Lemma 98.9.3. Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\eta = (R, \eta _ n, g_ n)$ be a formal object of $\mathcal{Y}$ and let $\xi _1$ be an object of $\mathcal{X}$ with $F(\xi _1) \cong \eta _1$. If $F$ is formally smooth on objects (see Criteria for Representability, Section 97.6), then there exists a formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{X}$ such that $F(\xi ) \cong \eta $.
Proof. Note that each of the morphisms $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \to \mathop{\mathrm{Spec}}(R/\mathfrak m^{n + 1})$ is a first order thickening of affine schemes over $S$. Hence the assumption on $F$ means that we can successively lift $\xi _1$ to objects $\xi _2, \xi _3, \ldots $ of $\mathcal{X}$ endowed with compatible isomorphisms $\eta _ n|_{\mathop{\mathrm{Spec}}(R/\mathfrak m^{n - 1})} \cong \eta _{n - 1}$ and $F(\eta _ n) \cong \xi _ n$. $\square$
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Your email address will not be published.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)
There are also: