Lemma 90.15.1. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$ such that (90.15.0.2) holds. Then $\xi $ is a minimal versal formal object. In particular, such $\xi $ are unique up to isomorphism.
90.15 Miniversal formal objects and tangent spaces
The general notion of minimality introduced in Definition 90.14.4 can sometimes be deduced from the behaviour on tangent spaces. Let $\xi $ be a formal object of the predeformation category $\mathcal{F}$ and let $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ be the corresponding morphism. Then we can consider the following the condition
90.15.0.1
\begin{equation} \label{formal-defos-equation-bijective} d\underline{\xi } : \text{Der}_\Lambda (R, k) \to T\mathcal{F} \text{ is bijective} \end{equation}
and the condition
90.15.0.2
\begin{equation} \label{formal-defos-equation-bijective-orbits} d\underline{\xi } : \text{Der}_\Lambda (R, k) \to T\mathcal{F} \text{ is bijective on }\text{Der}_\Lambda (k, k)\text{-orbits.} \end{equation}
Here we are using the identification $T\underline{R}|_{\mathcal{C}_\Lambda } = \text{Der}_\Lambda (R, k)$ of Example 90.11.11 and the action (90.12.6.2) of derivations on the tangent spaces. If $k' \subset k$ is separable, then $\text{Der}_\Lambda (k, k) = 0$ and the two conditions are equivalent. It turns out that, in the presence of condition (S2) a versal formal object is minimal if and only if $\underline{\xi }$ satisfies (90.15.0.2). Moreover, if $\underline{\xi }$ satisfies (90.15.0.1), then $\mathcal{F}$ satisfies (S2).
Proof. If $\xi $ is not minimal, then there exists a morphism $\xi ' \to \xi $ lying over $R' \to R$ such that $R = R'[[x_1, \ldots , x_ n]]$ with $n > 0$, see Lemma 90.14.5. Thus $d\underline{\xi }$ factors as
\[ \text{Der}_\Lambda (R, k) \to \text{Der}_\Lambda (R', k) \to T\mathcal{F} \]
and we see that (90.15.0.2) cannot hold because $D : f \mapsto \partial /\partial x_1(f) \bmod \mathfrak m_ R$ is an element of the kernel of the first arrow which is not in the image of $\text{Der}_\Lambda (k, k) \to \text{Der}_\Lambda (R, k)$. $\square$
Lemma 90.15.2. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$ such that (90.15.0.1) holds. Then
$\mathcal{F}$ satisfies (S1).
$\mathcal{F}$ satisfies (S2).
$\dim _ k T\mathcal{F}$ is finite.
Proof. Condition (S1) holds by Lemma 90.13.1. The first part of (S2) holds since (S1) holds. Let
\[ \vcenter { \xymatrix{ y \ar[r]_ c \ar[d]_ a & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[\epsilon ] \ar[r] \ar[d] & k[\epsilon ] \ar[d] \\ A \ar[r] & k } } \]
be diagrams as in the second part of (S2). As above we can find morphisms $b : \xi \to y$ and $b' : \xi \to y'$ such that
\[ \xymatrix{ \xi \ar[r]^{b'} \ar[d]_ b & y' \ar[d]^{a'} \\ y \ar[r]^{a} & x } \]
commutes. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ denote the structure morphism. Say $\widehat{p}(\xi ) = R$, i.e., $\xi $ lies over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. We see that the pushforward of $\xi $ via $p(c) \circ p(b)$ is $x_\epsilon $ and that the pushforward of $\xi $ via $p(c') \circ p(b')$ is $x_\epsilon $. Since $\xi $ satisfies (90.15.0.1), we see that $p(c) \circ p(b) = p(c') \circ p(b')$ as maps $R \to k[\epsilon ]$. Hence $p(b) = p(b')$ as maps from $R \to A \times _ k k[\epsilon ]$. Thus we see that $y$ and $y'$ are isomorphic to the pushforward of $\xi $ along this map and we get a unique morphism $y \to y'$ over $A \times _ k k[\epsilon ]$ compatible with $b$ and $b'$ as desired.
Finally, by Example 90.11.11 we see $\dim _ k T\mathcal{F} = \dim _ k T\underline{R}|_{\mathcal{C}_\Lambda }$ is finite. $\square$
Example 90.15.3. There exist predeformation categories which have a versal formal object satisfying (90.15.0.2) but which do not satisfy (S2). A quick example is to take $F = \underline{k[\epsilon ]}/G$ where $G \subset \text{Aut}_{\mathcal{C}_\Lambda }(k[\epsilon ])$ is a finite nontrivial subgroup. Namely, the map $\underline{k[\epsilon ]} \to F$ is smooth, but the tangent space of $F$ does not have a natural $k$-vector space structure (as it is a quotient of a $k$-vector space by a finite group).
Lemma 90.15.4. Let $\mathcal{F}$ be a predeformation category satisfying (S2) which has a versal formal object. Then its minimal versal formal object satisfies (90.15.0.2).
Proof. Let $\xi $ be a minimal versal formal object for $\mathcal{F}$, see Lemma 90.14.5. Say $\xi $ lies over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. In order to parse (90.15.0.2) we point out that $T\mathcal{F}$ has a natural $k$-vector space structure (see Lemma 90.12.2), that $d\underline{\xi } : \text{Der}_\Lambda (R, k) \to T\mathcal{F}$ is linear (see Lemma 90.12.4), and that the action of $\text{Der}_\Lambda (k, k)$ is given by addition (see Lemma 90.12.6). Consider the diagram
\[ \xymatrix{ & \mathop{\mathrm{Hom}}\nolimits _ k(\mathfrak m_ R/\mathfrak m_ R^2, k) \\ K \ar[r] & \text{Der}_\Lambda (R, k) \ar[r]^{d\underline{\xi }} \ar[u] & T\mathcal{F} \\ & \text{Der}_\Lambda (k, k) \ar[u] \ar[ru] } \]
The vector space $K$ is the kernel of $d\underline{\xi }$. Note that the middle column is exact in the middle as it is dual to the sequence (90.3.10.1). If (90.15.0.2) fails, then we can find a nonzero element $D \in K$ which does not map to zero in $\mathop{\mathrm{Hom}}\nolimits _ k(\mathfrak m_ R/\mathfrak m_ R^2, k)$. This means there exists an $t \in \mathfrak m_ R$ such that $D(t) = 1$. Set $R' = \{ a \in R \mid D(a) = 0\} $. As $D$ is a derivation this is a subring of $R$. Since $D(t) = 1$ we see that $R' \to k$ is surjective (compare with the proof of Lemma 90.3.12). Note that $\mathfrak m_{R'} = \mathop{\mathrm{Ker}}(D : \mathfrak m_ R \to k)$ is an ideal of $R$ and $\mathfrak m_ R^2 \subset \mathfrak m_{R'}$. Hence
\[ \mathfrak m_ R/\mathfrak m_ R^2 = \mathfrak m_{R'}/\mathfrak m_ R^2 + k\overline{t} \]
which implies that the map
\[ R'/\mathfrak m_ R^2 \times _ k k[\epsilon ] \to R/\mathfrak m_ R^2 \]
sending $\epsilon $ to $\overline{t}$ is an isomorphism. In particular there is a map $R/\mathfrak m_ R^2 \to R'/\mathfrak m_ R^2$.
Let $\xi \to y$ be a morphism lying over $R \to R/\mathfrak m_ R^2$. Let $y \to x$ be a morphism lying over $R/\mathfrak m_ R^2 \to R'/\mathfrak m_ R^2$. Let $y \to x_\epsilon $ be a morphism lying over $R/\mathfrak m_ R^2 \to k[\epsilon ]$. Let $x_0$ be the unique (up to unique isomorphism) object of $\mathcal{F}$ over $k$. By the axioms of a category cofibred in groupoids we obtain a commutative diagram
\[ \vcenter { \xymatrix{ y \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ R'/\mathfrak m_ R^2 \times _ k k[\epsilon ] \ar[r] \ar[d] & k[\epsilon ] \ar[d] \\ R'/\mathfrak m_ R^2 \ar[r] & k. } } \]
Because $D \in K$ we see that $x_\epsilon $ is isomorphic to $0 \in \mathcal{F}(k[\epsilon ])$, i.e., $x_\epsilon $ is the pushforward of $x_0$ via $k \to k[\epsilon ], a \mapsto a$. Hence by Lemma 90.10.7 we see that there exists a morphism $x \to y$. Since $\text{length}_\Lambda (R'/\mathfrak m_ R^2) < \text{length}_\Lambda (R/\mathfrak m_ R^2)$ the corresponding ring map $R'/\mathfrak m_ R^2 \to R/\mathfrak m_ R^2$ is not surjective. This contradicts the minimality of $\xi /R$, see part (1) of Lemma 90.14.2. This contradiction shows that such a $D$ cannot exist, hence we win. $\square$
Theorem 90.15.5. Let $\mathcal{F}$ be a predeformation category. Consider the following conditions
We always have
\[ (1) \Rightarrow (3) \Rightarrow (2). \]
If $k' \subset k$ is separable, then all three are equivalent.
Proof. Lemma 90.15.2 shows that (1) $\Rightarrow $ (3). Lemmas 90.13.4 and 90.15.4 show that (3) $\Rightarrow $ (2). If $k' \subset k$ is separable then $\text{Der}_\Lambda (k, k) = 0$ and we see that (90.15.0.1) $=$ (90.15.0.2), i.e., (1) is the same as (2).
An alternative proof of (3) $\Rightarrow $ (1) in the classical case is to add a few words to the proof of Lemma 90.13.4 to see that one can right away construct a versal object which satisfies (90.15.0.1) in this case. This avoids the use of Lemma 90.13.4 in the classical case. Details omitted. $\square$
Remark 90.15.6. Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a predeformation functor satisfying (S1) and (S2) and $\dim _ k TF < \infty $. Recall that these conditions correspond to the conditions (H1), (H2), and (H3) from Schlessinger's paper, see Remark 90.13.5. Now, in the classical case (or if $k' \subset k$ is separable) following Schlessinger we introduce the notion of a hull: a hull is a versal formal object $\xi \in \widehat{F}(R)$ such that $d\underline{\xi } : T\underline{R}|_{\mathcal{C}_\Lambda } \to TF$ is an isomorphism, i.e., (90.15.0.1) holds. Thus Theorem 90.15.5 tells us in the classical case. In other words, our theorem recovers Schlessinger's theorem on the existence of hulls.
\[ (H1) + (H2) + (H3) \Rightarrow \text{ there exists a hull} \]
Remark 90.15.7. Let $\mathcal{F}$ be a predeformation category. Recall that $\mathcal{F} \to \overline{\mathcal{F}}$ is smooth, see Remark 90.8.5. Hence if $\xi \in \widehat{\mathcal{F}}(R)$ is a versal formal object, then the composition is smooth (Lemma 90.8.7) and we conclude that the image $\overline{\xi }$ of $\xi $ in $\overline{\mathcal{F}}$ is a versal formal object. If (90.15.0.1) holds, then $\overline{\xi }$ induces an isomorphism $T\underline{R}|_{\mathcal{C}_\Lambda } \to T\overline{\mathcal{F}}$ because $\mathcal{F} \to \overline{\mathcal{F}}$ identifies tangent spaces. Hence in this case $\overline{\xi }$ is a hull for $\overline{\mathcal{F}}$, see Remark 90.15.6. By Theorem 90.15.5 we can always find such a $\xi $ if $k' \subset k$ is separable and $\mathcal{F}$ is a predeformation category satisfying (S1), (S2), and $\dim _ k T\mathcal{F} < \infty $.
\[ \underline{R}|_{\mathcal{C}_\Lambda } \longrightarrow \mathcal{F} \longrightarrow \overline{\mathcal{F}} \]
Example 90.15.8. In Lemma 90.9.5 we constructed objects $R \in \widehat{\mathcal{C}}_\Lambda $ such that $\underline{R}|_{\mathcal{C}_\Lambda }$ is smooth and such that Let us reinterpret this using the theorem above. Namely, consider $\mathcal{F} = \mathcal{C}_\Lambda $ as a category cofibred in groupoids over itself (using the identity functor). Then $\mathcal{F}$ is a predeformation category, satisfies (S1) and (S2), and we have $T\mathcal{F} = 0$. Thus $\mathcal{F}$ satisfies condition (3) of Theorem 90.15.5. The theorem implies that (2) holds, i.e., we can find a minimal versal formal object $\xi \in \widehat{\mathcal{F}}(S)$ over some $S \in \widehat{\mathcal{C}}_\Lambda $ satisfying (90.15.0.2). Lemma 90.9.3 shows that $\Lambda \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology (because $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F} = \mathcal{C}_\Lambda $ is smooth). Now condition (90.15.0.2) tells us that $\text{Der}_\Lambda (S, k) \to 0$ is bijective on $\text{Der}_\Lambda (k, k)$-orbits. This means the injection $\text{Der}_\Lambda (k, k) \to \text{Der}_\Lambda (S, k)$ is also surjective. In other words, we have $\Omega _{S/\Lambda } \otimes _ S k = \Omega _{k/\Lambda }$. Since $\Lambda \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology, we can apply More on Algebra, Lemma 15.40.4 to conclude the exact sequence (90.3.10.2) turns into a pair of identifications Reading the argument backwards, we find that the $R$ constructed in Lemma 90.9.5 carries a minimal versal object. By the uniqueness of minimal versal objects (Lemma 90.14.5) we also conclude $R \cong S$, i.e., the two constructions give the same answer.
\[ H_1(L_{k/\Lambda }) = \mathfrak m_ R/\mathfrak m_ R^2 \quad \text{and}\quad \Omega _{R/\Lambda } \otimes _ R k = \Omega _{k/\Lambda } \]
\[ H_1(L_{k/\Lambda }) = \mathfrak m_ S/\mathfrak m_ S^2 \quad \text{and}\quad \Omega _{S/\Lambda } \otimes _ S k = \Omega _{k/\Lambda } \]
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