In this section we transfer some of the notions already defined in the chapter “Formal Deformation Theory” to the current setting. In the following we will say “$R$ is an $S$-algebra” to indicate that $R$ is a ring endowed with a morphism of schemes $\mathop{\mathrm{Spec}}(R) \to S$.
Definition 98.9.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.
A formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{X}$ consists of a Noetherian complete local $S$-algebra $R$, objects $\xi _ n$ of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ n)$, and morphisms $f_ n : \xi _ n \to \xi _{n + 1}$ of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \to \mathop{\mathrm{Spec}}(R/\mathfrak m^{n + 1})$ such that $R/\mathfrak m$ is a field of finite type over $S$.
A morphism of formal objects $a : \xi = (R, \xi _ n, f_ n) \to \eta = (T, \eta _ n, g_ n)$ is given by morphisms $a_ n : \xi _ n \to \eta _ n$ such that for every $n$ the diagram
\[ \xymatrix{ \xi _ n \ar[r]_{f_ n} \ar[d]_{a_ n} & \xi _{n + 1} \ar[d]^{a_{n + 1}} \\ \eta _ n \ar[r]^{g_ n} & \eta _{n + 1} } \]
is commutative. Applying the functor $p$ we obtain a compatible collection of morphisms $\mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ n) \to \mathop{\mathrm{Spec}}(T/\mathfrak m_ T^ n)$ and hence a morphism $a_0 : \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(T)$ over $S$. We say that $a$ lies over $a_0$.
Thus we obtain a category of formal objects of $\mathcal{X}$.
If $F : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, then $F$ induces a functor between categories of formal objects as well.
Lemma 98.9.3. Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\eta = (R, \eta _ n, g_ n)$ be a formal object of $\mathcal{Y}$ and let $\xi _1$ be an object of $\mathcal{X}$ with $F(\xi _1) \cong \eta _1$. If $F$ is formally smooth on objects (see Criteria for Representability, Section 97.6), then there exists a formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{X}$ such that $F(\xi ) \cong \eta $.
Proof.
Note that each of the morphisms $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \to \mathop{\mathrm{Spec}}(R/\mathfrak m^{n + 1})$ is a first order thickening of affine schemes over $S$. Hence the assumption on $F$ means that we can successively lift $\xi _1$ to objects $\xi _2, \xi _3, \ldots $ of $\mathcal{X}$ endowed with compatible isomorphisms $\eta _ n|_{\mathop{\mathrm{Spec}}(R/\mathfrak m^{n - 1})} \cong \eta _{n - 1}$ and $F(\eta _ n) \cong \xi _ n$.
$\square$
Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Suppose that $x$ is an object of $\mathcal{X}$ over $R$, where $R$ is a Noetherian complete local $S$-algebra with residue field of finite type over $S$. Then we can consider the system of restrictions $\xi _ n = x|_{\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)}$ endowed with the natural morphisms $\xi _1 \to \xi _2 \to \ldots $ coming from transitivity of restriction. Thus $\xi = (R, \xi _ n, \xi _ n \to \xi _{n + 1})$ is a formal object of $\mathcal{X}$. This construction is functorial in the object $x$. Thus we obtain a functor
98.9.3.1
\begin{equation} \label{artin-equation-approximation} \left\{ \begin{matrix} \text{objects }x\text{ of }\mathcal{X} \text{ such that }p(x) = \mathop{\mathrm{Spec}}(R)
\\ \text{where }R\text{ is Noetherian complete local}
\\ \text{with }R/\mathfrak m\text{ of finite type over }S
\end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{formal objects of }\mathcal{X}
\end{matrix} \right\} \end{equation}
To be precise the left hand side is the full subcategory of $\mathcal{X}$ consisting of objects as indicated and the right hand side is the category of formal objects of $\mathcal{X}$ as in Definition 98.9.1.
Definition 98.9.4. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. A formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{X}$ is called effective if it is in the essential image of the functor (98.9.3.1).
If the category fibred in groupoids is an algebraic stack, then every formal object is effective as follows from the next lemma.
Lemma 98.9.5. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. The functor (98.9.3.1) is an equivalence.
Proof.
Case I: $\mathcal{X}$ is representable (by a scheme). Say $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ for some scheme $X$ over $S$. Unwinding the definitions we have to prove the following: Given a Noetherian complete local $S$-algebra $R$ with $R/\mathfrak m$ of finite type over $S$ we have
\[ \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R), X) \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R/\mathfrak m^ n), X) \]
is bijective. This follows from Formal Spaces, Lemma 87.33.2.
Case II. $\mathcal{X}$ is representable by an algebraic space. Say $\mathcal{X}$ is representable by $X$. Again we have to show that
\[ \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R), X) \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R/\mathfrak m^ n), X) \]
is bijective for $R$ as above. This is Formal Spaces, Lemma 87.33.3.
Case III: General case of an algebraic stack. A general remark is that the left and right hand side of (98.9.3.1) are categories fibred in groupoids over the category of affine schemes over $S$ which are spectra of Noetherian complete local rings with residue field of finite type over $S$. We will also see in the proof below that they form stacks for a certain topology on this category.
We first prove fully faithfulness. Let $R$ be a Noetherian complete local $S$-algebra with $k = R/\mathfrak m$ of finite type over $S$. Let $x, x'$ be objects of $\mathcal{X}$ over $R$. As $\mathcal{X}$ is an algebraic stack $\mathit{Isom}(x, x')$ is representable by an algebraic space $I$ over $\mathop{\mathrm{Spec}}(R)$, see Algebraic Stacks, Lemma 94.10.11. Applying Case II to $I$ over $\mathop{\mathrm{Spec}}(R)$ implies immediately that (98.9.3.1) is fully faithful on fibre categories over $\mathop{\mathrm{Spec}}(R)$. Hence the functor is fully faithful by Categories, Lemma 4.35.9.
Essential surjectivity. Let $\xi = (R, \xi _ n, f_ n)$ be a formal object of $\mathcal{X}$. Choose a scheme $U$ over $S$ and a surjective smooth morphism $f : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$. For every $n$ consider the fibre product
\[ (\mathit{Sch}/\mathop{\mathrm{Spec}}(R/\mathfrak m^ n))_{fppf} \times _{\xi _ n, \mathcal{X}, f} (\mathit{Sch}/U)_{fppf} \]
By assumption this is representable by an algebraic space $V_ n$ surjective and smooth over $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$. The morphisms $f_ n : \xi _ n \to \xi _{n + 1}$ induce cartesian squares
\[ \xymatrix{ V_{n + 1} \ar[d] & V_ n \ar[d] \ar[l] \\ \mathop{\mathrm{Spec}}(R/\mathfrak m^{n + 1}) & \mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \ar[l] } \]
of algebraic spaces. By Spaces over Fields, Lemma 72.16.2 we can find a finite separable extension $k'/k$ and a point $v'_1 : \mathop{\mathrm{Spec}}(k') \to V_1$ over $k$. Let $R \subset R'$ be the finite étale extension whose residue field extension is $k'/k$ (exists and is unique by Algebra, Lemmas 10.153.7 and 10.153.9). By the infinitesimal lifting criterion of smoothness (see More on Morphisms of Spaces, Lemma 76.19.6) applied to $V_ n \to \mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$ for $n = 2, 3, 4, \ldots $ we can successively find morphisms $v'_ n : \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n) \to V_ n$ over $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$ fitting into commutative diagrams
\[ \xymatrix{ \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^{n + 1}) \ar[d]_{v'_{n + 1}} & \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n) \ar[d]^{v'_ n} \ar[l] \\ V_{n + 1} & V_ n \ar[l] } \]
Composing with the projection morphisms $V_ n \to U$ we obtain a compatible system of morphisms $u'_ n : \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n) \to U$. By Case I the family $(u'_ n)$ comes from a unique morphism $u' : \mathop{\mathrm{Spec}}(R') \to U$. Denote $x'$ the object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(R')$ we get by applying the $1$-morphism $f$ to $u'$. By construction, there exists a morphism of formal objects
\[ (07X6)(x') = (R', x'|_{\mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n)}, \ldots ) \longrightarrow (R, \xi _ n, f_ n) \]
lying over $\mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)$. Note that $R' \otimes _ R R'$ is a finite product of spectra of Noetherian complete local rings to which our current discussion applies. Denote $p_0, p_1 : \mathop{\mathrm{Spec}}(R' \otimes _ R R') \to \mathop{\mathrm{Spec}}(R')$ the two projections. By the fully faithfulness shown above there exists a canonical isomorphism $\varphi : p_0^*x' \to p_1^*x'$ because we have such isomorphisms over $\mathop{\mathrm{Spec}}((R' \otimes _ R R')/\mathfrak m^ n(R' \otimes _ R R'))$. We omit the proof that the isomorphism $\varphi $ satisfies the cocycle condition (see Stacks, Definition 8.3.1). Since $\{ \mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)\} $ is an fppf covering we conclude that $x'$ descends to an object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(R)$. We omit the proof that $x_ n$ is the restriction of $x$ to $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$.
$\square$
Lemma 98.9.6. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If the functor (98.9.3.1) is an equivalence for $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, then it is an equivalence for $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.
Proof.
The left and the right hand side of (98.9.3.1) for $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ are simply the $2$-fibre products of the left and the right hand side of (98.9.3.1) for $\mathcal{X}$, $\mathcal{Z}$ over $\mathcal{Y}$. Hence the result follows as taking $2$-fibre products is compatible with equivalences of categories, see Categories, Lemma 4.31.7.
$\square$
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