Lemma 99.13.9. The stack in groupoids \mathcal{X} = \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} satisfies openness of versality over \mathop{\mathrm{Spec}}(\mathbf{Z}). Similarly, after base change (Remark 99.13.5) openness of versality holds over any Noetherian base scheme S.
Proof. For the “usual” proof of this fact, please see the discussion in the remark following this proof. We will prove this using Artin's Axioms, Lemma 98.20.3. We have already seen that \mathcal{X} has diagonal representable by algebraic spaces, has (RS*), and is limit preserving, see Lemmas 99.13.2, 99.13.7, and 99.13.6. Hence we only need to see that \mathcal{X} satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma 98.20.3.
Let (R_ n) be an inverse system of rings such that R_ n \to R_ m is surjective with square zero kernel for all n \geq m. Let X_ n \to \mathop{\mathrm{Spec}}(R_ n) be a finitely presented, flat, proper morphism where X_ n is an algebraic space and let X_{n + 1} \to X_ n be a morphism over \mathop{\mathrm{Spec}}(R_{n + 1}) inducing an isomorphism X_ n = X_{n + 1} \times _{\mathop{\mathrm{Spec}}(R_{n + 1})} \mathop{\mathrm{Spec}}(R_ n). We have to find a flat, proper, finitely presented morphism X \to \mathop{\mathrm{Spec}}(\mathop{\mathrm{lim}}\nolimits R_ n) whose source is an algebraic space such that X_ n is the base change of X for all n.
Let I_ n = \mathop{\mathrm{Ker}}(R_ n \to R_1). We may think of (X_1 \subset X_ n) \to (\mathop{\mathrm{Spec}}(R_1) \subset \mathop{\mathrm{Spec}}(R_ n)) as a morphism of first order thickenings. (Please read some of the material on thickenings of algebraic spaces in More on Morphisms of Spaces, Section 76.9 before continuing.) The structure sheaf of X_ n is an extension
0 \to \mathcal{O}_{X_1} \otimes _{R_1} I_ n \to \mathcal{O}_{X_ n} \to \mathcal{O}_{X_1} \to 0
over 0 \to I_ n \to R_ n \to R_1, see More on Morphisms of Spaces, Lemma 76.18.1. Let's consider the extension
0 \to \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{X_1} \otimes _{R_1} I_ n \to \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{X_ n} \to \mathcal{O}_{X_1} \to 0
over 0 \to \mathop{\mathrm{lim}}\nolimits I_ n \to \mathop{\mathrm{lim}}\nolimits R_ n \to R_1 \to 0. The displayed sequence is exact as the R^1\mathop{\mathrm{lim}}\nolimits of the system of kernels is zero by Derived Categories of Spaces, Lemma 75.5.4. Observe that the map
\mathcal{O}_{X_1} \otimes _{R_1} \mathop{\mathrm{lim}}\nolimits I_ n \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{X_1} \otimes _{R_1} I_ n
induces an isomorphism upon applying the functor DQ_ X, see Derived Categories of Spaces, Lemma 75.25.6. Hence we obtain a unique extension
0 \to \mathcal{O}_{X_1} \otimes _{R_1} \mathop{\mathrm{lim}}\nolimits I_ n \to \mathcal{O}' \to \mathcal{O}_{X_1} \to 0
over 0 \to \mathop{\mathrm{lim}}\nolimits I_ n \to \mathop{\mathrm{lim}}\nolimits R_ n \to R_1 \to 0 by the equivalence of categories of Deformation Theory, Lemma 91.14.4. The sheaf \mathcal{O}' determines a first order thickening of algebraic spaces X_1 \subset X over \mathop{\mathrm{Spec}}(R_1) \subset \mathop{\mathrm{Spec}}(\mathop{\mathrm{lim}}\nolimits R_ n) by More on Morphisms of Spaces, Lemma 76.9.7. Observe that X \to \mathop{\mathrm{Spec}}(\mathop{\mathrm{lim}}\nolimits R_ n) is flat by the already used More on Morphisms of Spaces, Lemma 76.18.1. By More on Morphisms of Spaces, Lemma 76.18.3 we see that X \to \mathop{\mathrm{Spec}}(\mathop{\mathrm{lim}}\nolimits R_ n) is proper and of finite presentation. This finishes the proof. \square
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