hcju/stacksqa huggingface dataset:
Note that $X \times_Y X$ is both the fibre product in the category of sheaves on $(\Sch/S)_{fppf}$ and the fibre product in the category of algebraic spaces over $S$, see Spaces, Lemma \ref{spaces-lemma-fibre-product-spaces}. The equivalence of (1) and (3) is a general characterization of injective maps of sheaves on any site. The equivalence of (2) and (3) is a characterization of monomorphisms in any category with fibre products.
We will use without further mention that a base change of a monomorphism is a monomorphism, see Lemma \ref{lemma-base-change-monomorphism}. In particular it is clear that (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4) (by taking $V$ to be a disjoint union of affine schemes \'etale over $Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}). Let $V$ be a scheme, and let $V \to Y$ be a surjective \'etale morphism. If $V \times_Y X \to V$ is a monomorphism, then it follows that $X \to Y$ is a monomorphism. Namely, given any cartesian diagram of sheaves $$ \vcenter{ \xymatrix{ \mathcal{F} \ar[r]_a \ar[d]_b & \mathcal{G} \ar[d]^c \\ \mathcal{H} \ar[r]^d & \mathcal{I} } } \quad \quad \mathcal{F} = \mathcal{H} \times_\mathcal{I} \mathcal{G} $$ if $c$ is a surjection of sheaves, and $a$ is injective, then also $d$ is injective. Thus (4) implies (1). Proof of the equivalence of (5) and (1) is omitted.
By Lemmas \ref{lemma-monomorphism-separated} and \ref{lemma-separated-over-separated} we see that $Z$ is a separated algebraic space. Hence there exists an open dense subspace $Z' \subset Z$ which is a scheme, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}. By Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field} we see that either $Z' = \emptyset$ or $Z' \cong \Spec(k)$. In the first case we conclude that $Z = \emptyset$ and in the second case we conclude that $Z' = Z = \Spec(k)$ as $Z \to \Spec(k)$ is a monomorphism which is an isomorphism over $Z'$.
We denote $\mathcal{F}$ also its extension to a sheaf of modules on $X_{spaces, \etale}$, see Properties of Spaces, Remark \ref{spaces-properties-remark-explain-equivalence}. Let $V \to Y$ be an object of $Y_\etale$. Then $V \times_Y X$ is an object of $X_{spaces, \etale}$, and by definition $f_*\mathcal{F}(V) = \mathcal{F}(V \times_Y X)$. Since $U \to X$ is surjective \'etale, we see that $\{V \times_Y U \to V \times_Y X\}$ is a covering. Also, we have $(V \times_Y U) \times_X (V \times_Y U) = V \times_Y R$. Hence, by the sheaf condition of $\mathcal{F}$ on $X_{spaces, \etale}$ we have a short exact sequence $$ 0 \to \mathcal{F}(V \times_Y X) \to \mathcal{F}(V \times_Y U) \to \mathcal{F}(V \times_Y R) $$ where the second arrow is the difference of restricting via $t$ or $s$. This exact sequence is functorial in $V$ and hence we obtain the lemma.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. We have to show that $f_*\mathcal{F}$ is a quasi-coherent sheaf on $Y$. For this it suffices to show that for any affine scheme $V$ and \'etale morphism $V \to Y$ the restriction of $f_*\mathcal{F}$ to $V$ is quasi-coherent, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent}. Let $f' : V \times_Y X \to V$ be the base change of $f$ by $V \to Y$. Note that $f'$ is also quasi-compact and quasi-separated, see Lemmas \ref{lemma-base-change-quasi-compact} and \ref{lemma-base-change-separated}. By (\ref{equation-representable-pushforward}) we know that the restriction of $f_*\mathcal{F}$ to $V$ is $f'_*$ of the restriction of $\mathcal{F}$ to $V \times_Y X$. Hence we may replace $f$ by $f'$, and assume that $Y$ is an affine scheme. \medskip\noindent Assume $Y$ is an affine scheme. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Thus we may choose 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}. By Lemma \ref{lemma-compute-pushforward} we get an exact sequence $$ 0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \to b_*(\mathcal{F}|_R). $$ where $R = U \times_X U$. As $X \to Y$ is quasi-separated we see that $R \to U \times_Y U$ is a quasi-compact monomorphism. This implies that $R$ is a quasi-compact separated scheme (as $U$ and $Y$ are affine at this point). Hence $a : U \to Y$ and $b : R \to Y$ are quasi-compact and quasi-separated morphisms of schemes. Thus by Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent-functorial} the sheaves $a_*(\mathcal{F}|_U)$ and $b_*(\mathcal{F}|_R)$ are quasi-coherent (see also the discussion preceding this lemma). This implies that $f_*\mathcal{F}$ is a kernel of quasi-coherent modules, and hence itself quasi-coherent, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-properties-quasi-coherent}.
Using that a base change of a closed immersion (resp.\ open immersion, resp.\ immersion) is another one it is clear that (1) implies (2) and (2) implies (3). Also (3) implies (4) since we can take $V$ to be a disjoint union of affines, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. \medskip\noindent Assume $V \to Y$ is as in (4). Let $\mathcal{P}$ be the property closed immersion (resp.\ open immersion, resp.\ immersion) of morphisms of schemes. Note that property $\mathcal{P}$ is preserved under any base change and fppf local on the base (see Section \ref{section-representable}). Moreover, morphisms of type $\mathcal{P}$ are separated and locally quasi-finite (in each of the three cases, see Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}, and Morphisms, Lemma \ref{morphisms-lemma-immersion-locally-quasi-finite}). Hence by More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} the morphisms of type $\mathcal{P}$ satisfy descent for fppf covering. Thus Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} applies and we see that $X \to Y$ is representable and has property $\mathcal{P}$, in other words (1) holds. \medskip\noindent The equivalence of (1) and (5) follows from the fact that $\mathcal{P}$ is Zariski local on the target (since we saw above that $\mathcal{P}$ is in fact fppf local on the target).
In each case the proof is to contemplate the commutative diagram $$ \xymatrix{ Z \ar[r] \ar[rd] & Y \times_X Z \ar[r] \ar[d] & Z \ar[d] \\ & Y \ar[r] & X } $$ where the composition of the top horizontal arrows is the identity. Let us prove (1). The first horizontal arrow is a section of $Y \times_X Z \to Z$, whence representable, locally of finite type, locally quasi-finite, separated, and a monomorphism by Lemma \ref{lemma-section-immersion}. The arrow $Y \times_X Z \to Y$ is a base change of $Z \to X$ hence is representable, locally of finite type, locally quasi-finite, separated, and a monomorphism (as each of these properties of morphisms of schemes is stable under base change, see Spaces, Remark \ref{spaces-remark-list-properties-stable-base-change}). Hence the same is true for the composition (as each of these properties of morphisms of schemes is stable under composition, see Spaces, Remark \ref{spaces-remark-list-properties-stable-composition}). This proves (1). The other results are proved in exactly the same manner.
The first statement is Properties of Spaces, Lemma \ref{spaces-properties-lemma-subspace-induced-topology}. Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. By assumption $T = U \times_X Z$ is a scheme and the morphism $j : T \to U$ is an immersion of schemes. By Lemma \ref{lemma-closed-immersion-local} the morphism $i$ is a closed immersion if and only if $j$ is a closed immersion. By Schemes, Lemma \ref{schemes-lemma-immersion-when-closed} this is true if and only if $j(T)$ is closed in $U$. However, the subset $j(T) \subset U$ is the inverse image of $|i|(|Z|) \subset |X|$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. This finishes the proof.
Let $U$ be a scheme and let $U \to X$ be surjective \'etale. As usual denote $R = U \times_X U$ with projections $s, t : R \to U$. Set $T = Z \times_U X$. Let $\overline{T} \subset U$ be the scheme theoretic image of $T \to U$. Note that $s^{-1}\overline{T} = t^{-1}\overline{T}$ as taking scheme theoretic images of quasi-compact morphisms commute with flat base change, see Morphisms, Lemma \ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}. Hence we obtain a closed subspace $\overline{Z} \subset X$ whose pullback to $U$ is $\overline{T}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-subspaces-presentation}. By Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-immersion} the morphism $T \to \overline{T}$ is an open immersion. It follows that $Z \to \overline{Z}$ is an open immersion and we win.
Let $U \to X$ be a surjective \'etale morphism whose source is a scheme. Consider the diagram $$ \xymatrix{ U \times_X Z \ar[r] \ar[d]_{i'} & Z \ar[d]^i \\ U \ar[r] & X } $$ By Lemma \ref{lemma-closed-immersion-local} we see that $i$ is a closed immersion if and only if $i'$ is a closed immersion. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules} we see that $i'_*\mathcal{O}_{U \times_X Z}$ is the restriction of $i_*\mathcal{O}_Z$ to $U$. Hence the assertions on $\mathcal{O}_X \to i_*\mathcal{O}_Z$ are equivalent to the corresponding assertions on $\mathcal{O}_U \to i'_*\mathcal{O}_{U \times_X Z}$. And since $i'$ is a closed immersion of schemes, these results follow from Morphisms, Lemma \ref{morphisms-lemma-closed-immersion}. \medskip\noindent Let us prove that given a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ the formula $$ Z(T) = \{h : T \to X \mid h^*\mathcal{I} \to \mathcal{O}_T \text{ is zero}\} $$ defines a closed subspace of $X$. It is clearly a subfunctor of $X$. To show that $Z \to X$ is representable by closed immersions, let $\varphi : U \to X$ be a morphism from a scheme towards $X$. Then $Z \times_X U$ is represented by the analogous subfunctor of $U$ corresponding to the sheaf of ideals $\Im(\varphi^*\mathcal{I} \to \mathcal{O}_U)$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pullback-quasi-coherent} the $\mathcal{O}_U$-module $\varphi^*\mathcal{I}$ is quasi-coherent on $U$, and hence $\Im(\varphi^*\mathcal{I} \to \mathcal{O}_U)$ is a quasi-coherent sheaf of ideals on $U$. By Schemes, Lemma \ref{schemes-lemma-characterize-closed-subspace} we conclude that $Z \times_X U$ is represented by the closed subscheme of $U$ associated to $\Im(\varphi^*\mathcal{I} \to \mathcal{O}_U)$. Thus $Z$ is a closed subspace of $X$. \medskip\noindent In the formula for $Z$ above the inputs $T$ are schemes since algebraic spaces are sheaves on $(\Sch/S)_{fppf}$. We omit the verification that the same formula remains true if $T$ is an algebraic space.
Let $U$ be a scheme and let $U \to X$ be surjective \'etale. Set $V = Z \times_X U$. Then $V$ is a scheme and $i' : V \to U$ is a closed immersion of schemes. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change} for any sheaf $\mathcal{G}$ on $Z$ we have $$ (i_{small}^{-1}i_{small, *}\mathcal{G})|_V = (i')_{small}^{-1}i'_{small, *}(\mathcal{G}|_V) $$ By \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-closed-immersion-pushforward} the map $(i')_{small}^{-1}i'_{small, *}(\mathcal{G}|_V) \to \mathcal{G}|_V$ is an isomorphism. Since $V \to Z$ is surjective and \'etale this implies that $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism. This clearly implies that $i_{small, *}$ is fully faithful, see Sites, Lemma \ref{sites-lemma-exactness-properties}. To prove the statement on the essential image, consider a sheaf of sets $\mathcal{F}$ on $X_\etale$ whose restriction to $X \setminus Z$ is isomorphic to $*$. As in the proof of \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-closed-immersion-pushforward} we consider the adjunction mapping $$ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F}. $$ As in the first part we see that the restriction of this map to $U$ is an isomorphism by the corresponding result for the case of schemes. Since $U$ is an \'etale covering of $X$ we conclude it is an isomorphism.
Choose an \'etale neighbourhood $(U, \overline{u})$ of $\overline{x}$. Then the stalk $(i_{small, *}\mathcal{F})_{\overline{z}}$ is the stalk of $i_{small, *}\mathcal{F}|_U$ at $\overline{u}$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change} we may replace $X$ by $U$ and $Z$ by $Z \times_X U$. Then $Z \to X$ is a closed immersion of schemes and the result is \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-stalk-pushforward-closed-immersion}.
The map is the map adjoint to the map $$ f^*\mathcal{F} \otimes_\mathcal{B} f^* f_*\mathcal{G} = f^*(\mathcal{F} \otimes_\mathcal{A} f_*\mathcal{G}) \longrightarrow f^*\mathcal{F} \otimes_\mathcal{B} \mathcal{G} $$ coming from $\text{id} : f^*\mathcal{F} \to f^*\mathcal{F}$ and the adjunction map $f^* f_*\mathcal{G} \to \mathcal{G}$. To see this map is an isomorphism, we may check on stalks (Properties of Spaces, Theorem \ref{spaces-properties-theorem-exactness-stalks}). Let $\overline{z} : \Spec(k) \to Z$ be a geometric point with image $\overline{x} = i \circ \overline{z} : \Spec(k) \to X$. Working out what our maps does on stalks, we see that we have to show $$ \mathcal{F}_{\overline{x}} \otimes_{\mathcal{A}_{\overline{x}}} \mathcal{G}_{\overline{z}} = (\mathcal{F}_{\overline{x}} \otimes_{\mathcal{A}_{\overline{x}}} \mathcal{B}_{\overline{z}}) \otimes_{\mathcal{B}_{\overline{z}}} \mathcal{G}_{\overline{z}} $$ which holds true. Here we have used that taking tensor products commutes with taking stalks, the behaviour of stalks under pullback Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback}, and the behaviour of stalks under pushforward along a closed immersion Lemma \ref{lemma-stalk-push-closed}.
Let $\mathcal{G}_a$, $a \in A$ be the set of quasi-coherent $\mathcal{O}_X$-submodules contained in $\mathcal{G}$. Then the image $\mathcal{G}'$ of $$ \bigoplus\nolimits_{a \in A} \mathcal{G}_a \longrightarrow \mathcal{F} $$ is quasi-coherent as the image of a map of quasi-coherent sheaves on $X$ is quasi-coherent and since a direct sum of quasi-coherent sheaves is quasi-coherent, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-properties-quasi-coherent}. The module $\mathcal{G}'$ is contained in $\mathcal{G}$. Hence this is the largest quasi-coherent $\mathcal{O}_X$-module contained in $\mathcal{G}$. \medskip\noindent To prove the formula, let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_X$-module and let $\alpha : \mathcal{H} \to \mathcal{G}$ be an $\mathcal{O}_X$-module map. The image of the composition $\mathcal{H} \to \mathcal{G} \to \mathcal{F}$ is quasi-coherent as the image of a map of quasi-coherent sheaves. Hence it is contained in $\mathcal{G}'$. Hence $\alpha$ factors through $\mathcal{G}'$ as desired.
Given quasi-coherent $\mathcal{O}_X$-module $\mathcal{G}$ we consider the subsheaf $\mathcal{H}_Z(\mathcal{G})$ of $\mathcal{G}$ of local sections annihilated by $\mathcal{I}$. By Lemma \ref{lemma-largest-quasi-coherent-subsheaf} there is a canonical largest quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{H}_Z(\mathcal{G})'$. By construction we have $$ \Hom_{\mathcal{O}_X}(i_*\mathcal{F}, \mathcal{H}_Z(\mathcal{G})') = \Hom_{\mathcal{O}_X}(i_*\mathcal{F}, \mathcal{G}) $$ for any quasi-coherent $\mathcal{O}_Z$-module $\mathcal{F}$. Hence we can set $i^!\mathcal{G} = i^*(\mathcal{H}_Z(\mathcal{G})')$. Details omitted.
The morphisms $Y \to Y \cup Z$ and $Z \to Y \cup Z$ are closed immersions by Lemma \ref{lemma-closed-immersion-ideals}. In the short exact sequence we use the equivalence of Lemma \ref{lemma-i-star-equivalence} to think of quasi-coherent modules on closed subspaces of $X$ as quasi-coherent modules on $X$. For the first map in the sequence we use the canonical maps $\mathcal{O}_{Y \cup Z} \to \mathcal{O}_Y$ and $\mathcal{O}_{Y \cup Z} \to \mathcal{O}_Z$ and for the second map we use the canonical map $\mathcal{O}_Y \to \mathcal{O}_{Y \cap Z}$ and the negative of the canonical map $\mathcal{O}_Z \to \mathcal{O}_{Y \cap Z}$. Then to check exactness we may work \'etale locally and deduce exactness from the case of schemes (Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-union}). \medskip\noindent To show the diagram is cocartesian, suppose we are given an algebraic space $T$ over $S$ and morphisms $f : Y \to T$, $g : Z \to T$ agreeing as morphisms $Y \cap Z \to T$. Goal: Show there exists a unique morphism $h : Y \cup Z \to T$ agreeing with $f$ and $g$. To construct $h$ we may work \'etale locally on $Y \cup Z$ (as $Y \cup Z$ is an \'etale sheaf being an algebraic space). Hence we may assume that $X$ is a scheme. In this case we know that $Y \cup Z$ is the pushout of $Y$ and $Z$ along $Y \cap Z$ in the category of schemes by Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-union}. Choose a scheme $T'$ and a surjective \'etale morphism $T' \to T$. Set $Y' = T' \times_{T, f} Y$ and $Z' = T' \times_{T, g} Z$. Then $Y'$ and $Z'$ are schemes and we have a canonical isomorphism $\varphi : Y' \times_Y (Y \cap Z) \to Z' \times_Z (Y \cap Z)$ of schemes. By More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-closed-immersions} the pushout $W' = Y' \amalg_{Y' \times_Y (Y \cap Z), \varphi} Z'$ exists in the category of schemes. The morphism $W' \to Y \cup Z$ is \'etale by More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-closed-immersions-properties-above}. It is surjective as $Y' \to Y$ and $Z' \to Z$ are surjective. The morphisms $f' : Y' \to T'$ and $g' : Z' \to T'$ glue to a unique morphism of schemes $h' : W' \to T'$. By uniqueness the composition $W' \to T' \to T$ descends to the desired morphism $h : Y \cup Z \to T$. Some details omitted.
Let $u\in U$ be a (usual) point and let $\overline{x}$ be a geometric point lying over $u$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-quasi-coherent} we have $(\varphi^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = \mathcal{F}_{\overline{x}}$. Since $\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}$ is the strict henselization by Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring} we see that it is faithfully flat (see More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization}). Thus we see that $(\varphi^*\mathcal{F})_u = 0$ if and only if $\mathcal{F}_{\overline{x}} = 0$. This proves the lemma.
Choose a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$. By Lemma \ref{lemma-support-covering} the inverse image of the support of $\mathcal{F}$ is the support of $\varphi^*\mathcal{F}$ which is closed by Morphisms, Lemma \ref{morphisms-lemma-support-finite-type}. Thus (1) follows from the definition of the topology on $|X|$. \medskip\noindent The first equivalence in (2) is the definition of support. The second equivalence follows from Nakayama's lemma, see Algebra, Lemma \ref{algebra-lemma-NAK}. \medskip\noindent Let $f : Y \to X$ be as in (3). Note that $f^*\mathcal{F}$ is of finite type by Properties of Spaces, Section \ref{spaces-properties-section-properties-modules}. For the final assertion, let $\overline{y}$ be a geometric point of $Y$ mapping to the geometric point $\overline{x}$ on $X$. Recall that $$ (f^*\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}}, $$ see Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback-quasi-coherent}. Hence $(f^*\mathcal{F})_{\overline{y}} \otimes \kappa(\overline{y})$ is nonzero if and only if $\mathcal{F}_{\overline{x}} \otimes \kappa(\overline{x})$ is nonzero. By (2) this implies $x \in \text{Supp}(\mathcal{F})$ if and only if $y \in \text{Supp}(f^*\mathcal{F})$, which is the content of assertion (3).
Choose a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$. Let $R = U \times_X U$ with projections $s, t : R \to U$. Let $i' : Z' \to U$ be the scheme theoretic support of $\varphi^*\mathcal{F}$ and let $\mathcal{G}'$ be the (unique up to unique isomorphism) finite type quasi-coherent $\mathcal{O}_{Z'}$-module with $i'_*\mathcal{G}' = \varphi^*\mathcal{F}$, see Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-support}. As $s^*\varphi^*\mathcal{F} = t^*\varphi^*\mathcal{F}$ we see that $R' = s^{-1}Z' = t^{-1}Z'$ as closed subschemes of $R$ by Morphisms, Lemma \ref{morphisms-lemma-flat-pullback-support}. Thus we may apply Properties of Spaces, Lemma \ref{spaces-properties-lemma-subspaces-presentation} to find a closed subspace $i : Z \to X$ whose pullback to $U$ is $Z'$. Writing $s', t' : R' \to Z'$ the projections and $j' : R' \to R$ the given closed immersion, we see that $$ j'_* (s')^*\mathcal{G}' = s^* i'_*\mathcal{G}' = s^*\varphi^*\mathcal{F} = t^*\varphi^*\mathcal{F} = t^*i'_*\mathcal{G}' = j'_*(t')^*\mathcal{G}' $$ (the first and the last equality by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}). Hence the uniqueness of Morphisms, Lemma \ref{morphisms-lemma-flat-pullback-support} applied to $R' \to R$ gives an isomorphism $\alpha : (t')^*\mathcal{G}' \to (s')^*\mathcal{G}'$ compatible with the canonical isomorphism $t^*\varphi^*\mathcal{F} = s^*\varphi^*\mathcal{F}$ via $j'_*$. Clearly $\alpha$ satisfies the cocycle condition, hence we may apply Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent} to obtain a quasi-coherent module $\mathcal{G}$ on $Z$ whose restriction to $Z'$ is $\mathcal{G}'$ compatible with $\alpha$. Again using the equivalence of the proposition mentioned above (this time for $X$) we conclude that $i_*\mathcal{G} \cong \mathcal{F}$. \medskip\noindent This proves existence. The other properties of the lemma follow by comparing with the result for schemes using Lemma \ref{lemma-support-covering}. Detailed proofs omitted.
Let $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$. If $\mathcal{I}$ is quasi-coherent then we just take $Z$ to be the closed subscheme determined by $\mathcal{I}$, see Lemma \ref{lemma-closed-immersion-ideals}. In general the lemma requires us to show that there exists a largest quasi-coherent sheaf of ideals $\mathcal{I}'$ contained in $\mathcal{I}$. This follows from Lemma \ref{lemma-largest-quasi-coherent-subsheaf}.
To prove (3) it suffices to prove (1) and (2) since the formation of $\mathcal{I}$ commutes with \'etale localization. If (1) holds then in the proof of Lemma \ref{lemma-scheme-theoretic-image} we showed (2). Let us prove that $\mathcal{I}$ is quasi-coherent. Since the property of being quasi-coherent is \'etale local we may assume $Y$ is an affine scheme. As $f$ is quasi-compact, we can find an affine scheme $U$ and a surjective \'etale morphism $U \to X$. Denote $f'$ the composition $U \to X \to Y$. Then $f_*\mathcal{O}_X$ is a subsheaf of $f'_*\mathcal{O}_U$, and hence $\mathcal{I} = \Ker(\mathcal{O}_Y \to \mathcal{O}_{X'})$. By Lemma \ref{lemma-pushforward} the sheaf $f'_*\mathcal{O}_U$ is quasi-coherent on $Y$. Hence $\mathcal{I}$ is quasi-coherent as a kernel of a map between coherent modules. Finally, part (4) follows from parts (1), (2), and (3) as the ideal $\mathcal{I}$ will be the unit ideal in any point of $|Y|$ which is not contained in the closure of $|f|(|X|)$.
Part (1) is true because the reduced induced algebraic space structure on $\overline{|f|(|X|)}$ is the smallest closed subspace of $Y$ through which $f$ factors, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-map-into-reduction}. Part (2) follows from (1), the fact that $|V| \to |Y|$ is open, and the fact that being reduced is preserved under \'etale localization.