hcju/stacksqa huggingface dataset:
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Set $U = V \times_Y X$. This is a scheme affine over $V$ with a surjective \'etale morphism $U \to X$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-thickening-equivalence} there exists a $U' \to X'$ surjective \'etale with $U = U' \times_{X'} X$. In particular the morphism of schemes $U \to U'$ is a thickening too. Apply More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening} to obtain a pushout $V' = V \amalg_U U'$ in the category of schemes. \medskip\noindent We repeat this procedure to construct a pushout $$ \xymatrix{ U \times_X U \ar[d] \ar[r] & U' \times_{X'} U' \ar[d] \\ V \times_Y V \ar[r] & R' } $$ in the category of schemes. Consider the morphisms $$ U \times_X U \to U \to V',\quad U' \times_{X'} U' \to U' \to V',\quad V \times_Y V \to V \to V' $$ where we use the first projection in each case. Clearly these glue to give a morphism $t' : R' \to V'$ which is \'etale by More on Morphisms, Lemma \ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}. Similarly, we obtain $s' : R' \to V'$ \'etale. The morphism $j' = (t', s') : R' \to V' \times_S V'$ is unramified (as $t'$ is \'etale) and a monomorphism when restricted to the closed subscheme $V \times_Y V \subset R'$. As $V \times_Y V \subset R'$ is a thickening it follows that $j'$ is a monomorphism too. Finally, $j'$ is an equivalence relation as we can use the functoriality of pushouts of schemes to construct a morphism $c' : R' \times_{s', V', t'} R' \to R'$ (details omitted). At this point we set $Y' = U'/R'$, see Spaces, Theorem \ref{spaces-theorem-presentation}. \medskip\noindent We have morphisms $X' = U'/U' \times_{X'} U' \to V'/R' = Y'$ and $Y = V/V \times_Y V \to V'/R' = Y'$. By construction these fit into the commutative diagram $$ \xymatrix{ X \ar[r] \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r] & Y' } $$ Since $Y \to Y'$ is a thickening we have $Y_\etale = (Y')_\etale$, see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-thickening-equivalence}. The commutativity of the diagram gives a map of sheaves $$ \mathcal{O}_{Y'} \longrightarrow \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'} $$ on this set. By More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening} this map is an isomorphism when we restrict to the scheme $V'$, hence it is an isomorphism. \medskip\noindent To finish the proof we show that the diagram above is a pushout in the category of algebraic spaces. To see this, let $Z$ be an algebraic space and let $a' : X' \to Z$ and $b : Y \to Z$ be morphisms of algebraic spaces. By Lemma \ref{lemma-pushout-along-thickening-schemes} we obtain a unique morphism $h : V' \to Z$ fitting into the commutative diagrams $$ \vcenter{ \xymatrix{ U' \ar[d] \ar[r] & V' \ar[d]^h \\ X' \ar[r]^{a'} & Z } } \quad\text{and}\quad \vcenter{ \xymatrix{ V \ar[r] \ar[d] & V' \ar[d]^h \\ Y \ar[r]^b & Z } } $$ The uniqueness shows that $h \circ t' = h \circ s'$. Hence $h$ factors uniquely as $V' \to Y' \to Z$ and we win.
The proof is completely formal. Since the morphisms $X \to X'$ and $X \to Y$ are representable it is clear that $F$ sends $(\Sch/Y')$ into $(\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')$. \medskip\noindent Let us construct $G$. Let $(V, U', \varphi)$ be an object of the fibre product category. Set $U = U' \times_{X'} X$. Note that $U \to U'$ is a thickening. Since $\varphi : V \times_Y X \to U' \times_{X'} X = U$ is an isomorphism we have a morphism $U \to V$ over $X \to Y$ which identifies $U$ with the fibre product $X \times_Y V$. In particular $U \to V$ is affine, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-affine}. Hence we can apply Lemma \ref{lemma-pushout-along-thickening} to get a pushout $V' = V \amalg_U U'$. Denote $V' \to Y'$ the morphism we obtain in virtue of the fact that $V'$ is a pushout and because we are given morphisms $V \to Y$ and $U' \to X'$ agreeing on $U$ as morphisms into $Y'$. Setting $G(V, U', \varphi) = V'$ gives the functor $G$. \medskip\noindent If $(V, U', \varphi)$ is an object of $(\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')$ then $U = U' \times_{X'} X$ is a scheme too and we can form the pushout $V' = V \amalg_U U'$ in the category of schemes by More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. By Lemma \ref{lemma-pushout-along-thickening-schemes} this is also a pushout in the category of schemes, hence $G$ sends $(\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')$ into $(\Sch/Y')$. \medskip\noindent Let us prove that $G$ is a left adjoint to $F$. Let $Z$ be an algebraic space over $Y'$. We have to show that $$ \Mor(V', Z) = \Mor((V, U', \varphi), F(Z)) $$ where the morphism sets are taking in their respective categories. Let $g' : V' \to Z$ be a morphism. Denote $\tilde g$, resp.\ $\tilde f'$ the composition of $g'$ with the morphism $V \to V'$, resp.\ $U' \to V'$. Base change $\tilde g$, resp.\ $\tilde f'$ by $Y \to Y'$, resp.\ $X' \to Y'$ to get a morphism $g : V \to Z \times_{Y'} Y$, resp.\ $f' : U' \to Z \times_{Y'} X'$. Then $(g, f')$ is an element of the right hand side of the equation above (details omitted). Conversely, suppose that $(g, f') : (V, U', \varphi) \to F(Z)$ is an element of the right hand side. We may consider the composition $\tilde g : V \to Z$, resp.\ $\tilde f' : U' \to Z$ of $g$, resp.\ $f$ by $Z \times_{Y'} X' \to Z$, resp.\ $Z \times_{Y'} Y \to Z$. Then $\tilde g$ and $\tilde f'$ agree as morphism from $U$ to $Z$. By the universal property of pushout, we obtain a morphism $g' : V' \to Z$, i.e., an element of the left hand side. We omit the verification that these constructions are mutually inverse.
A quasi-inverse functor assigns to the triple $(\mathcal{G}, \mathcal{F}', \varphi)$ the fibre product $$ (V \to V')_*\mathcal{G} \times_{(U \to V')_*\mathcal{F}} (U' \to V')_*\mathcal{F}' $$ where $\mathcal{F} = (U \to U')^*\mathcal{F}'$. This works, because on affines \'etale over $V'$ and $Y'$ we recover the equivalence of More on Algebra, Lemma \ref{more-algebra-lemma-relative-flat-module-over-fibre-product}. Details omitted. \medskip\noindent Parts (a) and (b) reduce by \'etale localization (Properties of Spaces, Section \ref{spaces-properties-section-properties-modules}) to the case where $V'$ and $Y'$ are affine in which case the result follows from More on Algebra, Lemmas \ref{more-algebra-lemma-relative-finite-module-over-fibre-product} and \ref{more-algebra-lemma-relative-finitely-presented-module-over-fibre-product}.
Choose a diagram (\ref{equation-cube}) as in the proof of Lemma \ref{lemma-equivalence-categories-spaces-over-pushout}. \medskip\noindent Proof of (1) -- (5). Let $(V, U', \varphi)$ be an object of $(\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X')$. Construct a diagram (\ref{equation-cube-over}) as in the proof of Lemma \ref{lemma-equivalence-categories-spaces-over-pushout}. Then the base change of $G(V, U', \varphi) \to Y'$ to $Y'_1$ is $G_1(V_1, U_1', \varphi_1) \to Y_1'$. Hence (1) -- (5) follow immediately from the corresponding statements of More on Morphisms, Lemma \ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat} for schemes. \medskip\noindent Suppose that $W' \to Y'$ is flat. Choose a scheme $W'_1$ and a surjective \'etale morphism $W'_1 \to Y_1' \times_{Y'} W'$. Observe that $W'_1 \to W'$ is surjective \'etale as a composition of surjective \'etale morphisms. We know that $G_1(F_1(W_1')) \to W_1'$ is an isomorphism by More on Morphisms, Lemma \ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat} applied to $W'_1$ over $Y'_1$ and the front of the diagram (with functors $G_1$ and $F_1$ as in the proof of Lemma \ref{lemma-equivalence-categories-spaces-over-pushout}). Then the construction of $G(F(W'))$ (as a pushout, i.e., as constructed in Lemma \ref{lemma-pushout-along-thickening}) shows that $G_1(F_1(W'_1)) \to G(F(W))$ is surjective \'etale. Whereupon we conclude that $G(F(W)) \to W$ is \'etale, see for example Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-local}. But $G(F(W)) \to W$ is an isomorphism on underlying reduced algebraic spaces (by construction), hence it is an isomorphism.
This is a consequence of Lemma \ref{lemma-colimit-agrees}, the proposition mentioned in the lemma and More on Morphisms, Lemmas \ref{more-morphisms-lemma-pushout-functor} and \ref{more-morphisms-lemma-pushout-functor-equivalence-flat}. Conditions (1) and (2) of Lemma \ref{lemma-colimit-agrees} follow immediately. To see (3) and (4) note that an \'etale morphism is locally quasi-finite and use that the equivalence of categories of More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-functor-equivalence-flat} is constructed using the pushout construction of More on Morphisms, Lemmas \ref{more-morphisms-lemma-pushout-functor}. Minor details omitted.
We are going to construct $W$ as the quotient of an \'etale equivalence relation $R$ on $U \amalg V$. Such a quotient is an algebraic space for example by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. Moreover, the proof of Lemma \ref{lemma-elementary-distinguished-square-pushout} tells us to take $$ R = U \amalg V' \amalg V' \amalg V \amalg (V' \times_U V' \setminus \Delta_{V'/U}(V')) $$ Since we assumed $V' \to U$ is separated, the image of $\Delta_{V'/U}$ is closed and hence the complement is an open subspace. The morphism $j : R \to (U \amalg V) \times_S (U \amalg V)$ is given by $$ u,\ v',\ v',\ v,\ (v'_1, v'_2) \mapsto (u, u),\ (f'(v'), v'),\ (v', f'(v')),\ (v, v),\ (v'_1, v'_2) $$ with obvious notation. It is immediately verified that this is a monomorphism, an equivalence relation, and that the induced morphisms $s, t : R \to U \amalg V$ are \'etale. Let $W = (U \amalg V)/R$ be the quotient algebraic space. We obtain a commutative diagram as in the statement of the lemma. To finish the proof it suffices to show that this diagram is an elementary distinguished square, since then Lemma \ref{lemma-elementary-distinguished-square-pushout} implies that it is a pushout. Thus we have to show that $U \to W$ is open and that $f$ is \'etale and is an isomorphism over $W \setminus U$. This follows from the choice of $R$; we omit the details.
Consider the commutative diagram of algebraic spaces $$ \xymatrix{ f^{-1}Z \ar[r]_{i'} \ar[d]_{f'} & Y \ar[d]_f \\ Z \ar[r]^i & X } $$ By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-complexes-with-support-on-closed} we can write $Q = Ri'_*K'$ for some object $K'$ of $D(f^{-1}Z_\etale)$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-universally-injective-push-pull} we have $K' = (f')^{-1}K$ with $K = Rf'_*K'$. Then we have $Rf_*Q = Rf_*Ri'_*K' = Ri_*Rf'_*K' = Ri_*K$. Let $\overline{z}$ be the geometric point of $Z$ corresponding to $\overline{x}$ and let $\overline{z}'$ be the geometric point of $f^{-1}Z$ corresponding to $\overline{y}$. We obtain the result of the lemma as follows $$ Q_{\overline{y}} = (Ri'_*K')_{\overline{y}} = K'_{\overline{z}'} = (f')^{-1}K_{\overline{z}'} = K_{\overline{z}} = Ri_*K_{\overline{x}} = Rf_*Q_{\overline{x}} $$ The middle equality holds because of the description of the stalk of a pullback given in Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback}.
Choose a distinguished triangle $$ \mathcal{G} \to Rj'_*\mathcal{G}|_V \to Q \to \mathcal{G}[1] $$ n $D(Y_\etale)$. The cohomology sheaves of $Q$ are supported on $|f^{-1}Z|$. We apply $Rf_*$ and we obtain $$ Rf_*\mathcal{G} \to Rf_*Rj'_*\mathcal{G}|_V \to Rf_*Q \to Rf_*\mathcal{G}[1] $$ Taking stalks at $\overline{x}$ we obtain an exact sequence $$ 0 \to (R^{-1}f_*Q)_{\overline{x}} \to f_*\mathcal{G}_{\overline{x}} \to (f \circ j')_*(\mathcal{G}|_V)_{\overline{x}} \to (R^0f_*Q)_{\overline{x}} $$ We can compare this with the exact sequence $$ 0 \to H^{-1}(Q)_{\overline{y}} \to \mathcal{G}_{\overline{y}} \to j'_*(\mathcal{G}|_V)_{\overline{y}} \to H^0(Q)_{\overline{y}} $$ Thus we see that the lemma follows because $Q_{\overline{y}} = Rf_*Q_{\overline{x}}$ by Lemma \ref{lemma-stalk-pushforward-with-support}.
Observe that $f_*\mathcal{G}_{\overline{x}} = \Gamma(\Spec(\mathcal{O}_{X, \overline{x}}), h^*f_*\mathcal{G})$ where $h : \Spec(\mathcal{O}_{X, \overline{x}}) \to X$. Hence the result is true because $h$ is flat so that Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology} applies.
Choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that $\overline{x}$ lifts to a geometric point $\overline{u}$ of $U$. Then we can replace $X$ by $U$, $Z$ by $U \times_X Z$, $Q$ by the restriction $Q|_U$, and $\overline{x}$ by $\overline{u}$. Thus we may assume that $X = \Spec(A)$ is affine. Let $I \subset A$ be the ideal defining $Z$. Since $i : Z \to X$ is of finite presentation, the ideal $I = (f_1, \ldots, f_r)$ is finitely generated. The object $Q$ comes from a complex of $A$-modules $M^\bullet$, see Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded}. Since the cohomology sheaves of $Q$ are supported on $Z$ we see that the localization $M^\bullet_f$ is acyclic for each $f \in I$. Take $x \in H^p(M^\bullet)$. By the above we can find $n_i$ such that $f_i^{n_i} x = 0$ in $H^p(M^\bullet)$ for each $i$. Then with $n = \sum n_i$ we see that $I^n$ annihilates $x$. Thus $H^p(M^\bullet)$ is $I$-power torsion. Since the ring map $A \to \mathcal{O}_{X, \overline{x}}$ is flat and since $I_{\overline{x}} = I\mathcal{O}_{X, \overline{x}}$ we conclude.
We show the canonical map $Lf^*Rf_*Q \to Q$ is an isomorphism by checking on stalks at $\overline{y}$. If $\overline{y}$ is not in $f^{-1}Z$, then both sides are zero and the result is true. Assume the image $\overline{x}$ of $\overline{y}$ is in $Z$. By Lemma \ref{lemma-stalk-pushforward-with-support} we have $Rf_*Q_{\overline{x}} = Q_{\overline{y}}$ and since $f$ is flat at $\overline{y}$ we see that $$ (Lf^*Rf_*Q)_{\overline{y}} = (Rf_*Q)_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} = Q_{\overline{y}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} $$ Thus we have to check that the canonical map $$ Q_{\overline{y}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} \longrightarrow Q_{\overline{y}} $$ is an isomorphism in the derived category. Let $I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}$ be the stalk of the ideal sheaf defining $Z$. Since $Z \to X$ is locally of finite presentation this ideal is finitely generated and the cohomology groups of $Q_{\overline{y}}$ are $I_{\overline{y}} = I_{\overline{x}}\mathcal{O}_{Y, \overline{y}}$-power torsion by Lemma \ref{lemma-stalk-of-module-with-support} applied to $Q$ on $Y$. It follows that they are also $I_{\overline{x}}$-power torsion. The ring map $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}$ is flat and induces an isomorphism after dividing by $I_{\overline{x}}$ and $I_{\overline{y}}$ because we assumed that $f^{-1}Z \to Z$ is an isomorphism. Hence we see that the cohomology modules of $Q_{\overline{y}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}}$ are equal to the cohomology modules of $Q_{\overline{y}}$ by More on Algebra, Lemma \ref{more-algebra-lemma-neighbourhood-isomorphism} which finishes the proof.
Since $f$ is quasi-compact and quasi-separated we see that $Rf_*$ defines a functor from $D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)$ to $D_{\QCoh, |Z|}(\mathcal{O}_X)$, see Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-quasi-coherence-direct-image}. By Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-quasi-coherence-pullback} we see that $Lf^*$ maps $D_{\QCoh, |Z|}(\mathcal{O}_X)$ into $D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)$. In Lemma \ref{lemma-formal-glueing-on-closed} we have seen that $Lf^*Rf_*Q = Q$ for $Q$ in $D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)$. By the dual of Derived Categories, Lemma \ref{derived-lemma-fully-faithful-adjoint-kernel-zero} to finish the proof it suffices to show that $Lf^*K = 0$ implies $K = 0$ for $K$ in $D_{\QCoh, |Z|}(\mathcal{O}_X)$. This follows from the fact that $f$ is flat at all points of $f^{-1}Z$ and the fact that $f^{-1}Z \to Z$ is surjective.
For the definition and general properties of fpqc coverings we refer to Topologies, Section \ref{topologies-section-fpqc}. In particular, we can first choose an \'etale covering $\{X_i \to X\}$ with $X_i$ affine and by base changing $Y$, $Z$, and $U$ to each $X_i$ we reduce to the case where $X$ is affine. In this case $U$ is quasi-compact and hence a finite union $U = U_1 \cup \ldots \cup U_n$ of affine opens.  Then $Z$ is quasi-compact hence also $f^{-1}Z$ is quasi-compact. Thus we can choose an affine scheme $W$ and an \'etale morphism $h : W \to Y$ such that $h^{-1}f^{-1}Z \to f^{-1}Z$ is surjective. Say $W = \Spec(B)$ and $h^{-1}f^{-1}Z = V(J)$ where $J \subset B$ is an ideal of finite type. By Pro-\'etale Cohomology, Lemma \ref{proetale-lemma-localization} there exists a localization $B \to B'$ such that points of $\Spec(B')$ correspond exactly to points of $W = \Spec(B)$ specializing to $h^{-1}f^{-1}Z = V(J)$. It follows that the composition $\Spec(B') \to \Spec(B) = W \to Y \to X$ is flat as by assumption $f : Y \to X$ is flat at all the points of $f^{-1}Z$. Then $\{\Spec(B') \to X, U_1 \to X, \ldots, U_n \to X\}$ is an fpqc covering by Topologies, Lemma \ref{topologies-lemma-recognize-fpqc-covering}.
The category of algebraic spaces affine over $X$ is equivalent to the category of quasi-coherent sheaves $\mathcal{A}$ of $\mathcal{O}_X$-algebras. The full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ affine is equivalent to the category of algebra objects of $\QCoh(Y \to X, Z)$. In both cases this follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-equivalence-algebras} with quasi-inverse given by the relative spectrum construction (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}) which commutes with arbitrary base change. Thus part (1) of the lemma follows from Proposition \ref{proposition-formal-glueing-modules}. \medskip\noindent Fully faithfulness in part (2) follows from part (1). For essential surjectivity, we reduce by part (1) to proving that $X' \to X$ is a closed immersion if and only if both $U \times_X X' \to U$ and $Y \times_X X' \to Y$ are closed immersions. By Lemma \ref{lemma-dominate-by-fpqc-covering} $\{U \to X, Y \to X\}$ can be refined by an fpqc covering. Hence the result follows from Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-closed-immersion}. \medskip\noindent For (3) use the argument proving (2) and Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-finite}.
By a formal argument with base change, this reduces to the following question: A morphism $a : X' \to X$ of algebraic spaces such that $U \times_X X' \to U$ and $Y \times_X X' \to Y$ are isomorphisms, is an isomorphism. The family $\{U \to X, Y \to X\}$ can be refined by an fpqc covering by Lemma \ref{lemma-dominate-by-fpqc-covering}. Hence the result follows from Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-isomorphism}.
Since $X'_2 \to X$ is separated, the graph $i : X'_1 \to X'_1 \times_X X'_2$ of a morphism $X'_1 \to X'_2$ over $X$ is a closed immersion, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-semi-diagonal}. Moreover a closed immersion $i : T \to X'_1 \times_X X'_2$ is the graph of a morphism if and only if $\text{pr}_1 \circ i$ is an isomorphism. The same is true for \begin{enumerate} \item the graph of a morphism $U \times_X X'_1 \to U \times_X X'_2$ over $U$, \item the graph of a morphism $V \times_X X'_1 \to V \times_X X'_2$ over $V$, and \item the graph of a morphism $Y \times_X X'_1 \to Y \times_X X'_2$ over $Y$. \end{enumerate} Moreover, if morphisms as in (1), (2), (3) fit together to form a morphism in the category $\textit{Spaces}(Y \to X, Z)$, then these graphs fit together to give an object of $\textit{Spaces}(Y \times_X (X'_1 \times_X X'_2) \to X'_1 \times_X X'_2, Z \times_X (X'_1 \times_X X'_2))$ whose triple of morphisms are closed immersions. The proof is finished by applying Lemmas \ref{lemma-equivalence-on-affine} and \ref{lemma-reflects-isomorphisms}.
It is immediate that (2) implies (1). Assume $\{Y_i \to Y\}$ is as in (1) and let $W \to Y$ be as in (2). Then $\{Y_i \times_Y W \to W\}_{i \in I}$ is an \'etale covering, which we may refine by an \'etale covering $\{W_j \to W\}_{j = 1, \ldots, m}$ with $W_j$ affine (Topologies, Lemma \ref{topologies-lemma-etale-affine}). Thus to finish the proof it suffices to show the following three algebraic statements: \begin{enumerate} \item if $R \to A \to B$ are ring maps with $A \to B$ \'etale and $A$ glueable as an $R$-module, then $B$ is glueable as an $R$-module, \item finite products of glueable $R$-modules are glueable, \item if $R \to A \to B$ are ring maps with $A \to B$ faithfully \'etale and $B$ glueable as an $R$-module, then $A$ is glueable as an $R$-module. \end{enumerate} Namely, the first of these will imply that $\Gamma(W_j, \mathcal{O}_{W_j})$ is a glueable $R$-module, the second will imply that $\prod \Gamma(W_j, \mathcal{O}_{W_j})$ is a glueable $R$-module, and the third will imply that $\Gamma(W, \mathcal{O}_W)$ is a glueable $R$-module. \medskip\noindent Consider an \'etale $R$-algebra homomorphism $A \to B$. Set $A' = A \otimes_R R'$ and $B' = B \otimes_R R' = A' \otimes_A B$. Statements (1) and (3) then follow from the following facts: (a) $A$, resp.\ $B$ is glueable if and only if the sequence $$ 0 \to A \to A_f \oplus A' \to A'_f \to 0, \quad\text{resp.}\quad 0 \to B \to B_f \oplus B' \to B'_f \to 0, $$ is exact, (b) the second sequence is equal to the functor $- \otimes_A B$ applied to the first and (c) (faithful) flatness of $A \to B$. We omit the proof of (2).
Let $(V, V', Y')$ be an object of $\textit{Spaces}(U \leftarrow U' \to X')$ with $V$, $V'$, $Y'$ affine. Write $V = \Spec(A_1)$ and $Y' = \Spec(A')$. By our definition of the category $\textit{Spaces}(U \leftarrow U' \to X')$ we find that $V'$ is the spectrum of $A_1 \otimes_{R_f} R'_f = A_1 \otimes_R R'$ and the spectrum of $A'_f$. Hence we get an isomorphism $\varphi : A'_f \to A_1 \otimes_R R'$ of $R'_f$-algebras. By More on Algebra, Theorem \ref{more-algebra-theorem-BL-glueing} there exists a unique glueable $R$-module $A$ and isomorphisms $A_f \to A_1$ and $A \otimes_R R' \to A'$ of modules compatible with $\varphi$. Since the sequence $$ 0 \to A \to A_1 \oplus A' \to A'_f \to 0 $$ is short exact, the multiplications on $A_1$ and $A'$ define a unique $R$-algebra structure on $A$ such that the maps $A \to A_1$ and $A \to A'$ are ring homomorphisms. We omit the verification that this construction defines a quasi-inverse to the functor (\ref{equation-beauville-laszlo-glueing-spaces}) restricted to the subcategories mentioned in the statement of the lemma.