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
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