Lemma 98.23.7. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}^{opp}$ be a category fibred in groupoids. Assume that $\mathcal{X}$ satisfies (RS*) and that $\mathcal{X}$ has a naive obstruction theory. Then openness of versality holds for $\mathcal{X}$ provided the complexes $E_ x$ of Definition 98.23.5 have finitely generated cohomology groups for pairs $(A, x)$ where $A$ is of finite type over $S$.
Proof. Let $U$ be a scheme locally of finite type over $S$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $u_0$ be a finite type point of $U$ such that $x$ is versal at $u_0$. We may first shrink $U$ to an affine scheme such that $u_0$ is a closed point and such that $U \to S$ maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$. Say $U = \mathop{\mathrm{Spec}}(A)$. Let $\xi _ x : E_ x \to \mathop{N\! L}\nolimits _{A/\Lambda }$ be the obstruction map. At this point we may apply Lemma 98.23.4 to conclude. $\square$
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Your email address will not be published.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)