The Stacks project

92.3 General outline

This section lays out the procedure for discussing the next few examples.

Step I. For each section we fix a Noetherian ring $\Lambda $ and we fix a finite ring map $\Lambda \to k$ where $k$ is a field. As usual we let $\mathcal{C}_\Lambda = \mathcal{C}_{\Lambda , k}$ be our base category, see Formal Deformation Theory, Definition 89.3.1.

Step II. In each section we define a category $\mathcal{F}$ cofibred in groupoids over $\mathcal{C}_\Lambda $. Occassionally we will consider instead a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$.

Step III. We explain to what extent $\mathcal{F}$ satisfies the Rim-Schlesssinger condition (RS) discussed in Formal Deformation Theory, Section 89.16. Similarly, we may discuss to what extent our $\mathcal{F}$ satisfies (S1) and (S2) or to what extent $F$ satisfies the corresponding Schlessinger's conditions (H1) and (H2). See Formal Deformation Theory, Section 89.10.

Step IV. Let $x_0$ be an object of $\mathcal{F}(k)$, in other words an object of $\mathcal{F}$ over $k$. In this chapter we will use the notation

\[ \mathcal{D}\! \mathit{ef}_{x_0} = \mathcal{F}_{x_0} \]

to denote the predeformation category constructed in Formal Deformation Theory, Remark 89.6.4. If $\mathcal{F}$ satisfies (RS), then $\mathcal{D}\! \mathit{ef}_{x_0}$ is a deformation category (Formal Deformation Theory, Lemma 89.16.11) and satisfies (S1) and (S2) (Formal Deformation Theory, Lemma 89.16.6). If (S1) and (S2) are satisfied, then an important question is whether the tangent space

\[ T\mathcal{D}\! \mathit{ef}_{x_0} = T_{x_0}\mathcal{F} = T\mathcal{F}_{x_0} \]

(see Formal Deformation Theory, Remark 89.12.5 and Definition 89.12.1) is finite dimensional. Namely, this insures that $\mathcal{D}\! \mathit{ef}_{x_0}$ has a versal formal object (Formal Deformation Theory, Lemma 89.13.4).

Step V. If $\mathcal{F}$ passes Step IV, then the next question is whether the $k$-vector space

\[ \text{Inf}(\mathcal{D}\! \mathit{ef}_{x_0}) = \text{Inf}_{x_0}(\mathcal{F}) \]

of infinitesimal automorphisms of $x_0$ is finite dimensional. Namely, if true, this implies that $\mathcal{D}\! \mathit{ef}_{x_0}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $, see Formal Deformation Theory, Theorem 89.26.4.

Comments (2)

Comment #6019 by Will Chen on

In "Step III" - "extend" should be "extent" (3 times)

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