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. $\square$
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)