hcju/stacksqa huggingface dataset:
The set $U$ is open by Modules, Lemmas \ref{modules-lemma-finite-type-surjective-on-stalk} and \ref{modules-lemma-finite-type-to-coherent-injective-on-stalk}. Observe that $j$ is quasi-compact by Properties, Lemma \ref{properties-lemma-immersion-into-noetherian}. To prove the final statement it suffices to show for every quasi-compact open $W \subset X$ there is an isomorphism $$ \colim \Gamma(W, \mathcal{F}^{[n]}) \longrightarrow \Gamma(U \cap W, \mathcal{O}_U) $$ of $\mathcal{O}_X(W)$-modules compatible with restriction maps. We will omit the verification of compatibilities. After replacing $X$ by $W$ and rewriting the above in terms of homs, we see that it suffices to construct an isomorphism $$ \colim \Hom_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{F}^{[n]}) \longrightarrow \Hom_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{O}_U) $$ Choose an open $V \subset X$ such that every irreducible component of $X \setminus V$ has codimension $\geq 2$ in $X$ and such that $\mathcal{F}|_V$ is invertible, see Lemma \ref{lemma-reflexive-over-normal}. Then restriction defines an equivalence of categories between rank $1$ coherent reflexive modules on $X$ and $V$ and between rank $1$ coherent reflexive modules on $U$ and $V \cap U$. See Lemma \ref{lemma-reflexive-S2-extend} and Serre's criterion Properties, Lemma \ref{properties-lemma-criterion-normal}. Thus it suffices to construct an isomorphism $$ \colim \Gamma(V, (\mathcal{F}|_V)^{\otimes n}) \longrightarrow \Gamma(V \cap U, \mathcal{O}_U) $$ Since $\mathcal{F}|_V$ is invertible and since $U \cap V$ is equal to the set of points where $s|_V$ generates this invertible module, this is a special case of Properties, Lemma \ref{properties-lemma-invert-s-sections} (there is an explicit formula for the map as well).
Let $\xi \in X$ be a generic point of an irreducible component $Z$ of $X \setminus U$. After replacing $X$ by an open neighbourhood of $\xi$ we may assume that $Z = X \setminus U$ is irreducible. Since $s : \mathcal{O}_U \to \mathcal{F}|_U$ is an isomorphism, if the codimension of $Z$ in $X$ is $\geq 2$, then $s : \mathcal{O}_X \to \mathcal{F}$ is an isomorphism by Lemma \ref{lemma-reflexive-S2-extend} and Serre's criterion Properties, Lemma \ref{properties-lemma-criterion-normal}. This would mean that $Z = \emptyset$, a contradiction.
Assume (1). Then for $x \in X \setminus U$ the inverse image $U_x$ of $U$ under the canonical morphism $f_x : \Spec(\mathcal{O}_{X, x}) \to X$ is affine and does not contain $x$. Thus $\mathfrak m_x \Gamma(U_x, \mathcal{O}_{U_x})$ is the unit ideal. In particular, we see that we can write $$ 1 = \sum f_i g_i $$ with $f_i \in \mathfrak m_x$ and $g_i \in \Gamma(U_x, \mathcal{O}_{U_x})$. By Lemma \ref{lemma-structure-sheaf-Xs} we have $\Gamma(U_x, \mathcal{O}_{U_x}) = \colim \mathcal{F}^{[n]}_x$ with transition maps given by multiplication by $s$. Hence for some $n > 0$ we have $$ s^n = \sum f_i t_i $$ for some $t_i = s^ng_i \in \mathcal{F}^{[n]}_x$. Thus (2) holds. \medskip\noindent Conversely, assume that (2) holds. To prove $j$ is affine is local on $X$, see Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. Thus we may and do assume that $X$ is affine. Our goal is to show that $U$ is affine. By Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-if-quasi-affine} it suffices to show that $H^p(U, \mathcal{O}_U) = 0$ for $p > 0$. Since $H^p(U, \mathcal{O}_U) = H^0(X, R^pj_*\mathcal{O}_U)$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images-application}) and since $R^pj_*\mathcal{O}_U$ is quasi-coherent (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}) it is enough to show the stalk $(R^pj_*\mathcal{O}_U)_x$ at a point $x \in X$ is zero. Consider the base change diagram $$ \xymatrix{ U_x \ar[d]_{j_x} \ar[r] & U \ar[d]^j \\ \Spec(\mathcal{O}_{X, x}) \ar[r] & X } $$ By Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} we have $(R^pj_*\mathcal{O}_U)_x = R^pj_{x, *}\mathcal{O}_{U_x}$. Hence we may assume $X$ is local with closed point $x$ and we have to show $U$ is affine (because this is equivalent to the desired vanishing by the reference given above). In particular $d = \dim(X)$ is finite (Algebra, Proposition \ref{algebra-proposition-dimension}). If $x \in U$, then $U = X$ and the result is clear. If $d = 0$ and $x \not \in U$, then $U = \emptyset$ and the result is clear. Now assume $d > 0$ and $x \not \in U$. Since $j_*\mathcal{O}_U = \colim \mathcal{F}^{[n]}$ our assumption means that we can write $$ 1 = \sum f_i g_i $$ for some $n > 0$, $f_i \in \mathfrak m_x$, and $g_i \in \mathcal{O}(U)$. By induction on $d$ we know that $D(f_i) \cap U$ is affine for all $i$: going through the whole argument just given with $X$ replaced by $D(f_i)$ we end up with Noetherian local rings whose dimension is strictly smaller than $d$. Hence $U$ is affine by Properties, Lemma \ref{properties-lemma-characterize-affine} as desired.
The question is local on the base, see Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}. Thus we may assume $S$ is affine. Say $S = \Spec(R)$ and $\mathcal{A}$ corresponds to the graded $R$-algebra $A$. Then $X = \text{Proj}(A)$, see Constructions, Section \ref{constructions-section-relative-proj-via-glueing}. In case (1) we may after possibly localizing more assume that $A$ is generated by homogeneous elements $f_1, \ldots, f_n \in A_{+}$ over $A_0$. Then $A_{+} = (f_1, \ldots, f_n)$ by Algebra, Lemma \ref{algebra-lemma-S-plus-generated}. In case (3) we see that $\mathcal{F} = \widetilde{M}$ for some finite type $A_0$-module $M \subset A_{+}$. Say $M = \sum A_0f_i$. Say $f_i = \sum f_{i, j}$ is the decomposition into homogeneous pieces. The condition in (3) signifies that $A_{+} \subset \sqrt{(f_{i, j})}$. Thus in both cases we conclude that $\text{Proj}(A)$ is quasi-compact by Constructions, Lemma \ref{constructions-lemma-proj-quasi-compact}. Finally, (2) follows from (1).
The assumption implies that $p$ is quasi-compact, see Lemma \ref{lemma-relative-proj-quasi-compact}. Hence it suffices to show that $p$ is locally of finite type. Thus the question is local on the base and target, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-type-characterize}. Say $S = \Spec(R)$ and $\mathcal{A}$ corresponds to the graded $R$-algebra $A$. After further localizing on $S$ we may assume that $A$ is a finite type $R$-algebra. The scheme $X$ is constructed out of glueing the spectra of the rings $A_{(f)}$ for $f \in A_{+}$ homogeneous. Each of these is of finite type over $R$ by Algebra, Lemma \ref{algebra-lemma-dehomogenize-finite-type} part (1). Thus $\text{Proj}(A)$ is of finite type over $R$. To see the statement on $\mathcal{O}_X(d)$ use part (2) of Algebra, Lemma \ref{algebra-lemma-dehomogenize-finite-type}.
The question is local on the base. Thus we may assume that $X = \Spec(R)$ is affine. Let $\mathcal{A}$ be the quasi-coherent $\mathcal{O}_X$-algebra associated to the graded $R$-algebra $A$. The assumption is that $R \to A_0$ is integral and $A$ is of finite type over $A_0$. Write $X \to \Spec(R)$ as the composition $X \to \Spec(A_0) \to \Spec(R)$. Since $R \to A_0$ is an integral ring map, we see that $\Spec(A_0) \to \Spec(R)$ is universally closed, see Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}. The quasi-compact (see Constructions, Lemma \ref{constructions-lemma-proj-quasi-compact}) morphism $$ X = \text{Proj}(A) \to \Spec(A_0) $$ satisfies the existence part of the valuative criterion by Constructions, Lemma \ref{constructions-lemma-proj-valuative-criterion} and hence it is universally closed by Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}. Thus $X \to \Spec(R)$ is universally closed as a composition of universally closed morphisms.
Assume that $\mathcal{A}_0$ is a finite type $\mathcal{O}_S$-module. Choose an affine open $U = \Spec(R) \subset X$ such that $\mathcal{A}$ corresponds to a graded $R$-algebra $A$ with $A_0$ a finite $R$-module. Condition (1) means that (after possibly localizing further on $S$) that $A$ is a finite type $A_0$-algebra and condition (2) means that (after possibly localizing further on $S$) that $A$ is a finite type $R$-algebra. Thus these conditions imply each other by Algebra, Lemma \ref{algebra-lemma-compose-finite-type}. \medskip\noindent A locally projective morphism is proper, see Morphisms, Lemma \ref{morphisms-lemma-locally-projective-proper}. Thus we may now assume that $S = \Spec(R)$ and $X = \text{Proj}(A)$ and that $A_0$ is finite over $R$ and $A$ of finite type over $R$. We will show that $X = \text{Proj}(A) \to \Spec(R)$ is projective. We urge the reader to prove this for themselves, by directly constructing a closed immersion of $X$ into a projective space over $R$, instead of reading the argument we give below. \medskip\noindent By Lemma \ref{lemma-relative-proj-finite-type} we see that $X$ is of finite type over $\Spec(R)$. Constructions, Lemma \ref{constructions-lemma-ample-on-proj} tells us that $\mathcal{O}_X(d)$ is ample on $X$ for some $d \geq 1$ (see Properties, Section \ref{properties-section-ample}). Hence $X \to \Spec(R)$ is quasi-projective (by Morphisms, Definition \ref{morphisms-definition-quasi-projective}). By Morphisms, Lemma \ref{morphisms-lemma-quasi-projective-open-projective} we conclude that $X$ is isomorphic to an open subscheme of a scheme projective over $\Spec(R)$. Therefore, to finish the proof, it suffices to show that $X \to \Spec(R)$ is universally closed (use Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). This follows from Lemma \ref{lemma-relative-proj-universally-closed}.
Namely, the morphism associated to the graded $\mathcal{O}_S$-algebra map $$ \text{Sym}_{\mathcal{O}_X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A} $$ is a closed immersion $X \to \mathbf{P}(\mathcal{A}_1)$, see Constructions, Lemma \ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}.
Affine locally flatness of $X$ over $S$ reduces to the following statement: Let $R$ be a ring, let $A$ be a graded $R$-algebra with $A_d$ flat over $R$ for $d \gg 0$, let $f \in A_d$ for some $d > 0$, then $A_{(f)}$ is flat over $R$. Since $A_{(f)} = \colim A_{nd}$ where the transition maps are given by multiplication by $f$, this follows from Algebra, Lemma \ref{algebra-lemma-colimit-flat}. Argue similarly to get flatness of $\mathcal{O}_X(d)$ over $S$.
Affine locally this reduces to the following statement: Let $R$ be a ring and let $A$ be a finitely presented graded $R$-algebra. Then $\text{Proj}(A) \to \Spec(R)$ is of finite presentation and $\mathcal{O}_{\text{Proj}(A)}(d)$ is a $\mathcal{O}_{\text{Proj}(A)}$-module of finite presentation. The finite presentation condition implies we can choose a presentation $$ A = R[X_1, \ldots, X_n]/(F_1, \ldots, F_m) $$ where $R[X_1, \ldots, X_n]$ is a polynomial ring graded by giving weights $d_i$ to $X_i$ and $F_1, \ldots, F_m$ are homogeneous polynomials of degree $e_j$. Let $R_0 \subset R$ be the subring generated by the coefficients of the polynomials $F_1, \ldots, F_m$. Then we set $A_0 = R_0[X_1, \ldots, X_n]/(F_1, \ldots, F_m)$. By construction $A = A_0 \otimes_{R_0} R$. Thus by Constructions, Lemma \ref{constructions-lemma-base-change-map-proj} it suffices to prove the result for $X_0 = \text{Proj}(A_0)$ over $R_0$. By Lemma \ref{lemma-relative-proj-finite-type} we know $X_0$ is of finite type over $R_0$ and $\mathcal{O}_{X_0}(d)$ is a quasi-coherent $\mathcal{O}_{X_0}$-module of finite type. Since $R_0$ is Noetherian (as a finitely generated $\mathbf{Z}$-algebra) we see that $X_0$ is of finite presentation over $R_0$ (Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}) and $\mathcal{O}_{X_0}(d)$ is of finite presentation by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}. This finishes the proof.
The morphism $p$ is separated by Constructions, Lemma \ref{constructions-lemma-relative-proj-separated}. As $p$ is quasi-compact, $p_*$ transforms quasi-coherent modules into quasi-coherent modules, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Hence $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_S$-module. In particular, $\mathcal{B} = \mathcal{A}/\mathcal{I}$ is a quasi-coherent graded $\mathcal{O}_S$-algebra. The functoriality morphism $Z' = \underline{\text{Proj}}_S(\mathcal{B}) \to \underline{\text{Proj}}_S(\mathcal{A})$ is everywhere defined and a closed immersion, see Constructions, Lemma \ref{constructions-lemma-surjective-graded-rings-map-relative-proj}. Hence it suffices to prove $Z = Z'$ as closed subschemes of $X$. \medskip\noindent Having said this, the question is local on the base and we may assume that $S = \Spec(R)$ and that $X = \text{Proj}(A)$ for some graded $R$-algebra $A$. Assume $\mathcal{I} = \widetilde{I}$ for $I \subset A$ a graded ideal. By Constructions, Lemma \ref{constructions-lemma-proj-quasi-compact} there exist $f_0, \ldots, f_n \in A_{+}$ such that $A_{+} \subset \sqrt{(f_0, \ldots, f_n)}$ in other words $X = \bigcup D_{+}(f_i)$. Therefore, it suffices to check that $Z \cap D_{+}(f_i) = Z' \cap D_{+}(f_i)$ for each $i$. By renumbering we may assume $i = 0$. Say $Z \cap D_{+}(f_0)$, resp.\ $Z' \cap D_{+}(f_0)$ is cut out by the ideal $J$, resp.\ $J'$ of $A_{(f_0)}$. \medskip\noindent The inclusion $J' \subset J$. Let $d$ be the least common multiple of $\deg(f_0), \ldots, \deg(f_n)$. Note that each of the twists $\mathcal{O}_X(nd)$ is invertible, trivialized by $f_i^{nd/\deg(f_i)}$ over $D_{+}(f_i)$, and that for any quasi-coherent module $\mathcal{F}$ on $X$ the multiplication maps $\mathcal{O}_X(nd) \otimes_{\mathcal{O}_X} \mathcal{F}(m) \to \mathcal{F}(nd + m)$ are isomorphisms, see Constructions, Lemma \ref{constructions-lemma-when-invertible}. Observe that $J'$ is the ideal generated by the elements $g/f_0^e$ where $g \in I$ is homogeneous of degree $e\deg(f_0)$ (see proof of Constructions, Lemma \ref{constructions-lemma-surjective-graded-rings-map-proj}). Of course, by replacing $g$ by $f_0^lg$ for suitable $l$ we may always assume that $d | e$. Then, since $g$ vanishes as a section of $\mathcal{O}_X(e\deg(f_0))$ restricted to $Z$ we see that $g/f_0^d$ is an element of $J$. Thus $J' \subset J$. \medskip\noindent Conversely, suppose that $g/f_0^e \in J$. Again we may assume $d | e$. Pick $i \in \{1, \ldots, n\}$. Then $Z \cap D_{+}(f_i)$ is cut out by some ideal $J_i \subset A_{(f_i)}$. Moreover, $$ J \cdot A_{(f_0f_i)} = J_i \cdot A_{(f_0f_i)} $$ The right hand side is the localization of $J_i$ with respect to $f_0^{\deg(f_i)}/f_i^{\deg(f_0)}$. It follows that $$ f_0^{e_i}g/f_i^{(e_i + e)\deg(f_0)/\deg(f_i)} \in J_i $$ for some $e_i \gg 0$ sufficiently divisible. This proves that $f_0^{\max(e_i)}g$ is an element of $I$, because its restriction to each affine open $D_{+}(f_i)$ vanishes on the closed subscheme $Z \cap D_{+}(f_i)$. Hence $g \in J'$ and we conclude $J \subset J'$ as desired.
Observe that the local rings of $X = \mathbf{P}^n_R$ are UFDs because $X$ is covered by affine pieces isomorphic to $\mathbf{A}^n_R$ and $R[x_1, \ldots, x_n]$ is a UFD (Algebra, Lemma \ref{algebra-lemma-polynomial-ring-UFD}). Thus $Z$ is an effective Cartier divisor by Lemma \ref{lemma-codimension-1-is-effective-Cartier}. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals corresponding to $Z$. Choose an isomorphism $\mathcal{O}(m) \to \mathcal{I}$ for some $m \in \mathbf{Z}$, see Lemma \ref{lemma-Pic-projective-space-UFD}. Then the composition $$ \mathcal{O}_X(m) \to \mathcal{I} \to \mathcal{O}_X $$ is nonzero. We conclude that $m \leq 0$ and that the corresponding section of $\mathcal{O}_X(m)^{\otimes -1} = \mathcal{O}_X(-m)$ is given by some $F \in R[T_0, \ldots, T_n]$ of degree $-m$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-cohomology-projective-space-over-ring}. Thus on the $i$th standard open $U_i = D_+(T_i)$ the closed subscheme $Z \cap U_i$ is cut out by the ideal $$ (F(T_0/T_i, \ldots, T_n/T_i)) \subset R[T_0/T_i, \ldots, T_n/T_i] $$ Thus the homogeneous elements of the graded ideal $I(Z) = \Ker(R[T_0, \ldots, T_n] \to \bigoplus \Gamma(\mathcal{O}_Z(m)))$ is the set of homogeneous polynomials $G$ such that $$ G(T_0/T_i, \ldots, T_n/T_i) \in (F(T_0/T_i, \ldots, T_n/T_i)) $$ for $i = 0, \ldots, n$. Clearing denominators, we see there exist $e_i \geq 0$ such that $$ T_i^{e_i}G \in (F) $$ for $i = 0, \ldots, n$. As $R$ is a UFD, so is $R[T_0, \ldots, T_n]$. Then $F | T_0^{e_0}G$ and $F | T_1^{e_1}G$ implies $F | G$ as $T_0^{e_0}$ and $T_1^{e_1}$ have no factor in common. Thus $I(Z) = (F)$.
By Lemma \ref{lemma-closed-subscheme-proj} we know there exists a quasi-coherent graded sheaf of ideals $\mathcal{I} \subset \mathcal{A}$ such that $Z = \underline{\text{Proj}}(\mathcal{A}/\mathcal{I})$. Since $S$ is quasi-compact we can choose a finite affine open covering $S = U_1 \cup \ldots \cup U_n$. Say $U_i = \Spec(R_i)$. Let $\mathcal{A}|_{U_i}$ correspond to the graded $R_i$-algebra $A_i$ and $\mathcal{I}|_{U_i}$ to the graded ideal $I_i \subset A_i$. Note that $p^{-1}(U_i) = \text{Proj}(A_i)$ as schemes over $R_i$. Since $p$ is quasi-compact we can choose finitely many homogeneous elements $f_{i, j} \in A_{i, +}$ such that $p^{-1}(U_i) = D_{+}(f_{i, j})$. The condition on $Z \to X$ means that the ideal sheaf of $Z$ in $\mathcal{O}_X$ is of finite type, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-finite-presentation}. Hence we can find finitely many homogeneous elements $h_{i, j, k} \in I_i \cap A_{i, +}$ such that the ideal of $Z \cap D_{+}(f_{i, j})$ is generated by the elements $h_{i, j, k}/f_{i, j}^{e_{i, j, k}}$. Choose $d > 0$ to be a common multiple of all the integers $\deg(f_{i, j})$ and $\deg(h_{i, j, k})$. By Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} there exists a finite type quasi-coherent $\mathcal{F} \subset \mathcal{I}_d$ such that all the local sections $$ h_{i, j, k}f_{i, j}^{(d - \deg(h_{i, j, k}))/\deg(f_{i, j})} $$ are sections of $\mathcal{F}$. By construction $\mathcal{F}$ is a solution.
Let $\mathcal{I} \subset \mathcal{A}$ be the sheaf of quasi-coherent graded ideals constructed in Lemma \ref{lemma-closed-subscheme-proj}. Let $U_i$, $R_i$, $A_i$, $I_i$, $f_{i, j}$, $h_{i, j, k}$, and $d$ be as constructed in the proof of Lemma \ref{lemma-closed-subscheme-proj-finite}. Since $U \cap p(i(Z)) = \emptyset$ we see that $\mathcal{I}_d|_U = \mathcal{A}_d|_U$ (by our construction of $\mathcal{I}$ as a kernel). Since $U$ is quasi-compact we can choose a finite affine open covering $U = W_1 \cup \ldots \cup W_m$. Since $\mathcal{A}_d$ is of finite type we can find finitely many sections $g_{t, s} \in \mathcal{A}_d(W_t)$ which generate $\mathcal{A}_d|_{W_t} = \mathcal{I}_d|_{W_t}$ as an $\mathcal{O}_{W_t}$-module. To finish the proof, note that by Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} there exists a finite type $\mathcal{F} \subset \mathcal{I}_d$ such that all the local sections $$ h_{i, j, k}f_{i, j}^{(d - \deg(h_{i, j, k}))/\deg(f_{i, j})} \quad\text{and}\quad g_{t, s} $$ are sections of $\mathcal{F}$. By construction $\mathcal{F}$ is a solution.
Recall that $\pi : \mathbf{P}(\mathcal{E}) \to X$ is the relative proj of the symmetric algebra on $\mathcal{E}$, see Constructions, Definition \ref{constructions-definition-projective-bundle}. Hence the descriptions of sections $\sigma$ follows immediately from the description of the functor of points of $\mathbf{P}(\mathcal{E})$ in Constructions, Lemma \ref{constructions-lemma-apply-relative}. Since $\pi$ is separated, any section is a closed immersion (Constructions, Lemma \ref{constructions-lemma-relative-proj-separated} and Schemes, Lemma \ref{schemes-lemma-section-immersion}). Let $U \subset X$ be an affine open and $k \in \mathcal{E}(U)$ and $s \in \mathcal{E}(U)$ be local sections such that $k$ maps to zero in $\mathcal{L}$ and $s$ maps to a generator $\overline{s}$ of $\mathcal{L}$. Then $f = k/s$ is a section of $\mathcal{O}_{\mathbf{P}(\mathcal{E})}$ defined in an open neighbourhood $D_+(s)$ of $s(U)$ in $\pi^{-1}(U)$. Moreover, since $k$ maps to zero in $\mathcal{L}$ we see that $f$ is a section of the ideal sheaf of $s(U)$ in $\pi^{-1}(U)$. Thus we can take the image $\overline{f}$ of $f$ in $\mathcal{C}_{\sigma(X)/\mathbf{P}(\mathcal{E})}(U)$. We claim (1) that the image $\overline{f}$ depends only on the sections $k$ and $\overline{s}$ and not on the choice of $s$ and (2) that we get an isomorphism over $U$ in this manner (see below). However, once (1) and (2) are established, we see that the construction is compatible with base change by $U' \to U$ where $U'$ is affine, which proves that these local maps glue and are compatible with arbitrary base change. \medskip\noindent To prove (1) and (2) we make explicit what is going on. Namely, say $U = \Spec(A)$ and say $\mathcal{E} \to \mathcal{L}$ corresponds to the map of $A$-modules $M \to N$. Then $k \in K = \Ker(M \to N)$ and $s \in M$ maps to a generator $\overline{s}$ of $N$. Hence $M = K \oplus A s$. Thus $$ \text{Sym}(M) = \text{Sym}(K)[s] $$ Consider the identification $\text{Sym}(K) \to \text{Sym}(M)_{(s)}$ via the rule $g \mapsto g/s^n$ for $g \in \text{Sym}^n(K)$. This gives an isomorphism $D_+(s) = \Spec(\text{Sym}(K))$ such that $\sigma$ corresponds to the ring map $\text{Sym}(K) \to A$ mapping $K$ to zero. Via this isomorphism we see that the quasi-coherent module corresponding to $K$ is identified with $\mathcal{C}_{\sigma(U)/D_+(s)}$ proving (2). Finally, suppose that $s' = k' + s$ for some $k' \in K$. Then $$ k/s' = (k/s) (s/s') = (k/s) (s'/s)^{-1} = (k/s) (1 + k'/s)^{-1} $$ in an open neighbourhood of $\sigma(U)$ in $D_+(s)$. Thus we see that $s'/s$ restricts to $1$ on $\sigma(U)$ and we see that $k/s'$ maps to the same element of the conormal sheaf as does $k/s$ thereby proving (1).
The first statement is clear from the construction of the relative Proj via glueing, see Constructions, Section \ref{constructions-section-relative-proj-via-glueing}. For $a \in I$ denote $a^{(1)}$ the element $a$ seen as an element of degree $1$ in the Rees algebra $\bigoplus_{n \geq 0} I^n$. Since these elements generate the Rees algebra over $A$ we see that $\text{Proj}(\bigoplus_{d \geq 0} I^d)$ is covered by the affine opens $D_{+}(a^{(1)})$. The affine scheme $D_{+}(a^{(1)})$ is the spectrum of the affine blowup algebra $A' = A[\frac{I}{a}]$, see Algebra, Definition \ref{algebra-definition-blow-up}. This finishes the proof.
Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$. Denote $g : X_1 \to X_2$ the given morphism. Then the ideal sheaf $\mathcal{I}_1$ of $Z_1$ is the image of $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (by definition of the inverse image, see Schemes, Definition \ref{schemes-definition-inverse-image-closed-subscheme}). By Constructions, Lemma \ref{constructions-lemma-relative-proj-base-change} we see that $X_1 \times_{X_2} X_2'$ is the relative Proj of $\bigoplus_{n \geq 0} g^*\mathcal{I}_2^n$. Because $g$ is flat the map $g^*\mathcal{I}_2^n \to \mathcal{O}_{X_1}$ is injective with image $\mathcal{I}_1^n$. Thus we see that $X_1 \times_{X_2} X_2' = X_1'$.