The Stacks project

97.6 Deformation categories

We match the notation introduced above with the notation from the chapter “Formal Deformation Theory”.

Lemma 97.6.1. 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 $ ( is a deformation category, see Formal Deformation Theory, Definition 89.16.8.

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

\[ \mathcal{F}(A_1 \times _ A A_2) \longrightarrow \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2) \]

is an equivalence, see Formal Deformation Theory, Lemma 89.16.4. 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 97.5.1. 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 97.6.2. 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 ( 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 89.16.6. Let $\overline{\mathcal{F}}$ be the functor of isomorphism classes, see Formal Deformation Theory, Remarks 89.5.2 (10). Then $\overline{\mathcal{F}}$ satisfies (S1) and (S2) as well, see Formal Deformation Theory, Lemma 89.10.5. This holds in particular in the situation of Lemma 97.6.1.

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