The Stacks project

Lemma 99.15.9. Consider the stack $\mathcal{C}\! \mathit{urves}$ over the base scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Then every formal object is effective.

Proof. For definitions of the notions in the lemma, please see Artin's Axioms, Section 98.9. Let $(A, \mathfrak m, \kappa )$ be a Noetherian complete local ring. Let $(X_ n \to \mathop{\mathrm{Spec}}(A/\mathfrak m^ n))$ be a formal object of $\mathcal{C}\! \mathit{urves}$ over $A$. By More on Morphisms of Spaces, Lemma 76.43.5 there exists a projective morphism $X \to \mathop{\mathrm{Spec}}(A)$ and a compatible system of ismomorphisms $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/\mathfrak m^ n) \cong X_ n$. By More on Morphisms, Lemma 37.12.4 we see that $X \to \mathop{\mathrm{Spec}}(A)$ is flat. By More on Morphisms, Lemma 37.30.6 we see that $X \to \mathop{\mathrm{Spec}}(A)$ has relative dimension $\leq 1$. This proves the lemma. $\square$

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