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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)