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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