The Stacks project

88.8 Smooth morphisms

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

Definition 88.8.1. 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 $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{G}(B)), x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$, and $y \to \varphi (x)$ be a morphism lying over $B \to A$. Then there exists $x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(B))$, 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 88.8.2. 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 88.8.1 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 $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{G}(B))$, $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$, and $y \to \varphi (x)$ be a morphism lying over $B \to A$. By Lemma 88.3.3 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 88.8.3. 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 88.8.2, whenever $B \to A$ is a small extension).

Remark 88.8.4. The characterization of smooth morphisms in Remark 88.8.3 is analogous to Schlessinger's notion of a smooth morphism of functors, cf. [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 88.5.2 (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 88.8.5. 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 $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$ and let $\overline{y} \in \overline{\mathcal{F}}(B)$ be the isomorphism class of $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(B))$ 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 88.8.1.

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 88.8.6. 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 88.8.4): 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 10.138.1.) 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 $f : S \to T = \mathop{\mathrm{lim}}\nolimits T/\mathfrak m_ T^ n$ 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 88.4.2 (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 88.4.2 again. Hence $s$ and $f$ are isomorphisms. $\square$

Smooth morphisms satisfy the following functorial properties.

Lemma 88.8.7. 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 88.8.3 and use that $\mathcal{F} \times _\mathcal {G} \mathcal{G}'$ is the $2$-fibre product, see Remarks 88.5.2 (13). $\square$

Lemma 88.8.8. 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 $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$. 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 $\xi _1 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. 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 $\xi _2 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(R/\mathfrak m_ R^2))$, 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 88.7.12 these morphisms correspond to certain formal objects of $\mathcal{F}$. More precisely, these are the so-called versal formal objects of $\mathcal{F}$.

Definition 88.8.9. Let $\mathcal{F}$ be a category cofibered in groupoids. Let $\xi $ be a formal object of $\mathcal{F}$ lying over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. We say $\xi $ is versal if the corresponding morphism $\underline{\xi }: \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ of Remark 88.7.12 is smooth.

Remark 88.8.10. 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 88.7.12.

Lemma 88.8.11. 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 88.7.12. By Lemma 88.8.8 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 $f$. $\square$


Comments (2)

Comment #2638 by Xiaowen Hu on

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


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