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## Tag 06HF

### 80.8. Smooth morphisms

In this section we discuss smooth morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.

Definition 80.8.1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. We say $\varphi$ is smooth if it satisfies the following condition: Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Let $y \in \mathop{\rm Ob}\nolimits(\mathcal{G}(B)), x \in \mathop{\rm Ob}\nolimits(\mathcal{F}(A))$, and $y \to \varphi(x)$ be a morphism lying over $B \to A$. Then there exists $x' \in \mathop{\rm Ob}\nolimits(\mathcal{F}(B))$, a morphism $x' \to x$ lying over $B \to A$, and a morphism $\varphi(x') \to y$ lying over $\text{id}: B \to B$, such that the diagram $$\xymatrix{ \varphi(x') \ar[r] \ar[dr] & y \ar[d] \\ & \varphi(x) }$$ commutes.

Lemma 80.8.2. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Then $\varphi$ is smooth if the condition in Definition 80.8.1 is assumed to hold only for small extensions $B \to A$.

Proof. Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Let $y \in \mathop{\rm Ob}\nolimits(\mathcal{G}(B))$, $x \in \mathop{\rm Ob}\nolimits(\mathcal{F}(A))$, and $y \to \varphi(x)$ be a morphism lying over $B \to A$. By Lemma 80.3.3 we can factor $B \to A$ into small extensions $B = B_n \to B_{n-1} \to \ldots \to B_0 = A$. We argue by induction on $n$. If $n = 1$ the result is true by assumption. If $n > 1$, then denote $f : B = B_n \to B_{n - 1}$ and denote $g : B_{n - 1} \to B_0 = A$. Choose a pushforward $y \to f_* y$ of $y$ along $f$, so that the morphism $y \to \varphi(x)$ factors as $y \to f_* y \to \varphi(x)$. By the induction hypothesis we can find $x_{n - 1} \to x$ lying over $g : B_{n - 1} \to A$ and $a : \varphi(x_{n - 1}) \to f_*y$ lying over $\text{id} : B_{n - 1} \to B_{n - 1}$ such that $$\xymatrix{ \varphi(x_{n - 1}) \ar[r]_-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }$$ commutes. We can apply the assumption to the composition $y \to \varphi(x_{n - 1})$ of $y \to f_*y$ with $a^{-1} : f_*y \to \varphi(x_{n - 1})$. We obtain $x_n \to x_{n - 1}$ lying over $B_n \to B_{n - 1}$ and $\varphi(x_n) \to y$ lying over $\text{id} : B_n \to B_n$ so that the diagram $$\xymatrix{ \varphi(x_n) \ar[r] \ar[d] & y \ar[d] \\ \varphi(x_{n - 1}) \ar[r]^-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }$$ commutes. Then the composition $x_n \to x_{n - 1} \to x$ and $\varphi(x_n) \to y$ are the morphisms required by the definition of smoothness. $\square$

Remark 80.8.3. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $B \to A$ be a ring map in $\mathcal{C}_\Lambda$. Choices of pushforwards along $B \to A$ for objects in the fiber categories $\mathcal{F}(B)$ and $\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$ and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative diagram $$\xymatrix{ \mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\ \mathcal{F}(A) \ar[r]^{\varphi} & \mathcal{G}(A) . }$$ Hence there is an induced functor $\mathcal{F}(B) \to \mathcal{F}(A) \times_{\mathcal{G}(A)} \mathcal{G}(B)$. Unwinding the definitions shows that $\varphi : \mathcal{F} \to \mathcal{G}$ is smooth if and only if this induced functor is essentially surjective whenever $B \to A$ is surjective (or equivalently, by Lemma 80.8.2, whenever $B \to A$ is a small extension).

Remark 80.8.4. The characterization of smooth morphisms in Remark 80.8.3 is analogous to Schlessinger's notion of a smooth morphism of functors, cf. [Sch, Definition 2.2.]. In fact, when $\mathcal{F}$ and $\mathcal{G}$ are cofibered in sets then our notion is equivalent to Schlessinger's. Namely, in this case let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding functors, see Remarks 80.5.2 (11). Then $F \to G$ is smooth if and only if for every surjection of rings $B \to A$ in $\mathcal{C}_\Lambda$ the map $F(B) \to F(A) \times_{G(A)} G(B)$ is surjective.

Remark 80.8.5. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Then the morphism $\mathcal{F} \to \overline{\mathcal{F}}$ is smooth. Namely, suppose that $f : B \to A$ is a ring map in $\mathcal{C}_\Lambda$. Let $x \in \mathop{\rm Ob}\nolimits(\mathcal{F}(A))$ and let $\overline{y} \in \overline{\mathcal{F}}(B)$ be the isomorphism class of $y \in \mathop{\rm Ob}\nolimits(\mathcal{F}(B))$ such that $\overline{f_*y} = \overline{x}$. Then we simply take $x' = y$, the implied morphism $x' = y \to x$ over $B \to A$, and the equality $\overline{x'} = \overline{y}$ as the solution to the problem posed in Definition 80.8.1.

If $R \to S$ is a ring map $\widehat{\mathcal{C}}_\Lambda$, then there is an induced morphism $\underline{S} \to \underline{R}$ between the functors $\underline{S}, \underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$. In this situation, smoothness of the restriction $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is a familiar notion:

Lemma 80.8.6. Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Then the induced morphism $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth if and only if $S$ is a power series ring over $R$.

Proof. Assume $S$ is a power series ring over $R$. Say $S = R[[x_1, \ldots, x_n]]$. Smoothness of $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ means the following (see Remark 80.8.4): Given a surjective ring map $B \to A$ in $\mathcal{C}_\Lambda$, a ring map $R \to B$, a ring map $S \to A$ such that the solid diagram $$\xymatrix{ S \ar[r] \ar@{..>}[rd] & A \\ R \ar[u] \ar[r] & B \ar[u] }$$ is commutative then a dotted arrow exists making the diagram commute. (Note the similarity with Algebra, Definition 10.136.1.) To construct the dotted arrow choose elements $b_i \in B$ whose images in $A$ are equal to the images of $x_i$ in $A$. Note that $b_i \in \mathfrak m_B$ as $x_i$ maps to an element of $\mathfrak m_A$. Hence there is a unique $R$-algebra map $R[[x_1, \ldots, x_n]] \to B$ which maps $x_i$ to $b_i$ and which can serve as our dotted arrow.

Conversely, assume $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth. Let $x_1, \ldots, x_n \in S$ be elements whose images form a basis in the relative cotangent space $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ of $S$ over $R$. Set $T = R[[X_1, \ldots, X_n]]$. Note that both $$S/(\mathfrak m_R S + \mathfrak m_S^2) \cong R/\mathfrak m_R[x_1, \ldots, x_n]/(x_ix_j)$$ and $$T/(\mathfrak m_R T + \mathfrak m_T^2) \cong R/\mathfrak m_R[X_1, \ldots, X_n]/(X_iX_j).$$ Let $S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)$ be the local $R$-algebra isomorphism given by mapping the class of $x_i$ to the class of $X_i$. Let $f_1 : S \to T/(\mathfrak m_R T + \mathfrak m_T^2)$ be the composition $S \to S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)$. The assumption that $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth means we can lift $f_1$ to a map $f_2 : S \to T/\mathfrak{m}_T^2$, then to a map $f_3 : S \to T/\mathfrak{m}_T^3$, and so on, for all $n \geq 1$. Thus we get an induced map $f : S \to T = \mathop{\rm lim}\nolimits T/\mathfrak m_T^n$ of local $R$-algebras. By our choice of $f_1$, the map $f$ induces an isomorphism $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2) \to \mathfrak m_T/(\mathfrak m_R T + \mathfrak m_T^2)$ of relative cotangent spaces. Hence $f$ is surjective by Lemma 80.4.2 (where we think of $f$ as a map in $\widehat{\mathcal{C}}_R$). Choose preimages $y_i \in S$ of $X_i \in T$ under $f$. As $T$ is a power series ring over $R$ there exists a local $R$-algebra homomorphism $s : T \to S$ mapping $X_i$ to $y_i$. By construction $f \circ s = \text{id}$. Then $s$ is injective. But $s$ induces an isomorphism on relative cotangent spaces since $f$ does, so it is also surjective by Lemma 80.4.2 again. Hence $s$ and $f$ are isomorphisms. $\square$

Smooth morphisms satisfy the following functorial properties.

Lemma 80.8.7. Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G} \to \mathcal{H}$ be morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.

1. If $\varphi$ and $\psi$ are smooth, then $\psi \circ \varphi$ is smooth.
2. If $\varphi$ is essentially surjective and $\psi \circ \varphi$ is smooth, then $\psi$ is smooth.
3. If $\mathcal{G}' \to \mathcal{G}$ is a morphism of categories cofibered in groupoids and $\varphi$ is smooth, then $\mathcal{F} \times_\mathcal{G} \mathcal{G}' \to \mathcal{G}'$ is smooth.

Proof. Statements (1) and (2) follow immediately from the definitions. Proof of (3) omitted. Hints: use the formulation of smoothness given in Remark 80.8.3 and use that $\mathcal{F} \times_\mathcal{G} \mathcal{G}'$ is the $2$-fibre product, see Remarks 80.5.2 (13). $\square$

Lemma 80.8.8. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Assume $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective. Then $\varphi : \mathcal{F} \to \mathcal{G}$ and $\widehat{\varphi} : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ are essentially surjective.

Proof. Let $y$ be an object of $\mathcal{G}$ lying over $A \in \mathop{\rm Ob}\nolimits(\mathcal{C}_\Lambda)$. Let $y \to y_0$ be a pushforward of $y$ along $A \to k$. By the assumption on essential surjectivity of $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ there exist an object $x_0$ of $\mathcal{F}$ lying over $k$ and an isomorphism $y_0 \to \varphi(x_0)$. Smoothness of $\varphi$ implies there exists an object $x$ of $\mathcal{F}$ over $A$ whose image $\varphi(x)$ is isomorphic to $y$. Thus $\varphi : \mathcal{F} \to \mathcal{G}$ is essentially surjective.

Let $\eta = (R, \eta_n, g_n)$ be an object of $\widehat{\mathcal{G}}$. We construct an object $\xi$ of $\widehat{\mathcal{F}}$ with an isomorphism $\eta \to \varphi(\xi)$. By the assumption on essential surjectivity of $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$, there exists a morphism $\eta_1 \to \varphi(\xi_1)$ in $\mathcal{G}(k)$ for some $\xi_1 \in \mathop{\rm Ob}\nolimits(\mathcal{F}(k))$. The morphism $\eta_2 \xrightarrow{g_1} \eta_1 \to \varphi(\xi_1)$ lies over the surjective ring map $R/\mathfrak m_R^2 \to k$, hence by smoothness of $\varphi$ there exists $\xi_2 \in \mathop{\rm Ob}\nolimits(\mathcal{F}(R/\mathfrak m_R^2))$, a morphism $f_1: \xi_2 \to \xi_1$ lying over $R/\mathfrak m_R^2 \to k$, and a morphism $\eta_2 \to \varphi(\xi_2)$ such that $$\xymatrix{ \varphi(\xi_2) \ar[r]^{\varphi(f_1)} & \varphi(\xi_{1}) \\ \eta_2 \ar[u] \ar[r]^{g_1} & \eta_1 \ar[u] \\ }$$ commutes. Continuing in this way we construct an object $\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ and a morphism $\eta \to \varphi(\xi) = (R, \varphi(\xi_n), \varphi(f_n))$ in $\widehat{\mathcal{G}}(R)$. $\square$

Later we are interested in producing smooth morphisms from prorepresentable functors to predeformation categories $\mathcal{F}$. By the discussion in Remark 80.7.12 these morphisms correspond to certain formal objects of $\mathcal{F}$ More precisely, these are the so-called versal formal objects of $\mathcal{F}$.

Definition 80.8.9. Let $\mathcal{F}$ be a category cofibered in groupoids. Let $\xi$ be a formal object of $\mathcal{F}$ lying over $R \in \mathop{\rm Ob}\nolimits(\widehat{\mathcal{C}}_\Lambda)$. We say $\xi$ is versal if the corresponding morphism $\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ of Remark 80.7.12 is smooth.

Remark 80.8.10. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$. It follows from the definition of smoothness that versality of $\xi$ is equivalent to the following condition: If $$\xymatrix{ & y \ar[d] \\ \xi \ar[r] & x }$$ is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a surjective map $B \to A$ of Artinian rings (we may assume it is a small extension), then there exists a morphism $\xi \to y$ such that $$\xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x }$$ commutes. In particular, the condition that $\xi$ be versal does not depend on the choices of pushforwards made in the construction of $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ in Remark 80.7.12.

Lemma 80.8.11. Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$. For any formal object $\eta$ of $\widehat{\mathcal{F}}$, there exists a morphism $\xi \to \eta$.

Proof. By assumption the morphism $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ is smooth. Then $\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$ is the completion of $\underline{\xi}$, see Remark 80.7.12. By Lemma 80.8.8 there exists an object $f$ of $\underline{R}$ such that $\iota(\xi)(f) = \eta$. Then $f$ is a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$. And $\iota(\xi)(f) = \eta$ means that $f_*\xi \cong \eta$ which means exactly that there is a morphism $\xi \to \eta$ lying over $f$. $\square$

The code snippet corresponding to this tag is a part of the file formal-defos.tex and is located in lines 1887–2249 (see updates for more information).

\section{Smooth morphisms}
\label{section-smooth-morphisms}

\noindent
In this section we discuss smooth morphisms of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.

\begin{definition}
\label{definition-smooth-morphism}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  We say  $\varphi$ is
{\it smooth} if it satisfies the following condition: Let $B \to A$ be
a surjective ring map in $\mathcal{C}_\Lambda$.  Let $y \in \Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y \to \varphi(x)$ be a morphism lying over $B \to A$.  Then there
exists $x' \in \Ob(\mathcal{F}(B))$, a morphism $x' \to x$
lying over $B \to A$, and a morphism $\varphi(x') \to y$ lying
over $\text{id}: B \to B$, such that the diagram
$$\xymatrix{ \varphi(x') \ar[r] \ar[dr] & y \ar[d] \\ & \varphi(x) }$$
commutes.
\end{definition}

\begin{lemma}
\label{lemma-smoothness-small-extensions}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  Then $\varphi$ is smooth
if the condition in Definition \ref{definition-smooth-morphism} is assumed to
hold only for small extensions $B \to A$.
\end{lemma}

\begin{proof}
Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.
Let $y \in \Ob(\mathcal{G}(B))$, $x \in \Ob(\mathcal{F}(A))$,
and $y \to \varphi(x)$ be a morphism lying over $B \to A$. By
Lemma \ref{lemma-factor-small-extension} we can factor $B \to A$ into
small extensions $B = B_n \to B_{n-1} \to \ldots \to B_0 = A$.
We argue by induction on $n$. If $n = 1$ the result is true by assumption.
If $n > 1$, then denote $f : B = B_n \to B_{n - 1}$ and denote
$g : B_{n - 1} \to B_0 = A$. Choose a pushforward
$y \to f_* y$ of $y$ along $f$, so that the morphism $y \to \varphi(x)$
factors as $y \to f_* y \to \varphi(x)$. By the induction hypothesis
we can find $x_{n - 1} \to x$ lying over $g : B_{n - 1} \to A$ and
$a : \varphi(x_{n - 1}) \to f_*y$ lying over
$\text{id} : B_{n - 1} \to B_{n - 1}$ such that
$$\xymatrix{ \varphi(x_{n - 1}) \ar[r]_-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }$$
commutes. We can apply the assumption to the composition
$y \to \varphi(x_{n - 1})$ of
$y \to f_*y$ with $a^{-1} : f_*y \to \varphi(x_{n - 1})$. We obtain
$x_n \to x_{n - 1}$ lying over $B_n \to B_{n - 1}$ and
$\varphi(x_n) \to y$ lying over  $\text{id} : B_n \to B_n$ so that the diagram
$$\xymatrix{ \varphi(x_n) \ar[r] \ar[d] & y \ar[d] \\ \varphi(x_{n - 1}) \ar[r]^-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }$$
commutes. Then the composition $x_n \to x_{n - 1} \to x$ and
$\varphi(x_n) \to y$ are the morphisms required by the definition of
smoothness.
\end{proof}

\begin{remark}
\label{remark-smoothness-2-categorical}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.  Let $B \to A$ be a
ring map in $\mathcal{C}_\Lambda$.  Choices of pushforwards along $B \to A$ for objects in the fiber categories $\mathcal{F}(B)$ and
$\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$
and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative
diagram
$$\xymatrix{ \mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\ \mathcal{F}(A) \ar[r]^{\varphi} & \mathcal{G}(A) . }$$
Hence there is an induced functor $\mathcal{F}(B) \to \mathcal{F}(A) \times_{\mathcal{G}(A)} \mathcal{G}(B)$.  Unwinding the definitions shows that
$\varphi : \mathcal{F} \to \mathcal{G}$ is smooth if and only if this
induced functor is essentially surjective whenever $B \to A$ is
surjective (or equivalently, by
Lemma \ref{lemma-smoothness-small-extensions},
whenever $B \to A$ is a small extension).
\end{remark}

\begin{remark}
\label{remark-compare-smooth-schlessinger}
The characterization of smooth morphisms in
Remark \ref{remark-smoothness-2-categorical}
is analogous to Schlessinger's notion of
a smooth morphism of functors, cf.\ \cite[Definition 2.2.]{Sch}. In
fact, when $\mathcal{F}$ and $\mathcal{G}$ are cofibered in sets
then our notion is equivalent to Schlessinger's. Namely, in this case
let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding
functors, see
Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-convention-cofibered-sets}).
Then $F \to G$ is smooth if and only if for every surjection of rings
$B \to A$ in $\mathcal{C}_\Lambda$ the map $F(B) \to F(A) \times_{G(A)} G(B)$
is surjective.
\end{remark}

\begin{remark}
\label{remark-smooth-to-iso-classes}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Then the morphism
$\mathcal{F} \to \overline{\mathcal{F}}$ is smooth.
Namely, suppose that $f : B \to A$ is a ring map in $\mathcal{C}_\Lambda$.
Let $x \in \Ob(\mathcal{F}(A))$ and let
$\overline{y} \in \overline{\mathcal{F}}(B)$
be the isomorphism class of $y \in \Ob(\mathcal{F}(B))$ such that
$\overline{f_*y} = \overline{x}$. Then we simply take $x' = y$, the
implied morphism $x' = y \to x$ over $B \to A$, and the equality
$\overline{x'} = \overline{y}$ as the solution to
the problem posed in Definition \ref{definition-smooth-morphism}.
\end{remark}

\noindent
If $R \to S$ is a ring map $\widehat{\mathcal{C}}_\Lambda$, then there
is an induced morphism $\underline{S} \to \underline{R}$ between the
functors $\underline{S}, \underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$.  In this situation, smoothness of the
restriction $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is a familiar notion:

\begin{lemma}
\label{lemma-smooth-morphism-power-series}
Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Then
the induced morphism
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth if and only if $S$ is a power series ring over $R$.
\end{lemma}

\begin{proof}
Assume $S$ is a power series ring over $R$. Say $S = R[[x_1, \ldots, x_n]]$.
Smoothness of
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
means the following (see Remark \ref{remark-compare-smooth-schlessinger}):
Given a surjective ring map $B \to A$ in
$\mathcal{C}_\Lambda$, a ring map $R \to B$, a ring map $S \to A$ such that
the solid diagram
$$\xymatrix{ S \ar[r] \ar@{..>}[rd] & A \\ R \ar[u] \ar[r] & B \ar[u] }$$
is commutative then a dotted arrow exists making the diagram commute.
(Note the similarity with
Algebra, Definition \ref{algebra-definition-formally-smooth}.)
To construct the dotted arrow choose elements $b_i \in B$ whose images
in $A$ are equal to the images of $x_i$ in $A$. Note that
$b_i \in \mathfrak m_B$ as $x_i$ maps to an element of $\mathfrak m_A$.
Hence there is a unique $R$-algebra map $R[[x_1, \ldots, x_n]] \to B$
which maps $x_i$ to $b_i$ and which can serve as our dotted arrow.

\medskip\noindent
Conversely, assume
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth. Let $x_1, \ldots, x_n \in S$ be elements whose images
form a basis in the relative cotangent space
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ of $S$ over $R$.
Set $T = R[[X_1, \ldots, X_n]]$. Note that both
$$S/(\mathfrak m_R S + \mathfrak m_S^2) \cong R/\mathfrak m_R[x_1, \ldots, x_n]/(x_ix_j)$$
and
$$T/(\mathfrak m_R T + \mathfrak m_T^2) \cong R/\mathfrak m_R[X_1, \ldots, X_n]/(X_iX_j).$$
Let
$S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)$
be the local $R$-algebra isomorphism given by mapping
the class of $x_i$ to the class of $X_i$. Let
$f_1 : S \to T/(\mathfrak m_R T + \mathfrak m_T^2)$ be the
composition
$S \to S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)$.
The assumption that
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth means we can lift $f_1$ to a map
$f_2 : S \to T/\mathfrak{m}_T^2$, then to a map
$f_3 : S \to T/\mathfrak{m}_T^3$, and so on, for all $n \geq 1$. Thus
we get an induced map $f : S \to T = \lim T/\mathfrak m_T^n$
of local $R$-algebras. By our choice of $f_1$, the map $f$ induces an
isomorphism
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2) \to \mathfrak m_T/(\mathfrak m_R T + \mathfrak m_T^2)$
of relative cotangent spaces.
Hence $f$ is surjective by
Lemma \ref{lemma-surjective-cotangent-space}
(where we think of $f$ as a map in $\widehat{\mathcal{C}}_R$).
Choose preimages $y_i \in S$ of $X_i \in T$ under $f$. As $T$ is a
power series ring over $R$ there exists a local
$R$-algebra homomorphism $s : T \to S$ mapping $X_i$ to $y_i$.
By construction $f \circ s = \text{id}$. Then $s$ is injective.
But $s$ induces an isomorphism on relative cotangent spaces since
$f$ does, so it is also surjective by
Lemma \ref{lemma-surjective-cotangent-space}
again. Hence $s$ and $f$ are isomorphisms.
\end{proof}

\noindent
Smooth morphisms satisfy the following functorial properties.

\begin{lemma}
\label{lemma-smooth-properties}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G} \to \mathcal{H}$ be morphisms of categories cofibered in groupoids over
$\mathcal{C}_\Lambda$.
\begin{enumerate}
\item If $\varphi$ and $\psi$ are smooth, then $\psi \circ \varphi$ is smooth.
\item If $\varphi$ is essentially surjective and $\psi \circ \varphi$ is
smooth, then $\psi$ is smooth.
\item If $\mathcal{G}' \to \mathcal{G}$ is a morphism of categories
cofibered in groupoids and $\varphi$ is smooth, then
$\mathcal{F} \times_\mathcal{G} \mathcal{G}' \to \mathcal{G}'$ is smooth.
\end{enumerate}
\end{lemma}

\begin{proof}
Statements (1) and (2) follow immediately from the definitions.
Proof of (3) omitted. Hints: use the formulation of smoothness given in
Remark \ref{remark-smoothness-2-categorical}
and use that $\mathcal{F} \times_\mathcal{G} \mathcal{G}'$
is the $2$-fibre product, see
Remarks \ref{remarks-cofibered-groupoids} (\ref{item-fibre-product}).
\end{proof}

\begin{lemma}
\label{lemma-smooth-morphism-essentially-surjective}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of
categories cofibered in groupoids over $\mathcal{C}_\Lambda$.  Assume
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective.
Then $\varphi : \mathcal{F} \to \mathcal{G}$ and
$\widehat{\varphi} : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$
are essentially surjective.
\end{lemma}

\begin{proof}
Let $y$ be an object of $\mathcal{G}$ lying over
$A \in \Ob(\mathcal{C}_\Lambda)$. Let $y \to y_0$ be a pushforward
of $y$ along $A \to k$. By the assumption on essential surjectivity of
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ there exist an object
$x_0$ of $\mathcal{F}$ lying over $k$ and an isomorphism
$y_0 \to \varphi(x_0)$. Smoothness of $\varphi$ implies there exists
an object $x$ of $\mathcal{F}$ over $A$ whose image $\varphi(x)$
is isomorphic to $y$. Thus $\varphi : \mathcal{F} \to \mathcal{G}$
is essentially surjective.

\medskip\noindent
Let $\eta = (R, \eta_n, g_n)$ be an object of $\widehat{\mathcal{G}}$.  We
construct an object $\xi$ of $\widehat{\mathcal{F}}$ with an isomorphism
$\eta \to \varphi(\xi)$. By the assumption on essential surjectivity of
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$, there exists a morphism
$\eta_1 \to \varphi(\xi_1)$ in $\mathcal{G}(k)$ for some
$\xi_1 \in \Ob(\mathcal{F}(k))$. The morphism
$\eta_2 \xrightarrow{g_1} \eta_1 \to \varphi(\xi_1)$
lies over the surjective ring map $R/\mathfrak m_R^2 \to k$, hence
by smoothness of $\varphi$ there exists
$\xi_2 \in \Ob(\mathcal{F}(R/\mathfrak m_R^2))$, a
morphism $f_1: \xi_2 \to \xi_1$ lying over
$R/\mathfrak m_R^2 \to k$, and a morphism
$\eta_2 \to \varphi(\xi_2)$ such that
$$\xymatrix{ \varphi(\xi_2) \ar[r]^{\varphi(f_1)} & \varphi(\xi_{1}) \\ \eta_2 \ar[u] \ar[r]^{g_1} & \eta_1 \ar[u] \\ }$$
commutes. Continuing in this way we construct an object
$\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ and a morphism
$\eta \to \varphi(\xi) = (R, \varphi(\xi_n), \varphi(f_n))$
in $\widehat{\mathcal{G}}(R)$.
\end{proof}

\noindent
Later we are interested in producing smooth morphisms from
prorepresentable functors to predeformation categories $\mathcal{F}$.
By the discussion in
Remark \ref{remark-formal-objects-yoneda}
these morphisms correspond to certain formal objects of $\mathcal{F}$
More precisely, these are the so-called versal formal objects of $\mathcal{F}$.

\begin{definition}
\label{definition-versal}
Let $\mathcal{F}$ be a category cofibered in groupoids.  Let $\xi$ be a formal
object of $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
We say $\xi$ is {\it versal} if the corresponding morphism
$\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
of Remark \ref{remark-formal-objects-yoneda} is smooth.
\end{definition}

\begin{remark}
\label{remark-versal-object}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$.  It follows
from the definition of smoothness that versality of $\xi$ is equivalent to the
following condition: If
$$\xymatrix{ & y \ar[d] \\ \xi \ar[r] & x }$$
is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a
surjective map $B \to A$ of Artinian rings (we may assume it is a small
extension),  then there exists a morphism $\xi \to y$ such that
$$\xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x }$$
commutes. In particular, the condition that $\xi$ be versal does not depend on
the choices of pushforwards made in the construction of
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ in
Remark \ref{remark-formal-objects-yoneda}.
\end{remark}

\begin{lemma}
\label{lemma-versal-object-quasi-initial}
Let $\mathcal{F}$ be a predeformation category.
Let $\xi$ be a versal formal object of $\mathcal{F}$.
For any formal object $\eta$ of $\widehat{\mathcal{F}}$,
there exists a morphism $\xi \to \eta$.
\end{lemma}

\begin{proof}
By assumption the morphism
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
is smooth. Then
$\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$
is the completion of $\underline{\xi}$, see
Remark \ref{remark-formal-objects-yoneda}.
By
Lemma \ref{lemma-smooth-morphism-essentially-surjective}
there exists an object $f$ of $\underline{R}$ such that
$\iota(\xi)(f) = \eta$. Then $f$ is
a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$. And
$\iota(\xi)(f) = \eta$ means that
$f_*\xi \cong \eta$ which means exactly that there is a morphism
$\xi \to \eta$ lying over $f$.
\end{proof}

Comment #2638 by Xiaowen Hu on July 9, 2017 a 1:51 pm UTC

A typo: in the paragraph before definition 79.8.9, a period mark is missing after the curled F.

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