hcju/stacksqa huggingface dataset:
Recall that a universal homeomorphism is the same thing as an integral, universally injective, surjective morphism, see Morphisms, Lemma \ref{morphisms-lemma-universal-homeomorphism}. In particular, the diagonal $\Delta : X \to X \times_Y X$ is a thickening by Morphisms, Lemma \ref{morphisms-lemma-universally-injective}. Thus by Lemma \ref{lemma-thickening} we see that given a finite \'etale morphism $U \to X$ there is a unique isomorphism $$ \varphi : U \times_Y X \to X \times_Y U $$ of schemes finite \'etale over $X \times_Y X$ which pulls back under $\Delta$ to $\text{id} : U \to U$ over $X$. Since $X \to X \times_Y X \times_Y X$ is a thickening as well (it is bijective and a closed immersion) we conclude that $(U, \varphi)$ is a descent datum relative to $X/Y$. By \'Etale Morphisms, Proposition \ref{etale-proposition-effective} we conclude that $U = X \times_Y V$ for some $V \to Y$ quasi-compact, separated, and \'etale. We omit the proof that $V \to Y$ is finite (hints: the morphism $U \to V$ is surjective and $U \to Y$ is integral). We conclude that $\textit{F\'Et}_Y \to \textit{F\'Et}_X$ is essentially surjective. \medskip\noindent Arguing in the same manner as above we see that given $V_1 \to Y$ and $V_2 \to Y$ in $\textit{F\'Et}_Y$ any morphism $a : X \times_Y V_1 \to X \times_Y V_2$ over $X$ is compatible with the canonical descent data. Thus $a$ descends to a morphism $V_1 \to V_2$ over $Y$ by \'Etale Morphisms, Lemma \ref{etale-lemma-fully-faithful-cases}.
The normal scheme $X$ is geometrically unibranch (Properties, Lemma \ref{properties-lemma-normal-geometrically-unibranch}). Hence Lemma \ref{lemma-irreducible-geometrically-unibranch} applies to $X$. Thus $\pi_1(\eta, \overline{\eta}) \to \pi_1(X, \overline{\eta})$ is surjective and top horizontal arrow of the commutative diagram $$ \xymatrix{ \textit{F\'Et}_X \ar[r] \ar[d] \ar[rd]_c & \textit{F\'Et}_\eta \ar[d] \\ \textit{Finite-}\pi_1(X, \overline{\eta})\textit{-sets} \ar[r] & \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-sets} } $$ is fully faithful. The left vertical arrow is the equivalence of Theorem \ref{theorem-fundamental-group} and the right vertical arrow is the equivalence of Lemma \ref{lemma-fundamental-group-Galois-group}. The lower horizontal arrow is induced by the map of the proposition. By Lemmas \ref{lemma-unramified-in-L} and \ref{lemma-finite-etale-covering-normal-unramified} we see that the essential image of $c$ consists of $\text{Gal}(K^{sep}/K)\textit{-Sets}$ isomorphic to sets of the form $$ S = \Hom_K(\prod\nolimits_{i = 1, \ldots, n} L_i, K^{sep}) = \coprod\nolimits_{i = 1, \ldots, n} \Hom_K(L_i, K^{sep}) $$ with $L_i/K$ finite separable such that $X$ is unramified in $L_i$. Thus if $M \subset K^{sep}$ is as in the statement of the lemma, then $\text{Gal}(K^{sep}/M)$ is exactly the subgroup of $\text{Gal}(K^{sep}/K)$ acting trivially on every object in the essential image of $c$. On the other hand, the essential image of $c$ is exactly the category of $S$ such that the $\text{Gal}(K^{sep}/K)$-action factors through the surjection $\text{Gal}(K^{sep}/K) \to \pi_1(X, \overline{\eta})$. We conclude that $\text{Gal}(K^{sep}/M)$ is the kernel. Hence $\text{Gal}(K^{sep}/M)$ is a normal subgroup, $M/K$ is Galois, and we have a short exact sequence $$ 1 \to \text{Gal}(K^{sep}/M) \to \text{Gal}(K^{sep}/K) \to \text{Gal}(M/K) \to 1 $$ by Galois theory (Fields, Theorem \ref{fields-theorem-inifinite-galois-theory} and Lemma \ref{fields-lemma-ses-infinite-galois}). The proof is done.
Let $Y \to X$ be a finite \'etale morphism. Consider the Stein factorization $$ \xymatrix{ Y \ar[d] \ar[r] & X \ar[d] \\ T \ar[r] & S } $$ of $Y \to S$. By Lemma \ref{lemma-stein-factorization-etale} the morphism $T \to S$ is finite \'etale. In this way we obtain a functor $\textit{F\'Et}_X \to \textit{F\'Et}_S$. For any finite \'etale morphism $U \to S$ a morphism $Y \to U \times_S X$ over $X$ is the same thing as a morphism $Y \to U$ over $S$ and such a morphism factors uniquely through the Stein factorization, i.e., corresponds to a unique morphism $T \to U$ (by the construction of the Stein factorization as a relative normalization in More on Morphisms, Lemma \ref{more-morphisms-lemma-stein-universally-closed} and factorization by Morphisms, Lemma \ref{morphisms-lemma-characterize-normalization}). Thus we see that the functors $\textit{F\'Et}_X \to \textit{F\'Et}_S$ and $\textit{F\'Et}_S \to \textit{F\'Et}_X$ are adjoints. Note that the Stein factorization of $U \times_S X \to S$ is $U$, because the fibres of $U \times_S X \to U$ are geometrically connected. \medskip\noindent By the discussion above and Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful} we conclude that $\textit{F\'Et}_S \to \textit{F\'Et}_X$ is fully faithful, i.e., $\pi_1(X) \to \pi_1(S)$ is surjective (Lemma \ref{lemma-functoriality-galois-surjective}). \medskip\noindent It is immediate that the composition $\textit{F\'Et}_S \to \textit{F\'Et}_X \to \textit{F\'Et}_{X_{\overline{s}}}$ sends any $U$ to a disjoint union of copies of $X_{\overline{s}}$. Hence $\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S)$ is trivial by Lemma \ref{lemma-composition-trivial}. \medskip\noindent Let $Y \to X$ be a finite \'etale morphism with $Y$ connected such that $Y \times_X X_{\overline{s}}$ contains a connected component $Z$ isomorphic to $X_{\overline{s}}$. Consider the Stein factorization $T$ as above. Let $\overline{t} \in T_{\overline{s}}$ be the point corresponding to the fibre $Z$. Observe that $T$ is connected (as the image of a connected scheme) and by the surjectivity above $T \times_S X$ is connected. Now consider the factorization $$ \pi : Y \longrightarrow T \times_S X $$ Let $\overline{x} \in X_{\overline{s}}$ be any closed point. Note that $\kappa(\overline{t}) = \kappa(\overline{s}) = \kappa(\overline{x})$ is an algebraically closed field. Then the fibre of $\pi$ over $(\overline{t}, \overline{x})$ consists of a unique point, namely the unique point $\overline{z} \in Z$ corresponding to $\overline{x} \in X_{\overline{s}}$ via the isomorphism $Z \to X_{\overline{s}}$. We conclude that the finite \'etale morphism $\pi$ has degree $1$ in a neighbourhood of $(\overline{t}, \overline{x})$. Since $T \times_S X$ is connected it has degree $1$ everywhere and we find that $Y \cong T \times_S X$. Thus $Y \times_X X_{\overline{s}}$ splits completely. Combining all of the above we see that Lemmas \ref{lemma-functoriality-galois-ses} and \ref{lemma-functoriality-galois-normal} both apply and the proof is complete.
By Lemma \ref{lemma-purity-and-completion} we may assume that $A$ is a complete local ring. By assumption we can write $A = B/(f_1, \ldots, f_r)$ where $B$ is a complete regular local ring and $f_1, \ldots, f_r$ is a regular sequence. We will finish the proof by induction on $r$. The base case is $r = 0$ which follows from Lemma \ref{lemma-local-purity} which applies to regular rings of dimension $\geq 2$. \medskip\noindent Assume that $A = B/(f_1, \ldots, f_r)$ and that the proposition holds for $r - 1$. Set $A' = B/(f_1, \ldots, f_{r - 1})$ and apply Lemma \ref{lemma-purity-inherited-by-hypersurface} to $f_r \in A'$. This is permissible: condition (1) holds as $f_1, \ldots, f_r$ is a regular sequence, condition (2) holds as $B$ and hence $A'$ is complete, condition (3) holds as $A = A'/f_r A'$ is Cohen-Macaulay of dimension $\dim(A) \geq 3$, see Dualizing Complexes, Lemma \ref{dualizing-lemma-depth}, condition (4) holds by induction hypothesis as $\dim((A'_{f_r})_\mathfrak p) \geq 3$ for a maximal prime $\mathfrak p$ of $A'_{f_r}$ and as $(A'_{f_r})_\mathfrak p = B_\mathfrak q/(f_1, \ldots, f_{r - 1})$ for some $\mathfrak q \subset B$, condition (5) holds by induction hypothesis.
A prime $\mathfrak q \subset B$ which is not contained in $V$ lies over $\mathfrak m_A$. In this case $A \to B_\mathfrak q$ is a flat local homomorphism and hence $\text{depth}(B_\mathfrak q) \geq 2$ (Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}). Thus the functor is fully faithful by Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian} combined with Local Cohomology, Lemma \ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}. \medskip\noindent Denote $A^\wedge$ and $B^\wedge$ the completions of $A$ and $B$ with respect to their maximal ideals. Observe that the assumptions of the proposition hold for $A^\wedge \to B^\wedge$, see More on Algebra, Lemmas \ref{more-algebra-lemma-completion-dimension}, \ref{more-algebra-lemma-completion-depth}, and \ref{more-algebra-lemma-formally-smooth-completion}. By the uniqueness and compatibility with flat base change of the construction of  Lemma \ref{lemma-extend-S2} it suffices to prove the essential surjectivity for $A^\wedge \to B^\wedge$ and the inverse image of $V$ (details omitted; compare with Lemma \ref{lemma-purity-and-completion} for the case where $V$ is the punctured spectrum). By More on Algebra, Proposition \ref{more-algebra-proposition-fs-regular} this means we may assume $A \to B$ is regular. \medskip\noindent Let $W \to V$ be a finite \'etale morphism. By Popescu's theorem (Smoothing Ring Maps, Theorem \ref{smoothing-theorem-popescu}) we can write $B = \colim B_i$ as a filtered colimit of smooth $A$-algebras. We can pick an $i$ and an open $V_i \subset \Spec(B_i)$ whose inverse image is $V$ (Limits, Lemma \ref{limits-lemma-descend-opens}). After increasing $i$ we may assume there is a finite \'etale morphism $W_i \to V_i$ whose base change to $V$ is $W \to V$, see Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}, \ref{limits-lemma-descend-finite-finite-presentation}, and \ref{limits-lemma-descend-etale}. We may assume the complement of $V_i$ is contained in the closed fibre of $\Spec(B_i) \to \Spec(A)$ as this is true for $V$ (either choose $V_i$ this way or use the lemma above to show this is true for $i$ large enough). Let $\eta$ be the generic point of the closed fibre of $\Spec(B) \to \Spec(A)$. Since $\eta \in V$, the image of $\eta$ is in $V_i$. Hence after replacing $V_i$ by an affine open neighbourhood of the image of the closed point of $\Spec(B)$, we may assume that the closed fibre of $\Spec(B_i) \to \Spec(A)$ is irreducible and that its generic point is contained in $V_i$ (details omitted; use that a scheme smooth over a field is a disjoint union of irreducible schemes). At this point we may apply Lemma \ref{lemma-purity-smooth-over-depth2} to see that $W_i \to V_i$ extends to a finite \'etale morphism $\Spec(C_i) \to \Spec(B_i)$ and pulling back to $\Spec(B)$ we conclude that $W$ is in the essential image of the functor $\textit{F\'Et}_Y \to \textit{F\'Et}_V$ as desired.
The first statement is a formal consequence of Lemma \ref{lemma-restriction-fully-faithful-special} and Algebraic and Formal Geometry, Proposition \ref{algebraization-proposition-lefschetz}. The second statement follows from Lemma \ref{lemma-restriction-fully-faithful-special} and Algebraic and Formal Geometry, Lemma \ref{algebraization-lemma-lefschetz-addendum}.
Fully faithfulness holds by Proposition \ref{proposition-lefschetz-fully-faithful}. By Proposition \ref{proposition-lefschetz-equivalence-general} any object of $\textit{F\'Et}_Y$ is isomorphic to the fibre product $U \times_V Y$ for some finite \'etale morphism $U \to V$ where $V \subset X$ is an open subscheme containing $Y$. The complement $T = X \setminus V$ is\footnote{Namely, $T$ is proper over $k$ (being closed in $X$) and affine (being closed in the affine scheme $X \setminus Y$, see Morphisms, Lemma \ref{morphisms-lemma-proper-ample-delete-affine}) and hence finite over $k$ (Morphisms, Lemma \ref{morphisms-lemma-finite-proper}). Thus $T$ is a finite set of closed points.} a finite set of closed points of $X \setminus Y$. Say $T = \{x_1, \ldots, x_n\}$. By assumption we can find finite \'etale morphisms $V'_i \to \Spec(\mathcal{O}_{X, x_i})$ agreeing with $U \to V$ over $V \times_X \Spec(\mathcal{O}_{X, x_i})$. By Limits, Lemma \ref{limits-lemma-glueing-near-closed-point} applied $n$ times we see that $U \to V$ extends to a finite \'etale morphism $U' \to X$ as desired. See Lemma \ref{lemma-what-equivalence-gives} for the final statement.
Let $Y \to Z$ be a finite \'etale morphism. It suffices to construct a finite \'etale morphism $f : U \to X$ such that $Y$ is isomorphic to an open and closed subscheme of $f^{-1}(Z)$, see Lemma \ref{lemma-functoriality-galois-injective}. Write $Y = \Spec(A)$ and $X = \Spec(R)$ so the closed immersion $Y \to X$ is given by a surjection $R \to A$. We may write $A = \colim A_i$ as the filtered colimit of its $\mathbf{F}_p$-subalgebras of finite type. By Lemma \ref{lemma-limit} we can find an $i$ and a finite \'etale morphism $Y_i \to Z_i = \Spec(A_i)$ such that $Y = Z \times_{Z_i} Y_i$. \medskip\noindent Choose a surjection $\mathbf{F}_p[x_1, \ldots, x_n] \to A_i$. This determines a closed immersion $$ Z_i = \Spec(A_i) \longrightarrow X_i = \mathbf{A}^n_{\mathbf{F}_p} = \Spec(\mathbf{F}_p[x_1, \ldots, x_n]) $$ By the universal property of polynomial algebras and since $R \to A$ is surjective, we can find a commutative diagram $$ \xymatrix{ \mathbf{F}_p[x_1, \ldots, x_n] \ar[r] \ar[d] & A_i \ar[d] \\ R \ar[r] & A } $$ of $\mathbf{F}_p$-algebras. Thus we have a commutative diagram $$ \xymatrix{ Y_i \ar[r] & Z_i \ar[r] & X_i \\ Y \ar[u] \ar[r] & Z \ar[u] \ar[r] & X \ar[u] } $$ whose right square is cartesian. Clearly, if we can find $f_i : U_i \to X_i$ finite \'etale such that $Y_i$ is isomorphic to an open and closed subscheme of $f_i^{-1}(Z_i)$, then the base change $f : U \to X$ of $f_i$ by $X \to X_i$ is a solution to our problem. Thus we conclude by applying Lemma \ref{lemma-dominate-affine-space} to $Y_i \to Z_i \to X_i = \mathbf{A}^n_{\mathbf{F}_p}$.
Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose a scheme $U$ and a surjective smooth morphism $U \to V \times_\mathcal{Y} \mathcal{X}$. Then $U \to \mathcal{X}$ is still smooth and surjective as a composition of two such morphisms. From the commutative diagram $$ \xymatrix{ U \ar[d] \ar[r]_{f'} & V \ar[d] \\ \mathcal{X} \ar[r]^f & \mathcal{Y} } $$ we obtain a commutative diagram $$ \xymatrix{ \QCoh(\mathcal{O}_U) & \QCoh(\mathcal{O}_V) \ar[l] \\ \QCoh(\mathcal{O}_\mathcal{X}) \ar[u] & \QCoh(\mathcal{O}_\mathcal{Y}) \ar[l] \ar[u] } $$ of abelian categories. Our proof that the bottom two categories in this diagram are abelian showed that the vertical functors are faithful exact functors (see proof of Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-quasi-coherent-algebraic-stack}). Since $f'$ is a flat morphism of schemes (by our definition of flat morphisms of algebraic stacks) we see that $(f')^*$ is an exact functor on quasi-coherent sheaves on $V$. Thus we win.
We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-composition-smooth}). The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-lqc}. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a surjective smooth morphism. Let $x$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation} there exists an \'etale covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$. This just means that $x$, $x_i$ live over schemes $V$, $V_i$, that $\{V_i \to V\}$ is an \'etale covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. The restriction $x_i^*\mathcal{F}|_{V_{i, \etale}}$ is equal to the restriction of $f^*\mathcal{F}$ to $V_{i, \etale}$, see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-comparison}. Hence $x^*\mathcal{F}|_{V_\etale}$ is a sheaf on the small \'etale site of $V$ which is quasi-coherent when restricted to $V_{i, \etale}$ for each $i$. This implies that it is quasi-coherent (as desired), for example by Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent}.
We will use Lemma \ref{lemma-general-pushforward} to prove this. We will check its assumptions (1) -- (4). Parts (1) and (2) follows from Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-lqc-colimits}. Part (3) follows from Lemma \ref{lemma-check-lqc-on-etale-covering}. Thus it suffices to show (4). \medskip\noindent Suppose $f : \mathcal{X} \to \mathcal{Y}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes $X$ and $Y$. Choose any object $y$ of $\mathcal{Y}$ lying over a scheme $V$. For clarity, denote $\mathcal{V} = (\Sch/V)_{fppf}$ the algebraic stack corresponding to $V$. Consider the cartesian diagram $$ \xymatrix{ \mathcal{Z} \ar[d] \ar[r]_g \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{V} \ar[r]^y & \mathcal{Y} } $$ Thus $\mathcal{Z}$ is representable by the scheme $Z = V \times_Y X$ and $f'$ is quasi-compact and separated (even affine). By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-compare-representable-morphism-cohomology} we have $$ R^if_*\mathcal{F}|_{V_\etale} = R^if'_{small, *}\big(g^*\mathcal{F}|_{Z_\etale}\big) $$ The right hand side is a quasi-coherent sheaf on $V_\etale$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-higher-direct-image}. This implies the left hand side is quasi-coherent which is what we had to prove.
First, suppose there is a morphism $a : \mathcal{U} \to \mathcal{X}$ which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that $a^*\mathcal{F}$ is locally quasi-coherent. Then there is an exact sequence $$ 0 \to \mathcal{F} \to a_*a^*\mathcal{F} \to b_*b^*\mathcal{F} $$ where $b$ is the morphism $b : \mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{X}$, see Sheaves on Stacks, Proposition \ref{stacks-sheaves-proposition-exactness-cech-complex} and Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}. Moreover, the pullback $b^*\mathcal{F}$ is the pullback of $a^*\mathcal{F}$ via one of the projection morphisms, hence is locally quasi-coherent (Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-lqc}). The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent by Lemma \ref{lemma-pushforward-locally-quasi-coherent}. (Note that $a_*$ and $b_*$ don't care about which topology is used to calculate them.) We conclude that $\mathcal{F}$ is locally quasi-coherent, see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-lqc-colimits}. \medskip\noindent We are going to reduce the proof of the general case the situation in the first paragraph. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. We have to show that $\mathcal{F}|_{U_\etale}$ is a quasi-coherent $\mathcal{O}_U$-module. It suffices to do this (Zariski) locally on $U$, hence we may assume that $U$ is affine. By Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation} there exists an fppf covering $\{a_i : U_i \to U\}$ such that each $x \circ a_i$ factors through some $f_j$. Hence $a_i^*\mathcal{F}$ is locally quasi-coherent on $(\Sch/U_i)_{fppf}$. After refining the covering we may assume $\{U_i \to U\}_{i = 1, \ldots, n}$ is a standard fppf covering. Then $x^*\mathcal{F}$ is an fppf module on $(\Sch/U)_{fppf}$ whose pullback by the morphism $a : U_1 \amalg \ldots \amalg U_n \to U$ is locally quasi-coherent. Hence by the first paragraph we see that $x^*\mathcal{F}$ is locally quasi-coherent, which certainly implies that $\mathcal{F}|_{U_\etale}$ is quasi-coherent.
We will use Lemma \ref{lemma-general-pushforward} to prove this. For every algebraic stack $\mathcal{X}$ let $\mathcal{M}_\mathcal{X}$ denote the full subcategory of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ consisting of locally quasi-coherent sheaves with the flat base change property. Once we verify conditions (1) -- (4) of Lemma \ref{lemma-general-pushforward} the lemma will follow. Properties (1), (2), and (3) follow from Sheaves on Stacks, Lemmas \ref{stacks-sheaves-lemma-pullback-lqc} and \ref{stacks-sheaves-lemma-lqc-colimits} and Lemmas \ref{lemma-check-lqc-on-etale-covering} and \ref{lemma-check-flat-comparison-on-etale-covering}. Thus it suffices to show part (4). \medskip\noindent Suppose $f : \mathcal{X} \to \mathcal{Y}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes $X$ and $Y$. In this case, suppose that $\psi : y \to y'$ is a morphism of $\mathcal{Y}$ lying over a flat morphism $b : V \to V'$ of schemes. For clarity denote $\mathcal{V} = (\Sch/V)_{fppf}$ and $\mathcal{V}' = (\Sch/V')_{fppf}$ the corresponding algebraic stacks. Consider the diagram of algebraic stacks $$ \xymatrix{ \mathcal{Z} \ar[d]_{f''} \ar[r]_a & \mathcal{Z}' \ar[r]_{x'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{V} \ar[r]^b & \mathcal{V}' \ar[r]^{y'} & \mathcal{Y} } $$ with both squares cartesian. As $f$ is representable by schemes (and quasi-compact and separated -- even affine) we see that $\mathcal{Z}$ and $\mathcal{Z}'$ are representable by schemes $Z$ and $Z'$ and in fact $Z = V \times_{V'} Z'$. Since $\mathcal{F}$ has the flat base change property we see that $$ a_{small}^*\big(\mathcal{F}|_{Z'_\etale}\big) \longrightarrow \mathcal{F}|_{Z_\etale} $$ is an isomorphism. Moreover, $$ R^if_*\mathcal{F}|_{V'_\etale} = R^i(f')_{small, *}\big(\mathcal{F}|_{Z'_\etale}\big) $$ and $$ R^if_*\mathcal{F}|_{V_\etale} = R^i(f'')_{small, *}\big(\mathcal{F}|_{Z_\etale}\big) $$ by Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-compare-representable-morphism-cohomology}. Hence we see that the comparison map $$ c_\psi : b_{small}^*(R^if_*\mathcal{F}|_{V'_\etale}) \longrightarrow R^if_*\mathcal{F}|_{V_\etale} $$ is an isomorphism by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology}. Thus $R^if_*\mathcal{F}$ has the flat base change property. Since $R^if_*\mathcal{F}$ is locally quasi-coherent by Lemma \ref{lemma-pushforward-locally-quasi-coherent} we win.