Lemma 76.43.5. Let (A, \mathfrak m, \kappa ) be a complete local Noetherian ring. Set S = \mathop{\mathrm{Spec}}(A) and S_ n = \mathop{\mathrm{Spec}}(A/\mathfrak m^ n). 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 algebraic spaces with cartesian squares. If \dim (X_1) \leq 1, then there exists a projective morphism of schemes X \to S and isomorphisms X_ n \cong X \times _ S S_ n compatible with i_ n.
Proof.
By Spaces over Fields, Lemma 72.9.3 the algebraic space X_1 is a scheme. Hence X_1 is a proper scheme of dimension \leq 1 over \kappa . By Varieties, Lemma 33.43.4 we see that X_1 is H-projective over \kappa . Let \mathcal{L}_1 be an ample invertible sheaf on X_1.
We are going to show that \mathcal{L}_1 lifts to a compatible system \{ \mathcal{L}_ n\} of invertible sheaves on \{ X_ n\} . Observe that X_ n is a scheme too by Lemma 76.9.5. Recall that X_1 \to X_ n induces homeomorphisms of underlying topological spaces. In the rest of the proof we do not distinguish between sheaves on X_ n and sheaves on X_1. Suppose, given a lift \mathcal{L}_ n to X_ n. We consider the exact sequence
1 \to (1 + \mathfrak m^ n\mathcal{O}_{X_{n + 1}})^* \to \mathcal{O}_{X_{n + 1}}^* \to \mathcal{O}_{X_ n}^* \to 1
of sheaves on X_{n + 1}. The class of \mathcal{L}_ n in H^1(X_ n, \mathcal{O}_{X_ n}^*) (see Cohomology, Lemma 20.6.1) can be lifted to an element of H^1(X_{n + 1}, \mathcal{O}_{X_{n + 1}}^*) if and only if the obstruction in H^2(X_{n + 1}, (1 + \mathfrak m^ n\mathcal{O}_{X_{n + 1}})^*) is zero. As X_1 is a Noetherian scheme of dimension \leq 1 this cohomology group vanishes (Cohomology, Proposition 20.20.7).
By Grothendieck's algebraization theorem (Cohomology of Schemes, Theorem 30.28.4) we find a projective morphism of schemes X \to S = \mathop{\mathrm{Spec}}(A) and a compatible system of isomorphisms X_ n = S_ n \times _ S X.
\square
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