The Stacks project

88.26 Presentations of deformation categories

According to the next lemma, a smooth morphism from a predeformation functor to a predeformation category $\mathcal{F}$ gives rise to a presentation of $\mathcal{F}$ by a smooth groupoid in functors.

Lemma 88.26.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a smooth morphism of categories cofibered in groupoids. Then:

  1. If $(U, R, s, t, c)$ is the groupoid in functors on $\mathcal{C}_\Lambda $ constructed from $f : U \to \mathcal{F}$ in Lemma 88.25.2, then $(U, R, s, t, c)$ is smooth.

  2. If $f : U(k) \to \mathcal{F}(k)$ is essentially surjective, then the morphism $[f] : [U/R] \to \mathcal{F}$ of Lemma 88.25.3 is an equivalence.

Proof. From the construction of Lemma 88.25.2 we have a commutative diagram

\[ \xymatrix{ R = U \times _{f, \mathcal{F}, f} U \ar[r]_-s \ar[d]_ t & U \ar[d]^ f \\ U \ar[r]^ f & \mathcal{F} } \]

where $t, s$ are the first and second projections. So $t, s$ are smooth by Lemma 88.8.7. Hence (1) holds.

If the assumption of (2) holds, then by Lemma 88.8.8 the morphism $f : U \to \mathcal{F}$ is essentially surjective. Hence by Lemma 88.25.3 the morphism $[f] : [U/R] \to \mathcal{F}$ is an equivalence. $\square$

Lemma 88.26.2. Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a deformation functor. Let $f: U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Then $U \times _{f, \mathcal{F}, f} U$ is a deformation functor with tangent space fitting into an exact sequence of $k$-vector spaces

\[ 0 \to \text{Inf}(\mathcal{F}) \to T(U \times _{f, \mathcal{F}, f} U) \to TU \oplus TU \]

Proof. Follows from Lemma 88.20.1 and the fact that $\text{Inf}(U) = (0)$. $\square$

Lemma 88.26.3. Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda $ constructed from $f : U \to \mathcal{F}$ in Lemma 88.25.2. If $\dim _ k \text{Inf}(\mathcal{F}) < \infty $, then $(U, R, s, t, c)$ is prorepresentable.

Proof. Note that $U$ is a deformation functor by Example 88.16.10. By Lemma 88.26.2 we see that $R = U \times _{f, \mathcal{F}, f} U$ is a deformation functor whose tangent space $TR = T(U \times _{f, \mathcal{F}, f} U)$ sits in an exact sequence $0 \to \text{Inf}(\mathcal{F}) \to TR \to TU \oplus TU$. Since we have assumed the first space has finite dimension and since $TU$ has finite dimension by Example 88.11.11 we see that $\dim TR < \infty $. The map $\gamma : \text{Der}_\Lambda (k, k) \to TR$ see (88.12.6.1) is injective because its composition with $TR \to TU$ is injective by Theorem 88.18.2 for the prorepresentable functor $U$. Thus $R$ is prorepresentable by Theorem 88.18.2. It follows from Lemma 88.22.4 that $(U, R, s, t, c)$ is prorepresentable. $\square$

Theorem 88.26.4. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Then $\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $ if and only if the following conditions hold:

  1. $\mathcal{F}$ is a deformation category.

  2. $\dim _ k T\mathcal{F}$ is finite.

  3. $\dim _ k \text{Inf}(\mathcal{F})$ is finite.

Proof. Recall that a prorepresentable functor is a deformation functor, see Example 88.16.10. Thus if $\mathcal{F}$ is equivalent to a smooth prorepresentable groupoid in functors, then conditions (1), (2), and (3) follow from Lemma 88.24.2 (1), (2), and (3).

Conversely, assume conditions (1), (2), and (3) hold. Condition (1) implies that (S1) and (S2) are satisfied, see Lemma 88.16.6. By Lemma 88.13.4 there exists a versal formal object $\xi $. Setting $U = \underline{R}|_{\mathcal{C}_\Lambda }$ the associated map $\underline{\xi } : U \to \mathcal{F}$ is smooth (this is the definition of a versal formal object). Let $(U, R, s, t, c)$ be the groupoid in functors constructed in Lemma 88.25.2 from the map $\underline{\xi }$. By Lemma 88.26.1 we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that $[U/R] \to \mathcal{F}$ is an equivalence. By Lemma 88.26.3 we see that $(U, R, s, t, c)$ is prorepresentable. Hence $[U/R] \to \mathcal{F}$ is the desired presentation of $\mathcal{F}$. $\square$


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