Lemma 99.14.11 (Strong formal effectiveness for polarized schemes). Let (R_ n) be an inverse system of rings with surjective transition maps whose kernels are locally nilpotent. Set R = \mathop{\mathrm{lim}}\nolimits R_ n. Set S_ n = \mathop{\mathrm{Spec}}(R_ n) and S = \mathop{\mathrm{Spec}}(R). Consider a commutative diagram
\xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots }
of schemes with cartesian squares. Suppose given (\mathcal{L}_ n, \varphi _ n) where each \mathcal{L}_ n is an invertible sheaf on X_ n and \varphi _ n : i_ n^*\mathcal{L}_{n + 1} \to \mathcal{L}_ n is an isomorphism. If
X_ n \to S_ n is proper, flat, of finite presentation, and
\mathcal{L}_1 is ample on X_1
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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