The Stacks project

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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