The Stacks project

91.3 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 88.3.1.

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 extend $\mathcal{F}$ satisfies the Rim-Schlesssinger condition (RS) discussed in Formal Deformation Theory, Section 88.16. Similarly, we may discuss to what extend our $\mathcal{F}$ satisfies (S1) and (S2) or to what extend $F$ satisfies the corresponding Schlessinger's conditions (H1) and (H2). See Formal Deformation Theory, Section 88.10.

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 88.6.4. If $\mathcal{F}$ satisfies (RS), then $\mathcal{D}\! \mathit{ef}_{x_0}$ is a deformation category (Formal Deformation Theory, Lemma 88.16.11) and satisfies (S1) and (S2) (Formal Deformation Theory, Lemma 88.16.6). 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 88.12.5 and Definition 88.12.1) is finite dimensional. Namely, this insures that $\mathcal{D}\! \mathit{ef}_{x_0}$ has a versal formal object (Formal Deformation Theory, Lemma 88.13.4).

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 88.26.4.


Comments (1)

Comment #6019 by Will Chen on

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


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