then there exists a morphism of schemes $X \to S$ proper, flat, and of finite presentation and an ample invertible $\mathcal{O}_ X$-module $\mathcal{L}$ and isomorphisms $X_ n \cong X \times _ S S_ n$ and $\mathcal{L}_ n \cong \mathcal{L}|_{X_ n}$ compatible with the morphisms $i_ n$ and $\varphi _ n$.
Proof.
Choose $d_0$ for $X_1 \to S_1$ and $\mathcal{L}_1$ as in More on Morphisms, Lemma 37.10.6. For any $n \geq 1$ set
\[ A_ n = R_ n \oplus \bigoplus \nolimits _{d \geq d_0} H^0(X_ n, \mathcal{L}_ n^{\otimes d}) \]
By the lemma each $A_ n$ is a finitely presented graded $R_ n$-algebra whose homogeneous parts $(A_ n)_ d$ are finite projective $R_ n$-modules such that $X_ n = \text{Proj}(A_ n)$ and $\mathcal{L}_ n = \mathcal{O}_{\text{Proj}(A_ n)}(1)$. The lemma also guarantees that the maps
\[ A_1 \leftarrow A_2 \leftarrow A_3 \leftarrow \ldots \]
induce isomorphisms $A_ n = A_ m \otimes _{R_ m} R_ n$ for $n \leq m$. We set
\[ B = \bigoplus \nolimits _{d \geq 0} B_ d \quad \text{with}\quad B_ d = \mathop{\mathrm{lim}}\nolimits _ n (A_ n)_ d \]
By More on Algebra, Lemma 15.13.4 we see that $B_ d$ is a finite projective $R$-module and that $B \otimes _ R R_ n = A_ n$. Thus the scheme
\[ X = \text{Proj}(B) \quad \text{and}\quad \mathcal{L} = \mathcal{O}_ X(1) \]
is flat over $S$ and $\mathcal{L}$ is a quasi-coherent $\mathcal{O}_ X$-module flat over $S$, see Divisors, Lemma 31.30.6. Because formation of Proj commutes with base change (Constructions, Lemma 27.11.6) we obtain canonical isomorphisms
\[ X \times _ S S_ n = X_ n \quad \text{and}\quad \mathcal{L}|_{X_ n} \cong \mathcal{L}_ n \]
compatible with the transition maps of the system. Thus we may think of $X_1 \subset X$ as a closed subscheme. Below we will show that $B$ is of finite presentation over $R$. By Divisors, Lemmas 31.30.4 and 31.30.7 this implies that $X \to S$ is of finite presentation and proper and that $\mathcal{L} = \mathcal{O}_ X(1)$ is of finite presentation as an $\mathcal{O}_ X$-module. Since the restriction of $\mathcal{L}$ to the base change $X_1 \to S_1$ is invertible, we see from More on Morphisms, Lemma 37.16.8 that $\mathcal{L}$ is invertible on an open neighbourhood of $X_1$ in $X$. Since $X \to S$ is closed and since $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical (More on Algebra, Lemma 15.11.3) we see that any open neighbourhood of $X_1$ in $X$ is equal to $X$. Thus $\mathcal{L}$ is invertible. Finally, the set of points in $S$ where $\mathcal{L}$ is ample on the fibre is open in $S$ (More on Morphisms, Lemma 37.50.3) and contains $S_1$ hence equals $S$. Thus $X \to S$ and $\mathcal{L}$ have all the properties required of them in the statement of the lemma.
We prove the claim above. Choose a presentation $A_1 = R_1[X_1, \ldots , X_ s]/(F_1, \ldots , F_ t)$ where $X_ i$ are variables having degrees $d_ i$ and $F_ j$ are homogeneous polynomials in $X_ i$ of degree $e_ j$. Then we can choose a map
\[ \Psi : R[X_1, \ldots , X_ s] \longrightarrow B \]
lifting the map $R_1[X_1, \ldots , X_ s] \to A_1$. Since each $B_ d$ is finite projective over $R$ we conclude from Nakayama's lemma (Algebra, Lemma 10.20.1 using again that $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical of $R$) that $\Psi $ is surjective. Since $- \otimes _ R R_1$ is right exact we can find $G_1, \ldots , G_ t \in \mathop{\mathrm{Ker}}(\Psi )$ mapping to $F_1, \ldots , F_ t$ in $R_1[X_1, \ldots , X_ s]$. Observe that $\mathop{\mathrm{Ker}}(\Psi )_ d$ is a finite projective $R$-module for all $d \geq 0$ as the kernel of the surjection $R[X_1, \ldots , X_ s]_ d \to B_ d$ of finite projective $R$-modules. We conclude from Nakayama's lemma once more that $\mathop{\mathrm{Ker}}(\Psi )$ is generated by $G_1, \ldots , G_ t$.
$\square$
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