The Stacks project

Lemma 97.23.7. 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 97.23.5 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 97.23.4 to conclude. $\square$


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