Lemma 98.6.1. Let S be a locally Noetherian scheme. Let \mathcal{X} be a category fibred in groupoids over (\mathit{Sch}/S)_{fppf} satisfying (RS). For any field k of finite type over S and any object x_0 of \mathcal{X} lying over k the predeformation category p : \mathcal{F}_{\mathcal{X}, k, x_0} \to \mathcal{C}_\Lambda (98.3.0.2) is a deformation category, see Formal Deformation Theory, Definition 90.16.8.
Proof. Set \mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}. Let f_1 : A_1 \to A and f_2 : A_2 \to A be ring maps in \mathcal{C}_\Lambda with f_2 surjective. We have to show that the functor
is an equivalence, see Formal Deformation Theory, Lemma 90.16.4. Set X = \mathop{\mathrm{Spec}}(A), X' = \mathop{\mathrm{Spec}}(A_2), Y = \mathop{\mathrm{Spec}}(A_1) and Y' = \mathop{\mathrm{Spec}}(A_1 \times _ A A_2). Note that Y' = Y \amalg _ X X' in the category of schemes, see More on Morphisms, Lemma 37.14.3. We know that in the diagram of functors of fibre categories
the top horizontal arrow is an equivalence by Definition 98.5.1. Since \mathcal{F}(B) is the category of objects of \mathcal{X}_{\mathop{\mathrm{Spec}}(B)} with an identification with x_0 over k we win. \square
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