hcju/stacksqa huggingface dataset:
Since $f$ is in particular separated and locally of finite type (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-finite-type}) we see that $(X/Y)_{fin}$ is an algebraic space by Proposition \ref{proposition-finite-algebraic-space}. To prove that $(X/Y)_{fin} \to Y$ is separated we have to show the following: Given a scheme $T$ and two pairs $(a, Z_1)$ and $(a, Z_2)$ over $T$ with identical first component satisfying \ref{equation-finite-conditions} there is a closed subscheme $V \subset T$ with the following property: For any morphism of schemes $h : T' \to T$ we have $$ h \text{ factors through } V \Leftrightarrow \Big(T' \times_T Z_1 = T' \times_T Z_2 \text{ as subspaces of }T' \times_Y X\Big) $$ In the proof of Lemma \ref{lemma-finite-diagonal} we have seen that $V = T' \setminus E$ is an open subscheme of $T'$ with closed complement $$ E = \text{pr}_0|_{Z_1}\left(Z_1 \setminus Z_1 \cap Z_2)\right) \cup \text{pr}_0|_{Z_2}\left(Z_2 \setminus Z_1 \cap Z_2)\right). $$ Thus everything comes down to showing that $E$ is also open. By Lemma \ref{lemma-finite-separated} we see that $Z_1$ and $Z_2$ are closed in $T' \times_Y X$. Hence $Z_1 \setminus Z_1 \cap Z_2$ is open in $Z_1$. As $f$ is flat and locally of finite presentation, so is $\text{pr}_0|_{Z_1}$. This is true as $Z_1$ is an open subspace of the base change $T' \times_Y X$, and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-finite-presentation} and Lemmas \ref{spaces-morphisms-lemma-base-change-flat}. Hence $\text{pr}_0|_{Z_1}$ is open, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}. Thus $\text{pr}_0|_{Z_1}\left(Z_1 \setminus Z_1 \cap Z_2)\right)$ is open and it follows that $E$ is open as desired. \medskip\noindent We have already seen that $(X/Y)_{fin} \to Y$ is \'etale, see Proposition \ref{proposition-finite-algebraic-space}. Hence now we know it is locally quasi-finite (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-locally-quasi-finite}) and separated, hence representable by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}. The final assertion is clear (if you like you can use Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}).
Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma \ref{lemma-finite-part-groupoid}. Recall that $R' = R \times_{(U \times_S U)} (U' \times_S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces $$ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') $$ (by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times_{s, U, g} U'$ is open and $R' \to R \times_{s, U, g} U'$ is \'etale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$. \medskip\noindent We think of $u$ as a morphism $\Spec(\kappa(u)) \to U$ as in the statement of the lemma. Set $F_u = R \times_{s, U} \Spec(\kappa(u))$. The set $\{r \in R : s(r) = u, t(r) = u\}$ is finite by assumption and $F_u \to \Spec(\kappa(u))$ is quasi-finite at each of its elements by assumption. Hence we can find a decomposition into open and closed subschemes $$ F_u = Z_u \amalg Rest $$ for some scheme $Z_u$ finite over $\kappa(u)$ whose support is $\{r \in R : s(r) = u, t(r) = u\}$. Note that $e(u) \in Z_u$. Hence by the construction of $U'$ in Section \ref{section-finite-part-groupoid} $(u, Z_u)$ defines a $\Spec(\kappa(u))$-valued point $u'$ of $U'$. \medskip\noindent We still have to show that the set $\{r' \in |R'| : s'(r') = u', t'(r') = u'\}$ is contained in $|Z_{univ}|$. Pick any point $r'$ in this set and represent it by a morphism $z' : \Spec(k) \to R'$. Denote $z : \Spec(k) \to R$ the composition of $z'$ with the map $R' \to R$. Clearly, $z$ defines an element of the set $\{r \in R : s(r) = u, t(r) = u\}$. Also, the compositions $s \circ z, t \circ z : \Spec(k) \to U$ factor through $u$, so we may think of $s \circ z, t \circ z$ as a morphism $\Spec(k) \to \Spec(\kappa(u))$. Then $z' = (z, u' \circ t \circ z, u'\circ s \circ u)$ as morphisms into $R' = R \times_{(U \times_S U)} (U' \times_S U')$. Consider the triple $$ (s \circ z, Z_u \times_{\Spec(\kappa(u)), s \circ z} \Spec(k), z) $$ where $Z_u$ is as above. This defines a $\Spec(k)$-valued point of $Z_{univ}$ whose image via $s', t'$ in $U'$ is $u'$ and whose image via $Z_{univ} \to R'$ is the point $r'$ by the relationship between $z$ and $z'$ mentioned above. This finishes the proof.
Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma \ref{lemma-finite-part-groupoid}. Recall that $R' = R \times_{(U \times_S U)} (U' \times_S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces $$ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') $$ (by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times_{s, U, g} U'$ is open and $R' \to R \times_{s, U, g} U'$ is \'etale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$. \medskip\noindent We think of $u$ as a morphism $\Spec(\kappa(u)) \to U$ as in the statement of the lemma. Set $F_u = R \times_{s, U} \Spec(\kappa(u))$. Let $G_u \subset F_u$ be the scheme theoretic fibre of $G \to U$ over $u$. By assumption $G_u$ is finite and $F_u \to \Spec(\kappa(u))$ is quasi-finite at each point of $G_u$ by assumption. Hence we can find a decomposition into open and closed subschemes $$ F_u = Z_u \amalg Rest $$ for some scheme $Z_u$ finite over $\kappa(u)$ whose support is $G_u$. Note that $e(u) \in Z_u$. Hence by the construction of $U'$ in Section \ref{section-finite-part-groupoid} $(u, Z_u)$ defines a $\Spec(\kappa(u))$-valued point $u'$ of $U'$. \medskip\noindent We still have to show that the set $\{g' \in |G'| : g'\text{ maps to }u'\}$ is contained in $|Z_{univ}|$. Pick any point $g'$ in this set and represent it by a morphism $z' : \Spec(k) \to G'$. Denote $z : \Spec(k) \to G$ the composition of $z'$ with the map $G' \to G$. Clearly, $z$ defines a point of $G_u$. In fact, let us write $\tilde u : \Spec(k) \to u \to U$ for the corresponding map to $u$ or $U$. Consider the triple $$ (\tilde u, Z_u \times_{u, \tilde u} \Spec(k), z) $$ where $Z_u$ is as above. This defines a $\Spec(k)$-valued point of $Z_{univ}$ whose image via $s', t'$ in $U'$ is $u'$ and whose image via $Z_{univ} \to R'$ is the point $z'$ (because the image in $R$ is $z$). This finishes the proof.
Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma \ref{lemma-finite-part-groupoid}. Recall that $R' = R \times_{(U \times_S U)} (U' \times_S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces $$ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') $$ (by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times_{s, U, g} U'$ is open and $R' \to R \times_{s, U, g} U'$ is \'etale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$. \medskip\noindent We think of $u$ as a morphism $\Spec(\kappa(u)) \to U$ as in the statement of the lemma. Set $F_u = R \times_{s, U} \Spec(\kappa(u))$. The morphism $F_u \to \Spec(\kappa(u))$ is quasi-finite at $e(u)$ by assumption. Hence we can find a decomposition into open and closed subschemes $$ F_u = Z_u \amalg Rest $$ for some scheme $Z_u$ finite over $\kappa(u)$ whose support is $e(u)$. Hence by the construction of $U'$ in Section \ref{section-finite-part-groupoid} $(u, Z_u)$ defines a $\Spec(\kappa(u))$-valued point $u'$ of $U'$. To finish the proof we have to show that $e'(u') \in Z_{univ}$ which is clear.
Let $U' \to U$ and $u' \in U'$ be the separated \'etale morphism of schemes we found in Lemma \ref{lemma-strong-splitting-scheme}. Let $P \subset R'$ be the strong splitting of $R'$ over $u'$. By More on Groupoids, Lemma \ref{more-groupoids-lemma-restrict-preserves-type} the morphisms $s', t' : R' \to U'$ are flat and locally of finite presentation. They are finite by assumption. Hence $s', t'$ are finite locally free, see Morphisms, Lemma \ref{morphisms-lemma-finite-flat}. In particular $t(s^{-1}(u'))$ is a finite set of points $\{u'_1, u'_2, \ldots, u'_n\}$ of $U'$. Choose a quasi-compact open $W \subset U'$ containing each $u'_i$. As $U$ is affine the morphism $W \to U$ is quasi-compact (see Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}). The morphism $W \to U$ is also locally quasi-finite (see Morphisms, Lemma \ref{morphisms-lemma-etale-locally-quasi-finite}) and separated. Hence by More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine} (a version of Zariski's Main Theorem) we conclude that $W$ is quasi-affine. By Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine} we see that $\{u'_1, \ldots, u'_n\}$ are contained in an affine open of $U'$. Thus we may apply Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine} to conclude that there exists an affine $P$-invariant open $U'' \subset U'$ which contains $u'$. \medskip\noindent To finish the proof denote $R'' = R|_{U''}$ the restriction of $R$ to $U''$. This is the same as the restriction of $R'$ to $U''$. As $P \subset R'$ is an open and closed subscheme, so is $P|_{U''} \subset R''$. By construction the open subscheme $U'' \subset U'$ is $P$-invariant which means that $P|_{U''} = (s'|_P)^{-1}(U'') = (t'|_P)^{-1}(U'')$ (see discussion in Groupoids, Section \ref{groupoids-section-invariant}) so the restrictions of $s''$ and $t''$ to $P|_{U''}$ are still finite. The sub groupoid scheme $P|_{U''}$ is still a strong splitting of $R''$ over $u''$; above we verified (a), (b) and (c) holds as $\{r' \in R': t'(r') = u', s'(r') = u'\} = \{r'' \in R'': t''(r'') = u', s''(r'') = u'\}$ trivially. The lemma is proved.
By Lemma \ref{lemma-finite-lives-on-locally-quasi-finite-part} we may replace $X$ by the open subscheme which is locally quasi-finite over $Y$. Hence we may assume that $f$ is separated and locally quasi-finite. We will check the three conditions of Spaces, Definition \ref{spaces-definition-algebraic-space}. Condition (1) follows from Lemma \ref{lemma-finite-sheaf}. Condition (2) follows from Lemma \ref{lemma-finite-diagonal}. Finally, condition (3) follows from Lemma \ref{lemma-finite-surjective-etale-cover}. Thus $(X/Y)_{fin}$ is an algebraic space. Moreover, that lemma shows that there exists a commutative diagram $$ \xymatrix{ U \ar[rr] \ar[rd] & & (X/Y)_{fin} \ar[ld] \\ & Y } $$ with horizontal arrow surjective and \'etale and south-east arrow \'etale. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-local} this implies that the south-west arrow is \'etale as well.
By Homology, Definition \ref{homology-definition-abelian-category} we have to show that image and coimage agree. By Sheaves, Lemma \ref{sheaves-lemma-points-exactness} it is enough to show that image and coimage have the same stalk at every $x \in X$. By the constructions of kernels and cokernels above these stalks are the coimage and image in the categories of $\mathcal{O}_{X, x}$-modules. Thus we get the result from the fact that the category of modules over a ring is abelian.
As $\textit{Mod}(\mathcal{O}_X)$ is abelian (Lemma \ref{lemma-abelian}) it has all finite limits and colimits (Homology, Lemma \ref{homology-lemma-colimit-abelian-category}). Thus the existence of limits and colimits and their description follows from the existence of products and coproducts and their description (see discussion above) and Categories, Lemmas \ref{categories-lemma-limits-products-equalizers} and \ref{categories-lemma-colimits-coproducts-coequalizers}. Since sheafification commutes with taking stalks we see that colimits commute with taking stalks. Part (3) signifies that given a system $0 \to \mathcal{F}_i \to \mathcal{G}_i \to \mathcal{H}_i \to 0$ of exact sequences of $\mathcal{O}_X$-modules over a directed set $I$ the sequence $0 \to \colim \mathcal{F}_i \to \colim \mathcal{G}_i \to \colim \mathcal{H}_i \to 0$ is exact as well. Since we can check exactness on stalks (Lemma \ref{lemma-abelian}) this follows from the case of modules which is Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}. We omit the proof of (4).
Parts (1) and (2) hold because $(f^*, f_*)$ is an adjoint pair of functors, see Sheaves, Lemma \ref{sheaves-lemma-adjoint-pullback-pushforward-modules} and Categories, Section \ref{categories-section-adjoint}. Part (3) holds because exactness can be checked on stalks (Lemma \ref{lemma-abelian}) and the description of stalks of the pullback, see Sheaves, Lemma \ref{sheaves-lemma-pullback-abelian-stalk}.
If $s$ is an element of the right hand side, then there exists an open covering $U = \bigcup_{j \in J} U_j$ such that $s|_{U_j}$ is a finite sum $\sum_{i \in I_j} s_{ji}$ with $s_{ji} \in \mathcal{F}_i(U_j)$. Because $U$ is quasi-compact we may assume that the covering is finite, i.e., that $J$ is finite. Then $I' = \bigcup_{j \in J} I_j$ is a finite subset of $I$. Clearly, $s$ is a section of the subsheaf $\bigoplus_{i \in I'} \mathcal{F}_i$. The result follows from the fact that for a finite direct sum sheafification is not needed, see Lemma \ref{lemma-limits-colimits} above.
Consider the subpresheaf of $\mathcal{O}_X$-modules defined by the rule $$ U \longmapsto \{ \text{sums } \sum\nolimits_{i \in J} f_i (s_i|_U) \text{ where } J \text{ is finite, } U \subset U_i \text{ for } i\in J, \text{ and } f_i \in \mathcal{O}_X(U) \} $$ Let $\mathcal{G}$ be the sheafification of this subpresheaf. This is a subsheaf of $\mathcal{F}$ by Sheaves, Lemma \ref{sheaves-lemma-characterize-epi-mono}. Since all the finite sums clearly have to be in $\mathcal{G}$ this is the smallest subsheaf as desired.
Given an open subspace $V$ of $Y$ we may consider the commutative diagram of ringed spaces $$ \xymatrix{ (f^{-1}V, \mathcal{O}_{f^{-1}V}) \ar[r]_{j'} \ar[d]_{f'} & (X, \mathcal{O}_X) \ar[d]^f \\ (V, \mathcal{O}_V) \ar[r]^j & (Y, \mathcal{O}_Y) } $$ We know that $f^*\mathcal{G}|_{f^{-1}V} \cong (f')^*(\mathcal{G}|_V)$, see Sheaves, Lemma \ref{sheaves-lemma-push-pull-composition-modules}. Thus we may assume that $\mathcal{G}$ is globally generated. \medskip\noindent We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma \ref{lemma-exactness-pushforward-pullback}. Thus if we have a surjection $$ \bigoplus\nolimits_{i \in I} \mathcal{O}_Y \to \mathcal{G} \to 0 $$ then upon applying $f^*$ we obtain the surjection $$ \bigoplus\nolimits_{i \in I} \mathcal{O}_X \to f^*\mathcal{G} \to 0. $$ This implies the lemma.
Arguing as in the proof of Lemma \ref{lemma-pullback-locally-generated} we may assume $\mathcal{G}$ is globally generated by finitely many sections. We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma \ref{lemma-exactness-pushforward-pullback}. Thus if we have a surjection $$ \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_Y \to \mathcal{G} \to 0 $$ then upon applying $f^*$ we obtain the surjection $$ \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_X \to f^*\mathcal{G} \to 0. $$ This implies the lemma.
Arguing as in the proof of Lemma \ref{lemma-pullback-locally-generated} we may assume $\mathcal{G}$ has a global presentation by direct sums of copies of $\mathcal{O}_Y$. We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma \ref{lemma-exactness-pushforward-pullback}. Thus if we have an exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{O}_Y \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_Y \longrightarrow \mathcal{G} \longrightarrow 0 $$ then upon applying $f^*$ we obtain the exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{O}_X \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_X \longrightarrow f^*\mathcal{G} \longrightarrow 0. $$ This implies the lemma.
The isomorphism between $\mathcal{F}_1$ and $\mathcal{F}_3$ comes from the fact that $\pi^*$ is defined as the sheafification of the presheaf in (3), see Sheaves, Section \ref{sheaves-section-ringed-spaces-functoriality-modules}. The isomorphism between the constructions in (2) and (1) comes from the fact that the functor $\pi^*$ is right exact, so $\pi^*(\bigoplus_{j \in J} R) \to \pi^*(\bigoplus_{i \in I} R) \to \pi^*M \to 0$ is exact, $\pi^*$ commutes with arbitrary direct sums, see Lemma \ref{lemma-exactness-pushforward-pullback}, and finally the fact that $\pi^*(R) = \mathcal{O}_X$. \medskip\noindent Assertion (1) is clear from construction (2). Assertion (2) is clear since $\pi^*$ has these properties. Assertion (3) follows from the description of stalks of pullback sheaves, see Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}. Assertion (4) follows from adjointness of $\pi_*$ and $\pi^*$.
The assertion follows from the first description of $\mathcal{F}_M$ in Lemma \ref{lemma-construct-quasi-coherent-sheaves} as $\pi^*M$, and the following commutative diagram of ringed spaces $$ \xymatrix{ (Y, \mathcal{O}_Y) \ar[r]_-\pi \ar[d]_g & (\{*\}, \Gamma(Y, \mathcal{O}_Y)) \ar[d]^{\text{induced by }g^\sharp} \\ (X, \mathcal{O}_X) \ar[r]^-\pi & (\{*\}, \Gamma(X, \mathcal{O}_X)) } $$ (Also use Sheaves, Lemma \ref{sheaves-lemma-push-pull-composition-modules}.)
First we may replace $X$ by an open neighbourhood of $x$ and assume that $\mathcal{F}$ is isomorphic to the cokernel of a map $$ \Psi : \bigoplus\nolimits_{j \in J} \mathcal{O}_X \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_X. $$ The problem is that this map may not be given by a ``matrix'', because the module of global sections of a direct sum is in general different from the direct sum of the modules of global sections. \medskip\noindent Let $x \in E \subset X$ be a quasi-compact neighbourhood of $x$ (note: $E$ may not be open). Let $x \in U \subset E$ be an open neighbourhood of $x$ contained in $E$. Next, we proceed as in the proof of Lemma \ref{lemma-section-direct-sum-quasi-compact}. For each $j \in J$ denote $s_j \in \Gamma(X, \bigoplus\nolimits_{i \in I} \mathcal{O}_X)$ the image of the section $1$ in the summand $\mathcal{O}_X$ corresponding to $j$. There exists a finite collection of opens $U_{jk}$, $k \in K_j$ such that $E \subset \bigcup_{k \in K_j} U_{jk}$ and such that each restriction $s_j|_{U_{jk}}$ is a finite sum $\sum_{i \in I_{jk}} f_{jki}$ with $I_{jk} \subset I$, and $f_{jki}$ in the summand $\mathcal{O}_X$ corresponding to $i \in I$. Set $I_j = \bigcup_{k \in K_j} I_{jk}$. This is a finite set. Since $U \subset E \subset \bigcup_{k \in K_j} U_{jk}$ the section $s_j|_U$ is a section of the finite direct sum $\bigoplus_{i \in I_j} \mathcal{O}_X$. By Lemma \ref{lemma-limits-colimits} we see that actually $s_j|_U$ is a sum $\sum_{i \in I_j} f_{ij}$ and $f_{ij} \in \mathcal{O}_X(U) = \Gamma(U, \mathcal{O}_U)$. \medskip\noindent At this point we can define a module $M$ as the cokernel of the map $$ \bigoplus\nolimits_{j \in J} \Gamma(U, \mathcal{O}_U) \longrightarrow \bigoplus\nolimits_{i \in I} \Gamma(U, \mathcal{O}_U) $$ with matrix given by the $(f_{ij})$. By construction (2) of Lemma \ref{lemma-construct-quasi-coherent-sheaves} we see that $\mathcal{F}_M$ has the same presentation as $\mathcal{F}|_U$ and therefore $\mathcal{F}_M \cong \mathcal{F}|_U$.
Choose $s_1, \ldots, s_r \in \mathcal{F}_x$ mapping to a basis of $\mathcal{O}_{X, x}^{\oplus r}$ by the isomorphism. Choose an open neighbourhood $U$ of $x$ such that $s_i$ lifts to $s_i \in \mathcal{F}(U)$. After shrinking $U$ we see that the induced map $\psi : \mathcal{O}_U^{\oplus r} \to \mathcal{F}|_U$ is surjective (Lemma \ref{lemma-finite-type-surjective-on-stalk}). By Lemma \ref{lemma-kernel-surjection-finite-free-onto-finite-presentation} we see that $\Ker(\psi)$ is of finite type. Then $\Ker(\psi)_x = 0$ implies that $\Ker(\psi)$ becomes zero after shrinking $U$ once more (Lemma \ref{lemma-finite-type-stalk-zero}).
Denote $\mathcal{K} \subset \mathcal{G}$ the kernel of $\varphi$. By Lemma \ref{lemma-coherent-abelian} we see that $\mathcal{K}$ is a finite type $\mathcal{O}_X$-module. Our assumption is that $\mathcal{K}_x = 0$. By Lemma \ref{lemma-finite-type-stalk-zero} there exists an open neighbourhood $U$ of $x$ such that $\mathcal{K}|_U = 0$. Then $U$ works.
This is true because $i_*\mathcal{O}_Z$ is quasi-coherent by definition. And locally on $Z$ the sheaf $\mathcal{F}$ is a cokernel of a map between direct sums of copies of $\mathcal{O}_Z$. Moreover, any direct sum of copies of the {\it the same} quasi-coherent sheaf is quasi-coherent. And finally, $i_*$ commutes with arbitrary colimits, see Lemma \ref{lemma-i-star-right-adjoint}. Some details omitted.
We claim that for a $\mathcal{O}_Z$-module $\mathcal{F}$ the canonical map $$ i^*i_*\mathcal{F} \longrightarrow \mathcal{F} $$ is an isomorphism. We check this on stalks. Say $z \in Z$ and $x = i(z)$. We have $$ (i^*i_*\mathcal{F})_z = (i_*\mathcal{F})_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{Z, z} = \mathcal{F}_z \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{Z, z} = \mathcal{F}_z $$ by Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}, the fact that $\mathcal{O}_{Z, z}$ is a quotient of $\mathcal{O}_{X, x}$, and Sheaves, Lemma \ref{sheaves-lemma-stalks-closed-pushforward}. It follows that $i_*$ is fully faithful. \medskip\noindent Let $\mathcal{G}$ be a $\mathcal{O}_X$-module with $\mathcal{I}\mathcal{G} = 0$. We will prove the canonical map $$ \mathcal{G} \longrightarrow i_*i^*\mathcal{G} $$ is an isomorphism. This proves that $\mathcal{G} = i_*\mathcal{F}$ with $\mathcal{F} = i^*\mathcal{G}$ which finishes the proof. We check the displayed map induces an isomorphism on stalks. If $x \in X$, $x \not \in i(Z)$, then $\mathcal{G}_x = 0$ because $\mathcal{I}_x = \mathcal{O}_{X, x}$ in this case. As above $(i_*i^*\mathcal{G})_x = 0$ by Sheaves, Lemma \ref{sheaves-lemma-stalks-closed-pushforward}. On the other hand, if $x \in Z$, then we obtain the map $$ \mathcal{G}_x \longrightarrow \mathcal{G}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{Z, x} $$ by Sheaves, Lemmas \ref{sheaves-lemma-stalk-pullback-modules} and \ref{sheaves-lemma-stalks-closed-pushforward}. This map is an isomorphism because $\mathcal{O}_{Z, x} = \mathcal{O}_{X, x}/\mathcal{I}_x$ and because $\mathcal{G}_x$ is annihilated by $\mathcal{I}_x$ by assumption.
Assume $\varphi$ is surjective. Pick $x \in X$. There exists an open neighbourhood $U$ of $x$ such that both $\mathcal{F}|_U$ and $\mathcal{G}|_U$ are isomorphic to $\mathcal{O}_U^{\oplus r}$. Pick lifts of the free generators of $\mathcal{G}|_U$ to obtain a map $\psi : \mathcal{G}|_U \to \mathcal{F}|_U$ such that $\varphi|_U \circ \psi = \text{id}$. Hence we conclude that the map $\Gamma(U, \mathcal{F}) \to \Gamma(U, \mathcal{G})$ induced by $\varphi$ is surjective. Since both $\Gamma(U, \mathcal{F})$ and $\Gamma(U, \mathcal{G})$ are isomorphic to $\Gamma(U, \mathcal{O}_U)^{\oplus r}$ as an $\Gamma(U, \mathcal{O}_U)$-module we may apply Algebra, Lemma \ref{algebra-lemma-fun} to see that $\Gamma(U, \mathcal{F}) \to \Gamma(U, \mathcal{G})$ is injective. This finishes the proof.
Assume $\mathcal{F}$ is a direct summand of the finite locally free $\mathcal{O}_X$-module $\mathcal{H}$. Let $x \in X$ be a point. Then $\mathcal{H}_x$ is a finite free $\mathcal{O}_{X, x}$-module. Because $\mathcal{O}_{X, x}$ is local, we see that $\mathcal{F}_x \cong \mathcal{O}_{X, x}^{\oplus r}$ for some $r$, see Algebra, Lemma \ref{algebra-lemma-finite-projective}. By Lemma \ref{lemma-finite-presentation-stalk-free} we see that $\mathcal{F}$ is free of rank $r$ in an open neighbourhood of $x$. (Note that $\mathcal{F}$ is of finite presentation as a summand of $\mathcal{H}$.)