Lemma 98.9.3 (07X5)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 98: Artin's Axioms
Section 98.9: Formal objects
Lemma 98.9.3 (cite)

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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 $F(\eta _ n) \cong \xi _ n$ . $\square$

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