Lemma 76.43.6. Let (A, \mathfrak m, \kappa ) be a complete Noetherian local ring. Let X be an algebraic space over \mathop{\mathrm{Spec}}(A). If X \to \mathop{\mathrm{Spec}}(A) is proper and \dim (X_\kappa ) \leq 1, then X is a scheme projective over A.
Proof. Set X_ n = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/\mathfrak m^ n). By Lemma 76.43.5 there exists a projective morphism Y \to \mathop{\mathrm{Spec}}(A) and compatible isomorphisms Y \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/\mathfrak m^ n) \cong X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/\mathfrak m^ n). By Lemma 76.43.3 we see that X \cong Y and the proof is complete. \square
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