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.
98.6 Deformation categories
We match the notation introduced above with the notation from the chapter “Formal Deformation Theory”.
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
\mathcal{F}(A_1 \times _ A A_2) \longrightarrow \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)
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
\xymatrix{ \mathcal{X}_{Y'} \ar[r] \ar[d] & \mathcal{X}_ Y \times _{\mathcal{X}_ X} \mathcal{X}_{X'} \ar[d] \\ \mathcal{X}_{\mathop{\mathrm{Spec}}(k)} \ar@{=}[r] & \mathcal{X}_{\mathop{\mathrm{Spec}}(k)} }
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
Remark 98.6.2. Let S be a locally Noetherian scheme. Let \mathcal{X} be fibred in groupoids over (\mathit{Sch}/S)_{fppf}. Let k be a field of finite type over S and x_0 an object of \mathcal{X} over k. Let p : \mathcal{F} \to \mathcal{C}_\Lambda be as in (98.3.0.2). If \mathcal{F} is a deformation category, i.e., if \mathcal{F} satisfies the Rim-Schlessinger condition (RS), then we see that \mathcal{F} satisfies Schlessinger's conditions (S1) and (S2) by Formal Deformation Theory, Lemma 90.16.6. Let \overline{\mathcal{F}} be the functor of isomorphism classes, see Formal Deformation Theory, Remarks 90.5.2 (10). Then \overline{\mathcal{F}} satisfies (S1) and (S2) as well, see Formal Deformation Theory, Lemma 90.10.5. This holds in particular in the situation of Lemma 98.6.1.
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