Proof.
Since the squares in the diagram are cartesian and since the morphisms S_ n \to S_{n + 1} are closed immersions, we see that the morphisms i_ n are closed immersions too. In particular we may think of X_ m as a closed subscheme of X_ n for m < n. In fact X_ m is the closed subscheme cut out by the quasi-coherent sheaf of ideals I^ m\mathcal{O}_{X_ n}. Moreover, the underlying topological spaces of the schemes X_1, X_2, X_3, \ldots are all identified, hence we may (and do) think of sheaves \mathcal{O}_{X_ n} as living on the same underlying topological space; similarly for coherent \mathcal{O}_{X_ n}-modules. Set
\mathcal{F}_ n = \mathop{\mathrm{Ker}}(\mathcal{O}_{X_{n + 1}} \to \mathcal{O}_{X_ n})
so that we obtain short exact sequences
0 \to \mathcal{F}_ n \to \mathcal{O}_{X_{n + 1}} \to \mathcal{O}_{X_ n} \to 0
By the above we have \mathcal{F}_ n = I^ n\mathcal{O}_{X_{n + 1}}. It follows \mathcal{F}_ n is a coherent sheaf on X_{n + 1} annihilated by I, hence we may (and do) think of it as a coherent module \mathcal{O}_{X_1}-module. Observe that for m > n the sheaf
I^ n\mathcal{O}_{X_ m}/I^{n + 1}\mathcal{O}_{X_ m}
maps isomorphically to \mathcal{F}_ n under the map \mathcal{O}_{X_ m} \to \mathcal{O}_{X_{n + 1}}. Hence given n_1, n_2 \geq 0 we can pick an m > n_1 + n_2 and consider the multiplication map
I^{n_1}\mathcal{O}_{X_ m} \times I^{n_2}\mathcal{O}_{X_ m} \longrightarrow I^{n_1 + n_2}\mathcal{O}_{X_ m} \to \mathcal{F}_{n_1 + n_2}
This induces an \mathcal{O}_{X_1}-bilinear map
\mathcal{F}_{n_1} \times \mathcal{F}_{n_2} \longrightarrow \mathcal{F}_{n_1 + n_2}
which in turn defines the structure of a graded \mathcal{O}_{X_1}-algebra on \mathcal{F} = \bigoplus _{n \geq 0} \mathcal{F}_ n.
Set B = \bigoplus I^ n/I^{n + 1}; this is a finitely generated graded A/I-algebra. Set \mathcal{B} = (X_1 \to S_1)^*\widetilde{B}. The discussion above provides us with a canonical surjection
\mathcal{B} \longrightarrow \mathcal{F}
of graded \mathcal{O}_{X_1}-algebras. In particular we see that \mathcal{F} is a finite type quasi-coherent graded \mathcal{B}-module. By Lemma 30.19.3 we can find an integer d_0 such that H^1(X_1, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0 for all d \geq d_0. Pick a d \geq d_0 such that there exist sections s_{0, 1}, \ldots , s_{N, 1} \in \Gamma (X_1, \mathcal{L}_1^{\otimes d}) which induce an immersion
\psi _1 : X_1 \to \mathbf{P}^ N_{S_1}
over S_1, see Morphisms, Lemma 29.39.4. As X_1 is proper over S_1 we see that \psi _1 is a closed immersion, see Morphisms, Lemma 29.41.7 and Schemes, Lemma 26.10.4. We are going to “lift” \psi _1 to a compatible system of closed immersions of X_ n into \mathbf{P}^ N.
Upon tensoring the short exact sequences of the first paragraph of the proof by \mathcal{L}_{n + 1}^{\otimes d} we obtain short exact sequences
0 \to \mathcal{F}_ n \otimes \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to 0
Using the isomorphisms \varphi _ n we obtain isomorphisms \mathcal{L}_{n + 1} \otimes \mathcal{O}_{X_ l} = \mathcal{L}_ l for l \leq n. Whence the sequence above becomes
0 \to \mathcal{F}_ n \otimes \mathcal{L}_1^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_ n^{\otimes d} \to 0
The vanishing of H^1(X, \mathcal{F}_ n \otimes \mathcal{L}_1^{\otimes d}) implies we can inductively lift s_{0, 1}, \ldots , s_{N, 1} \in \Gamma (X_1, \mathcal{L}_1^{\otimes d}) to sections s_{0, n}, \ldots , s_{N, n} \in \Gamma (X_ n, \mathcal{L}_ n^{\otimes d}). Thus we obtain a commutative diagram
\xymatrix{ X_1 \ar[r]_{i_1} \ar[d]_{\psi _1} & X_2 \ar[r]_{i_2} \ar[d]_{\psi _2} & X_3 \ar[r] \ar[d]_{\psi _3} & \ldots \\ \mathbf{P}^ N_{S_1} \ar[r] & \mathbf{P}^ N_{S_2} \ar[r] & \mathbf{P}^ N_{S_3} \ar[r] & \ldots }
where \psi _ n = \varphi _{(\mathcal{L}_ n, (s_{0, n}, \ldots , s_{N, n}))} in the notation of Constructions, Section 27.13. As the squares in the statement of the theorem are cartesian we see that the squares in the above diagram are cartesian. We win by applying Lemma 30.28.1.
\square
Comments (0)