hcju/stacksqa huggingface dataset:
If $x \in D_{+}(fg)$ with $\deg(f) = \deg(g) + d$ then on $D_{+}(fg)$ the sheaves $\mathcal{O}_X(dn)$ are generated by the element $(f/g)^n = f^{2n}/(fg)^n$. This implies $x$ is in the open subset $W$ defined in (1) by arguing as in the proof of Lemma \ref{lemma-when-invertible}. \medskip\noindent Conversely, suppose that $\mathcal{O}_X(d)$ is free of rank 1 in an open neighbourhood $V$ of $x \in X$ and all the multiplication maps $\mathcal{O}_X(nd)|_V \otimes_{\mathcal{O}_V} \mathcal{O}_X(md)|_V \to \mathcal{O}_X(nd + md)|_V$ are isomorphisms. We may choose $h \in S_{+}$ homogeneous such that $D_{+}(h) \subset V$. By the definition of the twists of the structure sheaf we conclude there exists an element $s$ of $(S_h)_d$ such that $s^n$ is a basis of $(S_h)_{nd}$ as a module over $S_{(h)}$ for all $n \in \mathbf{Z}$. We may write $s = f/h^m$ for some $m \geq 1$ and $f \in S_{d + m \deg(h)}$. Set $g = h^m$ so $s = f/g$. Note that $x \in D(g)$ by construction. Note that $g^d \in (S_h)_{-d\deg(g)}$. By assumption we can write this as a multiple of $s^{\deg(g)} = f^{\deg(g)}/g^{\deg(g)}$, say $g^d = a/g^e \cdot f^{\deg(g)}/g^{\deg(g)}$. Then we conclude that $g^{d + e + \deg(g)} = a f^{\deg(g)}$ and hence also $x \in D_{+}(f)$. So $x$ is an element of the set defined in (2). \medskip\noindent The existence of the generating section $s = f/g$ over the affine open $D_{+}(fg)$ whose powers freely generate the sheaves of modules $\mathcal{O}_X(nd)$ easily implies that the multiplication maps $\widetilde M(nd)|_W = \widetilde M|_W \otimes_{\mathcal{O}_W} \mathcal{O}_X(nd)|_W \to \widetilde{M(nd)}|_W$ (see \ref{equation-multiply-more-generally}) are isomorphisms. Compare with the proof of Lemma \ref{lemma-when-invertible}.
Let $D_{+}(ab) \subset W$ be a standard affine open with $a, b \in S$ homogeneous and $\deg(a) = \deg(b) + d$. Note that $D_{+}(ab) \cap D_{+}(f) = D_{+}(abf)$. On the other hand the restriction of $s$ to $D_{+}(ab)$ corresponds to the element $f/1 = b^nf/a^n (a/b)^n \in (S_{ab})_{nd}$. We have seen in the proof of Lemma \ref{lemma-where-invertible} that $(a/b)^n$ is a generator for $\mathcal{O}_W(nd)$ over $D_{+}(ab)$. We conclude that $W_s \cap D_{+}(ab)$ is the principal open associated to $b^nf/a^n \in \mathcal{O}_X(D_{+}(ab))$. Thus the result of the lemma is clear.
Since $Y$ is quasi-compact there exist finitely many homogeneous $f_i \in S_{+}$, $i = 1, \ldots, n$ such that the standard opens $D_{+}(f_i)$ give an open covering of $Y$. Let $d_i = \deg(f_i)$ and set $d = d_1 \ldots d_n$. Note that $D_{+}(f_i) = D_{+}(f_i^{d/d_i})$ and hence we see immediately that $Y \subset W_d$, by characterization (2) in Lemma \ref{lemma-where-invertible} or by (1) using Lemma \ref{lemma-when-invertible}. Note that (1) implies (2), (3) and (4) by Lemma \ref{lemma-where-invertible}. (Note that (3) is a special case of (4).) Assertion (5) follows from Lemma \ref{lemma-principal-open}. Assertions (6) and (7) follow because the open subsets $D_{+}(f)$ form a basis for the topology of $X$ and are affine.
Let $f \in S$ be homogeneous of degree $d > 0$. Recall that $\widetilde{M}|_{D_{+}(f)}$ corresponds to the $S_{(f)}$-module $M_{(f)}$ by Lemma \ref{lemma-proj-sheaves}. Thus we can define a canonical map $$ M_{(f)} \longrightarrow \Gamma(D_+(f), \mathcal{F}),\quad m/f^n \longmapsto m|_{D_+(f)} \otimes f|_{D_+(f)}^{-n} $$ which makes sense because $f|_{D_+(f)}$ is a trivializing section of the invertible sheaf $\mathcal{O}_X(d)|_{D_+(f)}$, see Lemma \ref{lemma-when-invertible} and its proof. Since $\widetilde{M}$ is quasi-coherent, this leads to a canonical map $$ \widetilde{M}|_{D_+(f)} \longrightarrow \mathcal{F}|_{D_+(f)} $$ via Schemes, Lemma \ref{schemes-lemma-compare-constructions}. We obtain a global map if we prove that the displayed maps glue on overlaps. Proof of this is omitted. We also omit the proof of the final statement.
We have (1) by Lemma \ref{lemma-surjective-graded-rings-map-proj}. Let $f \in A_{+}$ be homogeneous. The assumption on $\psi$ implies that $A_f \to B_f$ is an isomorphism (details omitted). Thus it is clear that $r_\psi$ and $\theta$ restrict to isomorphisms over $D_{+}(f)$. The lemma follows.
By Lemmas \ref{lemma-eventual-iso-graded-rings-map-proj} and \ref{lemma-morphism-proj-transitive} we may replace $B$ by the image of $A \to B$ without changing $X$ or the sheaves $\mathcal{O}_X(n)$. Thus we may assume that $A \to B$ is surjective. By Lemma \ref{lemma-surjective-graded-rings-map-proj} we get (1) and (2) and surjectivity in (3). By Lemma \ref{lemma-apply-modules} we see that both $\mathcal{O}_X(n)$ and $\mathcal{O}_Y(n)$ are invertible. Hence $\theta$ is an isomorphism.
The injective ring map $S' \to S$ (which is not a homomorphism of graded rings due to our conventions), induces a map $j : \Spec(S) \to \Spec(S')$. Given a graded prime ideal $\mathfrak p \subset S$ we see that $\mathfrak p' = j(\mathfrak p) = S' \cap \mathfrak p$ is a graded prime ideal of $S'$. Moreover, if $f \in S_+$ is homogeneous and $f \not \in \mathfrak p$, then $f^d \in S'_+$ and $f^d \not \in \mathfrak p'$. Conversely, if $\mathfrak p' \subset S'$ is a graded prime ideal not containing some homogeneous element $f \in S'_+$, then $\mathfrak p = \{g \in S \mid g^d \in \mathfrak p'\}$ is a graded prime ideal of $S$ not containing $f$ whose image under $j$ is $\mathfrak p'$. To see that $\mathfrak p$ is an ideal, note that if $g, h \in \mathfrak p$, then $(g + h)^{2d} \in \mathfrak p'$ by the binomial formula and hence $g + h \in \mathfrak p'$ as $\mathfrak p'$ is a prime. In this way we see that $j$ induces a homeomorphism $i : X \to X'$. Moreover, given $f \in S_+$ homogeneous, then we have $S_{(f)} \cong S'_{(f^d)}$. Since these isomorphisms are compatible with the restrictions mappings of Lemma \ref{lemma-standard-open}, we see that there exists an isomorphism $i^\sharp : i^{-1}\mathcal{O}_{X'} \to \mathcal{O}_X$ of structure sheaves on $X$ and $X'$, hence $i$ is an isomorphism of schemes. \medskip\noindent Let $M$ be a graded $S$-module. Given $f \in S_+$ homogeneous, we have $M_{(f)} \cong M'_{(f^d)}$, hence in exactly the same manner as above we obtain the isomorphism in (1). The isomorphisms in (2) are a special case of (1) for $M = S(nd)$ which gives $M' = S'(n)$. Let $M$ and $N$ be graded $S$-modules. Then we have $$ M' \otimes_{S'} N' = (M \otimes_S N)^{(d)} = (M \otimes_S N)' $$ as can be verified directly from the definitions. Having said this the compatibility with the multiplication maps of Lemma \ref{lemma-widetilde-tensor} is the commutativity of the diagram $$ \xymatrix{ \widetilde M \otimes_{\mathcal{O}_X} \widetilde N \ar[d]_{(1) \otimes (1)} \ar[r] & \widetilde{M \otimes_S N} \ar[d]^{(1)} \\ i^*(\widetilde{M'} \otimes_{\mathcal{O}_{X'}} \widetilde{N'}) \ar[r] & i^*(\widetilde{M' \otimes_{S'} N'}) } $$ This can be seen by looking at the construction of the maps over the open $D_+(f) = D_+(f^d)$ where the top horizontal arrow is given by the map $M_{(f)} \times N_{(f)} \to (M \otimes_S N)_{(f)}$ and the lower horizontal arrow by the map $M'_{(f^d)} \times N'_{(f^d)} \to (M' \otimes_{S'} N')_{(f^d)}$. Since these maps agree via the identifications $M_{(f)} = M'_{(f^d)}$, etc, we get the desired compatibility. We omit the proof of the other compatibilities.
Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Also (3) implies (4) by the uniqueness statement in Lemma \ref{lemma-converse-construction}. Thus we have to prove that (4) implies (1). Assume (4), in other words $\varphi = \varphi'$. Note that this implies that we may write $\mathcal{L} = \varphi^*\mathcal{O}_X(d)$ and $\mathcal{L}' = \varphi^*\mathcal{O}_X(d')$. Moreover, via these identifications we have that the graded ring maps $\psi$ and $\psi'$ correspond to the restriction of the canonical graded ring map $$ S \longrightarrow \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{O}_X(n)) $$ to $S^{(d)}$ and $S^{(d')}$ composed with pullback by $\varphi$ (by Lemma \ref{lemma-converse-construction} again). Hence taking $\beta$ to be the isomorphism $$ (\varphi^*\mathcal{O}_X(d))^{\otimes a} = \varphi^*\mathcal{O}_X(n) = (\varphi^*\mathcal{O}_X(d'))^{\otimes a'} $$ works.
This is a special case of Lemma \ref{lemma-apply} above. Namely, for any graded ring $A$ we have \begin{eqnarray*} \Mor_{graded rings}(\mathbf{Z}[T_0, \ldots, T_n], A) & = & A_1 \times \ldots \times A_1 \\ \psi & \mapsto & (\psi(T_0), \ldots, \psi(T_n)) \end{eqnarray*} and the degree $1$ part of $\Gamma_*(Y, \mathcal{L})$ is just $\Gamma(Y, \mathcal{L})$.
All these properties are stable under base change (this is clear for the last two and for the other two see Schemes, Lemmas \ref{schemes-lemma-separated-permanence} and \ref{schemes-lemma-quasi-compact-preserved-base-change}). Hence it suffices to prove them for the morphism $\mathbf{P}^n_{\mathbf{Z}} \to \Spec(\mathbf{Z})$. Separatedness is Lemma \ref{lemma-proj-separated}. Quasi-compactness follows from Lemma \ref{lemma-standard-covering-projective-space}. Existence and uniqueness of the valuative criterion follow from Lemma \ref{lemma-proj-valuative-criterion}. Universally closed follows from the above and Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}.
It suffices to prove this when $S = \Spec(\mathbf{Z})$. Hence we will drop the index $S$ and work in the absolute setting. Write $\mathbf{P}^n = \text{Proj}(\mathbf{Z}[X_0, \ldots, X_n])$, $\mathbf{P}^m = \text{Proj}(\mathbf{Z}[Y_0, \ldots, Y_m])$, and $\mathbf{P}^{nm + n + m} = \text{Proj}(\mathbf{Z}[Z_0, \ldots, Z_{nm + n + m}])$. In order to map into $\mathbf{P}^{nm + n + m}$ we have to write down an invertible sheaf $\mathcal{L}$ on the left hand side and $(n + 1)(m + 1)$ sections $s_i$ which generate it. See Lemma \ref{lemma-projective-space}. The invertible sheaf we take is $$ \mathcal{L} = \text{pr}_1^*\mathcal{O}_{\mathbf{P}^n}(1) \otimes \text{pr}_2^*\mathcal{O}_{\mathbf{P}^m}(1) $$ The sections we take are $$ s_0 = X_0Y_0, \ s_1 = X_1Y_0, \ldots, \ s_n = X_nY_0, \ s_{n + 1} = X_0Y_1, \ldots, \ s_{nm + n + m} = X_nY_m. $$ These generate $\mathcal{L}$ since the sections $X_i$ generate $\mathcal{O}_{\mathbf{P}^n}(1)$ and the sections $Y_j$ generate $\mathcal{O}_{\mathbf{P}^m}(1)$. The induced morphism $\varphi$ has the property that $$ \varphi^{-1}(D_{+}(Z_{i + (n + 1)j})) = D_{+}(X_i) \times D_{+}(Y_j). $$ Hence it is an affine morphism. The corresponding ring map in case $(i, j) = (0, 0)$ is the map $$ \mathbf{Z}[Z_1/Z_0, \ldots, Z_{nm + n + m}/Z_0] \longrightarrow \mathbf{Z}[X_1/X_0, \ldots, X_n/X_0, Y_1/Y_0, \ldots, Y_n/Y_0] $$ which maps $Z_i/Z_0$ to the element $X_i/X_0$ for $i \leq n$ and the element $Z_{(n + 1)j}/Z_0$ to the element $Y_j/Y_0$. Hence it is surjective. A similar argument works for the other affine open subsets. Hence the morphism $\varphi$ is a closed immersion (see Schemes, Lemma \ref{schemes-lemma-closed-local-target} and Example \ref{schemes-example-closed-immersion-affines}.)
It is clear that $I$ is a graded ideal. Set $Z' = \text{Proj}(R[T_0, \ldots, T_n]/I)$. By Lemma \ref{lemma-surjective-graded-rings-generated-degree-1-map-proj} we see that $Z'$ is a closed subscheme of $\mathbf{P}^n_R$. To see the equality $Z = Z'$ it suffices to check on an standard affine open $D_{+}(T_i)$. By renumbering the homogeneous coordinates we may assume $i = 0$. Say $Z \cap D_{+}(T_0)$, resp.\ $Z' \cap D_{+}(T_0)$ is cut out by the ideal $J$, resp.\ $J'$ of $R[T_1/T_0, \ldots, T_n/T_0]$. Then $J'$ is the ideal generated by the elements $F/T_0^{\deg(F)}$ where $F \in I$ is homogeneous. Suppose the degree of $F \in I$ is $d$. Since $F$ vanishes as a section of $\mathcal{O}_{\mathbf{P}^n_R}(d)$ restricted to $Z$ we see that $F/T_0^d$ is an element of $J$. Thus $J' \subset J$. \medskip\noindent Conversely, suppose that $f \in J$. If $f$ has total degree $d$ in $T_1/T_0, \ldots, T_n/T_0$, then we can write $f = F/T_0^d$ for some $F \in R[T_0, \ldots, T_n]_d$. Pick $i \in \{1, \ldots, n\}$. Then $Z \cap D_{+}(T_i)$ is cut out by some ideal $J_i \subset R[T_0/T_i, \ldots, T_n/T_i]$. Moreover, $$ J \cdot R\left[ \frac{T_1}{T_0}, \ldots, \frac{T_n}{T_0}, \frac{T_0}{T_i}, \ldots, \frac{T_n}{T_i} \right] = J_i \cdot R\left[ \frac{T_1}{T_0}, \ldots, \frac{T_n}{T_0}, \frac{T_0}{T_i}, \ldots, \frac{T_n}{T_i} \right] $$ The left hand side is the localization of $J$ with respect to the element $T_i/T_0$ and the right hand side is the localization of $J_i$ with respect to the element $T_0/T_i$. It follows that $T_0^{d_i}F/T_i^{d + d_i}$ is an element of $J_i$ for some $d_i$ sufficiently large. This proves that $T_0^{\max(d_i)}F$ is an element of $I$, because its restriction to each standard affine open $D_{+}(T_i)$ vanishes on the closed subscheme $Z \cap D_{+}(T_i)$. Hence $f \in J'$ and we conclude $J \subset J'$ as desired.
Suppose that we have two triples $(U, r : U \to X, \theta)$ and $(U', r' : U' \to X, \theta')$ satisfying (1) and (2). Property (1) implies that $U = U' = U(\psi)$ and that $r = r'$ as maps of underlying topological spaces, since the opens $D_{+}(f)$ form a basis for the topology on $X$, and since $X$ is a sober topological space (see Algebra, Section \ref{algebra-section-proj} and Schemes, Lemma \ref{schemes-lemma-scheme-sober}). Let $f \in A_{+}$ be homogeneous. Note that $\Gamma(D_{+}(f), \bigoplus_{n \in \mathbf{Z}} \mathcal{O}_X(n)) = A_f$ as a $\mathbf{Z}$-graded algebra. Consider the two $\mathbf{Z}$-graded ring maps $$ \theta, \theta' : A_f \longrightarrow \Gamma(T_{\psi(f)}, \bigoplus \mathcal{L}^{\otimes n}). $$ We know that multiplication by $f$ (resp.\ $\psi(f)$) is an isomorphism on the left (resp.\ right) hand side. We also know that $\theta(x/1) = \theta'(x/1) = \psi(x)|_{T_{\psi(f)}}$ by (2) for all $x \in A$. Hence we deduce easily that $\theta = \theta'$ as desired. Considering the degree $0$ parts we deduce that $r^\sharp = (r')^\sharp$, i.e., that $r = r'$ as morphisms of schemes. This proves the uniqueness. \medskip\noindent Now we come to existence. By the uniqueness just proved, it is enough to construct the pair $(r, \theta)$ locally on $T$. Hence we may assume that $T = \Spec(R)$ is affine, that $\mathcal{L} = \mathcal{O}_T$ and that for some $f \in A_{+}$ homogeneous we have $\psi(f)$ generates $\mathcal{O}_T = \mathcal{O}_T^{\otimes \deg(f)}$. In other words, $\psi(f) = u \in R^*$ is a unit. In this case the map $\psi$ is a graded ring map $$ A \longrightarrow R[x] = \Gamma_*(T, \mathcal{O}_T) $$ which maps $f$ to $ux^{\deg(f)}$. Clearly this extends (uniquely) to a $\mathbf{Z}$-graded ring map $\theta : A_f \to R[x, x^{-1}]$ by mapping $1/f$ to $u^{-1}x^{-\deg(f)}$. This map in degree zero gives the ring map $A_{(f)} \to R$ which gives the morphism $r : T = \Spec(R) \to \Spec(A_{(f)}) = D_{+}(f) \subset X$. Hence we have constructed $(r, \theta)$ in this special case. \medskip\noindent Let us show the last statement of the lemma. According to Lemma \ref{lemma-converse-construction} the morphism constructed there is the unique one such that the displayed diagram in its statement commutes. The commutativity of the diagram in the lemma implies the commutativity when restricted to $V$ and $A^{(d)}$. Whence the result.
Let $R = \mathcal{O}_S(U)$ and $R' = \mathcal{O}_S(U')$. Note that the map $R \otimes_{R'} A' \to A$ is an isomorphism as $\mathcal{A}$ is quasi-coherent (see Schemes, Lemma \ref{schemes-lemma-widetilde-pullback} for example). Hence the lemma follows from Lemma \ref{lemma-base-change-map-proj}.
We are going to use Lemma \ref{lemma-relative-glueing-sheaves} to glue the sheaves of $\mathbf{Z}$-graded algebras $\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n)$ for $A = \mathcal{A}(U)$, $U \subset S$ affine open over the scheme $\underline{\text{Proj}}_S(\mathcal{A})$. We have constructed the data necessary for this in Lemma \ref{lemma-proj-inclusion} and we have checked condition (d) of Lemma \ref{lemma-relative-glueing-sheaves} in Lemma \ref{lemma-transitive-proj}. Hence we get the sheaf of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}$-algebras $\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)$ together with the isomorphisms $\theta_U$ for all $U \subset S$ affine open and all $n \in \mathbf{Z}$. For every affine open $U \subset S$ with $A = \mathcal{A}(U)$ we have a map $A \to \Gamma(\text{Proj}(A), \bigoplus_{n \geq 0} \mathcal{O}_{\text{Proj}(A)}(n))$. Hence the map $\psi$ exists by functoriality of relative glueing, see Remark \ref{remark-relative-glueing-functorial}. The diagram of the lemma commutes by construction. This characterizes the sheaf of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}$-algebras $\bigoplus \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)$ because the proof of Lemma \ref{lemma-morphism-proj} shows that having these diagrams commute uniquely determines the maps $\theta_U$. Some details omitted.
Write $S = \Spec(R)$ and $A = \Gamma(S, \mathcal{A})$. Then $A$ is a graded $R$-algebra and $\mathcal{A} = \widetilde A$. To prove the lemma we have to identify the functor $F_d$ with the functor $F_d^{triples}$ of triples defined in Section \ref{section-morphisms-proj}. \medskip\noindent Let $(d, f : T \to S, \mathcal{L}, \psi)$ be a quadruple. We may think of $\psi$ as a $\mathcal{O}_S$-module map $\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n}$. Since $\mathcal{A}^{(d)}$ is quasi-coherent this is the same thing as an $R$-linear homomorphism of graded rings $A^{(d)} \to \Gamma(S, \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n})$. Clearly, $\Gamma(S, \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n}) = \Gamma_*(T, \mathcal{L})$. Thus we may associate to the quadruple the triple $(d, \mathcal{L}, \psi)$. \medskip\noindent Conversely, let $(d, \mathcal{L}, \psi)$ be a triple. The composition $R \to A_0 \to \Gamma(T, \mathcal{O}_T)$ determines a morphism $f : T \to S = \Spec(R)$, see Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. With this choice of $f$ the map $A^{(d)} \to \Gamma(S, \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n})$ is $R$-linear, and hence corresponds to a $\psi$ which we can use for a quadruple $(d, f : T \to S, \mathcal{L}, \psi)$. We omit the verification that this establishes an isomorphism of functors $F_d = F_d^{triples}$.
We are going to use Schemes, Lemma \ref{schemes-lemma-glue-functors}. \medskip\noindent First we check that $F_d$ satisfies the sheaf property for the Zariski topology. Namely, suppose that $T$ is a scheme, that $T = \bigcup_{i \in I} U_i$ is an open covering, and that $(d, f_i, \mathcal{L}_i, \psi_i) \in F_d(U_i)$ such that $(d, f_i, \mathcal{L}_i, \psi_i)|_{U_i \cap U_j}$ and $(d, f_j, \mathcal{L}_j, \psi_j)|_{U_i \cap U_j}$ are strictly equivalent. This implies that the morphisms $f_i : U_i \to S$ glue to a morphism of schemes $f : T \to S$ such that $f|_{I_i} = f_i$, see Schemes, Section \ref{schemes-section-glueing-schemes}. Thus $f_i^*\mathcal{A}^{(d)} = f^*\mathcal{A}^{(d)}|_{U_i}$. It also implies there exist isomorphisms $\beta_{ij} : \mathcal{L}_i|_{U_i \cap U_j} \to \mathcal{L}_j|_{U_i \cap U_j}$ such that $\beta_{ij} \circ \psi_i = \psi_j$ on $U_i \cap U_j$. Note that the isomorphisms $\beta_{ij}$ are uniquely determined by this requirement because the maps $f_i^*\mathcal{A}_d \to \mathcal{L}_i$ are surjective. In particular we see that $\beta_{jk} \circ \beta_{ij} = \beta_{ik}$ on $U_i \cap U_j \cap U_k$. Hence by Sheaves, Section \ref{sheaves-section-glueing-sheaves} the invertible sheaves $\mathcal{L}_i$ glue to an invertible $\mathcal{O}_T$-module $\mathcal{L}$ and the morphisms $\psi_i$ glue to morphism of $\mathcal{O}_T$-algebras $\psi : f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}$. This proves that $F_d$ satisfies the sheaf condition with respect to the Zariski topology. \medskip\noindent Let $S = \bigcup_{i \in I} U_i$ be an affine open covering. Let $F_{d, i} \subset F_d$ be the subfunctor consisting of those pairs $(f : T \to S, \varphi)$ such that $f(T) \subset U_i$. \medskip\noindent We have to show each $F_{d, i}$ is representable. This is the case because $F_{d, i}$ is identified with the functor associated to $U_i$ equipped with the quasi-coherent graded $\mathcal{O}_{U_i}$-algebra $\mathcal{A}|_{U_i}$ by Lemma \ref{lemma-proj-base-change}. Thus the result follows from Lemma \ref{lemma-relative-proj-affine}. \medskip\noindent Next we show that $F_{d, i} \subset F_d$ is representable by open immersions. Let $(f : T \to S, \varphi) \in F_d(T)$. Consider $V_i = f^{-1}(U_i)$. It follows from the definition of $F_{d, i}$ that given $a : T' \to T$ we gave $a^*(f, \varphi) \in F_{d, i}(T')$ if and only if $a(T') \subset V_i$. This is what we were required to show. \medskip\noindent Finally, we have to show that the collection $(F_{d, i})_{i \in I}$ covers $F_d$. Let $(f : T \to S, \varphi) \in F_d(T)$. Consider $V_i = f^{-1}(U_i)$. Since $S = \bigcup_{i \in I} U_i$ is an open covering of $S$ we see that $T = \bigcup_{i \in I} V_i$ is an open covering of $T$. Moreover $(f, \varphi)|_{V_i} \in F_{d, i}(V_i)$. This finishes the proof of the lemma.
Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Assume (3) for some integer $m = ad = a'd'$. Let $m_0 = \text{lcm}(d, d')$ and write it as $m_0 = a_0d = a'_0d'$. We are given an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property described in (3). We want to find an isomorphism $\beta_0 : \mathcal{L}^{\otimes a_0} \to (\mathcal{L}')^{\otimes a'_0}$ having that property as well. Since by assumption the maps $\psi : f^*\mathcal{A}_d \to \mathcal{L}$ and $\psi' : (f')^*\mathcal{A}_{d'} \to \mathcal{L}'$ are surjective the same is true for the maps $\psi : f^*\mathcal{A}_{m_0} \to \mathcal{L}^{\otimes a_0}$ and $\psi' : (f')^*\mathcal{A}_{m_0} \to (\mathcal{L}')^{\otimes a_0}$. Hence if $\beta_0$ exists it is uniquely determined by the condition that $\beta_0 \circ \psi = \psi'$. This means that we may work locally on $T$. Hence we may assume that $f = f' : T \to S$ maps into an affine open, in other words we may assume that $S$ is affine. In this case the result follows from the corresponding result for triples (see Lemma \ref{lemma-equivalent}) and the fact that triples and quadruples correspond in the affine base case (see proof of Lemma \ref{lemma-relative-proj-affine}).
Let $U_d \to S$ be the scheme representing the functor $F_d$ defined above. Let $\mathcal{L}_d$, $\psi^d : \pi_d^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} \mathcal{L}_d^{\otimes n}$ be the universal object. If $d | d'$, then we may consider the quadruple $(d', \pi_d, \mathcal{L}_d^{\otimes d'/d}, \psi^d|_{\mathcal{A}^{(d')}})$ which determines a canonical morphism $U_d \to U_{d'}$ over $S$. By construction this morphism corresponds to the transformation of functors $F_d \to F_{d'}$ defined above. \medskip\noindent For every affine open $\Spec(R) = V \subset S$ setting $A = \Gamma(V, \mathcal{A})$ we have a canonical identification of the base change $U_{d, V}$ with the corresponding open subscheme of $\text{Proj}(A)$, see Lemma \ref{lemma-relative-proj-affine}. Moreover, the morphisms $U_{d, V} \to U_{d', V}$ constructed above correspond to the inclusions of opens in $\text{Proj}(A)$. Thus we conclude that $U_d \to U_{d'}$ is an open immersion. \medskip\noindent This allows us to construct $X$ by glueing the schemes $U_d$ along the open immersions $U_d \to U_{d'}$. Technically, it is convenient to choose a sequence $d_1 | d_2 | d_3 | \ldots$ such that every positive integer divides one of the $d_i$ and to simply take $X = \bigcup U_{d_i}$ using the open immersions above. It is then a simple matter to prove that $X$ represents the functor $F$.