fibre-product}. We omit the verification that this property then holds for every \'etale morphism from a scheme to $X$.
Properties (1), (2), and (3) follow from the corresponding results for schemes (Morphisms, Lemma \ref{morphisms-lemma-normalization-normal}) combined with the fact that a point of a scheme is a generic point of an irreducible component if and only if the dimension of the local ring is zero (Properties, Lemma \ref{properties-lemma-generic-point}). \medskip\noindent Let $Z \to X$ be a morphism as in (4). Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. Choose a scheme $V$ and a surjective \'etale morphism $V \to U \times_X Z$. The condition on geometric points assures us that $V \to U$ maps generic points of irreducible components of $V$ to generic points of irreducible components of $U$. Thus we obtain a unique factorization $V \to U^\nu \to U$ by Morphisms, Lemma \ref{morphisms-lemma-normalization-normal}. The uniqueness guarantees us that the two maps $$ V \times_{U \times_X Z} V \to V \to U^\nu $$ agree because these maps are the unique factorization of the map $V \times_{U \times_X Z} V \to V \to U$. Since the algebraic space $U \times_X Z$ is equal to the quotient $V/V \times_{U \times_X Z} V$ (see Spaces, Section \ref{spaces-section-presentations}) we find a canonical morphism $U \times_X Z \to U^\nu$. Picture $$ \xymatrix{ U \times_X Z \ar[r] \ar[d] & U^\nu \ar[r] \ar[d] & U \ar[d] \\ Z \ar@/_/[rr] \ar@{..>}[r] & X^\nu \ar[r] & X } $$ To obtain the dotted arrow we note that the construction of the arrow $U \times_X Z$ is functorial in the \'etale morphism $U \to X$ (precise formulation and proof omitted). Hence if we set $R = U \times_X U$ with projections $s, t : R \to U$, then we obtain a morphism $R \times_X Z \to R^\nu$ commuting with $s, t : R \to U$ and $s^\nu, t^\nu : R^\nu \to U^\nu$. Recall that $X^\nu = U^\nu/R^\nu$, see proof of Lemma \ref{lemma-normalization}. Since $X = U/R$ a simple sheaf theoretic argument shows that $Z = (U \times_X Z)/(R \times_X Z)$. Thus the morphisms $U \times_X Z \to U^\nu$ and $R \times_X Z \to R^\nu$ define a morphism $Z \to X^\nu$ as desired.
Let $T \to Y$ be a morphism from a scheme into $Y$. If we can show that $X \times_Y T \to T$ is an open immersion, then we are done. Since being \'etale and being universally injective are properties of morphisms stable under base change (see Lemmas \ref{lemma-base-change-etale} and \ref{lemma-base-change-universally-injective}) we may assume that $Y$ is a scheme. Note that the diagonal $\Delta_{X/Y} : X \to X \times_Y X$ is \'etale, a monomorphism, and surjective by Lemma \ref{lemma-universally-injective}. Hence we see that $\Delta_{X/Y}$ is an isomorphism (see Spaces, Lemma \ref{spaces-lemma-surjective-flat-locally-finite-presentation}), in particular we see that $X$ is separated over $Y$. It follows that $X$ is a scheme too, by Proposition \ref{proposition-locally-quasi-finite-separated-over-scheme}. Finally, $X \to Y$ is an open immersion by the fundamental theorem for \'etale morphisms of schemes, see \'Etale Morphisms, Theorem \ref{etale-theorem-etale-radicial-open}.
Let $W \to Y$ be a surjective \'etale morphism where $W$ is a scheme. Then $W \times_Y X$ is a scheme as well. By Lemma \ref{lemma-properties-normalization} the algebraic space $W \times_Y Y'$ is representable and is the normalization of the scheme $W$ in the scheme $W \times_Y X$. Picture $$ \xymatrix{ W \times_Y X \ar[rd]_{(1, f)} \ar[rr]_{(1, f')} & & W \times_Y Y' \ar[ld]^{(1, \nu)} \\ & W & } $$ By More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-type-separated} the result of the lemma holds over $W$. Let $V' \subset W \times_Y Y'$ be the open subscheme such that \begin{enumerate} \item $(1, f')^{-1}(V') \to V'$ is an isomorphism, and \item $(1, f')^{-1}(V') \subset W \times_Y X$ is the set of points at which $(1, f)$ is quasi-finite. \end{enumerate} By Lemma \ref{lemma-locally-finite-type-quasi-finite-part} there is a maximal open set of points $U \subset X$ where $f$ is quasi-finite and $W \times_Y U = (1, f')^{-1}(V')$. The morphism $f'|_U : U \to Y'$ is an open immersion by Lemma \ref{lemma-closed-immersion-local} as its base change to $W$ is the isomorphism $(1, f')^{-1}(V') \to V'$ followed by the open immersion $V' \to W \times_Y Y'$. Setting $U' = \Im(U \to Y')$ finishes the proof (omitted: the verification that $(f')^{-1}(U') = U$).
By Lemma \ref{lemma-locally-quasi-finite-separated-representable} the morphism $f$ is representable. Hence we may apply Lemma \ref{lemma-finite-type-separated}. Thus there exists an open subspace $U' \subset Y'$ such that $(f')^{-1}(U') = X$ (!) and $X \to U'$ is an isomorphism! In other words, $f'$ is an open immersion. Note that $f'$ is quasi-compact as $f$ is quasi-compact and $\nu : Y' \to Y$ is separated (Lemma \ref{lemma-quasi-compact-permanence}). Hence for every affine scheme $Z$ and morphism $Z \to Y$ the fibre product $Z \times_Y X$ is a quasi-compact open subscheme of the affine scheme $Z \times_Y Y'$. Hence $f$ is quasi-affine by definition.
If for every morphism of algebraic spaces $Z \to Y$ the base change map $Z \times_Y X \to Z$ induces a homeomorphism $|Z \times_Y X| \to |Z|$, then the same is true whenever $Z$ is a scheme, which formally implies that $f$ is a universal homeomorphism in the sense of Section \ref{section-representable}. Conversely, if $f$ is a universal homeomorphism in the sense of Section \ref{section-representable} then $X \to Y$ is integral, universally injective and surjective (by Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication} and Morphisms, Lemma \ref{morphisms-lemma-universal-homeomorphism}). Hence $f$ is universally closed, see Lemma \ref{lemma-integral-universally-closed} and universally injective and (universally) surjective, i.e., $f$ is a universal homeomorphism.
Since $f$ is integral it is universally closed (Lemma \ref{lemma-integral-universally-closed}). In particular, $f(|Y|)$ is a closed subset of $|X|$ and the statements make sense. The rest of the proof is identical to the proof of Lemma \ref{lemma-closed-immersion-push-pull} except that we use \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-integral-universally-injective-pushforward} instead of \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-closed-immersion-pushforward}.
Let $V$ be a scheme and let $V \to Y$ be a surjective \'etale morphism. Let $X_V = V \times_Y X$ and let $\mathcal{F}_V$ be the restriction of $\mathcal{F}$ to $X_V$. Suppose that the result holds for the morphism $X_V \to V$ and the sheaf $\mathcal{F}_V$. Then there exists an open subscheme $V' \subset V$ such that $X_{V'} \to V'$ is flat and of finite presentation and $\mathcal{F}_{V'}$ is an $\mathcal{O}_{X_{V'}}$-module of finite presentation flat over $V'$. Let $W \subset Y$ be the image of the \'etale morphism $V' \to Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-image-open}. Then $V' \to W$ is a surjective \'etale morphism, hence we see that $X_W \to W$ is flat, locally of finite presentation, and quasi-compact by Lemmas \ref{lemma-finite-presentation-local}, \ref{lemma-flat-local}, and \ref{lemma-quasi-compact-local}. By the discussion in Properties of Spaces, Section \ref{spaces-properties-section-properties-modules} we see that $\mathcal{F}_W$ is of finite presentation as a $\mathcal{O}_{X_W}$-module and by Lemma \ref{lemma-base-change-module-flat} we see that $\mathcal{F}_W$ is flat over $W$. This argument reduces the proposition to the case where $Y$ is a scheme. \medskip\noindent Suppose we can prove the proposition when $Y$ is an affine scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$ with $Y$ a scheme, and let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Choose an affine open covering $Y = \bigcup V_j$. By assumption we can find dense open $W_j \subset V_j$ such that $X_{W_j} \to W_j$ is flat, locally of finite presentation, and quasi-compact and such that $\mathcal{F}|_{X_{W_j}}$ is flat over $W_j$ and of finite presentation as an $\mathcal{O}_{X_{W_j}}$-module. In this situation we simply take $W = \bigcup W_j$ and we win. Hence we reduce the proposition to the case where $Y$ is an affine scheme. \medskip\noindent Let $Y$ be an affine scheme over $S$, let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$, and let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Since $f$ is of finite type it is quasi-compact, hence $X$ is quasi-compact. Thus we can find an affine scheme $U$ and a surjective \'etale morphism $U \to X$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. Note that $U \to Y$ is of finite type (this is what it means for $f$ to be of finite type in this case). Hence we can apply Morphisms, Proposition \ref{morphisms-proposition-generic-flatness-reduced} to see that there exists a dense open $W \subset Y$ such that $U_W \to W$ is flat and of finite presentation and such that $\mathcal{F}|_{U_W}$ is flat over $W$ and of finite presentation as an $\mathcal{O}_{U_W}$-module. According to our definitions this means that the base change $X_W \to W$ of $f$ is flat, locally of finite presentation, and quasi-compact and $\mathcal{F}|_{X_W}$ is flat over $W$ and of finite presentation over $\mathcal{O}_{X_W}$.