## Tag `07X3`

## 88.9. Formal objects

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 $\mathop{\rm Spec}(R) \to S$.

Definition 88.9.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ 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 $\mathop{\rm Spec}(R/\mathfrak m_R^n)$, and morphisms $f_n : \xi_n \to \xi_{n + 1}$ of $\mathcal{X}$ lying over $\mathop{\rm Spec}(R/\mathfrak m^n) \to \mathop{\rm Spec}(R/\mathfrak m^{n + 1})$ 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 $$ \xymatrix{ \xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n } $$ is commutative. Applying the functor $p$ we obtain a compatible collection of morphisms $\mathop{\rm Spec}(R/\mathfrak m_R^n) \to \mathop{\rm Spec}(T/\mathfrak m_T^n)$ and hence a morphism $a_0 : \mathop{\rm Spec}(R) \to \mathop{\rm Spec}(T)$ over $S$. We say that $a$lies over$a_0$.

Thus we obtain a category of formal objects of $\mathcal{X}$.

Remark 88.9.2. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\xi = (R, \xi_n, f_n)$ be a formal object. Set $k = R/\mathfrak m$ and $x_0 = \xi_1$. The formal object $\xi$ defines a formal object $\xi$ of the predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$. This follows immediately from Definition 88.9.1 above, Formal Deformation Theory, Definition 80.7.1, and our construction of the predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$ in Section 88.3.

If $F : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$, then $F$ induces a functor between categories of formal objects as well.

Lemma 88.9.3. Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. 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 87.6), 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 $\mathop{\rm Spec}(R/\mathfrak m^n) \to \mathop{\rm Spec}(R/\mathfrak m^{n + 1})$ 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 $\eta_n|_{\mathop{\rm Spec}(R/\mathfrak m^{n - 1})} \cong \eta_{n - 1}$ and $F(\eta_n) \cong \xi_n$. $\square$Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ 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 $\xi_n = x|_{\mathop{\rm Spec}(R/\mathfrak m^n)}$ 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 \begin{equation} \tag{88.9.3.1} \left\{ \begin{matrix} \text{objects }x\text{ of }\mathcal{X} \text{ such that }p(x) = \mathop{\rm 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 88.9.1.

Definition 88.9.4. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. A formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{X}$ is called

effectiveif it is in the essential image of the functor (88.9.3.1).If the category fibred in groupoids is an algebraic stack, then every formal object is effective as follows from the next lemma.

Lemma 88.9.5. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. The functor (88.9.3.1) is an equivalence.

Proof.Case I: $\mathcal{X}$ is representable (by a scheme). Say $\mathcal{X} = (\textit{Sch}/X)_{fppf}$ 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 $$ \mathop{\rm Mor}\nolimits_S(\mathop{\rm Spec}(R), X) \longrightarrow \mathop{\rm lim}\nolimits \mathop{\rm Mor}\nolimits_S(\mathop{\rm Spec}(R/\mathfrak m^n), X) $$ is bijective. This follows from Formal Spaces, Lemma 77.26.2.Case II. $\mathcal{X}$ is representable by an algebraic space. Say $\mathcal{X}$ is representable by $X$. Again we have to show that $$ \mathop{\rm Mor}\nolimits_S(\mathop{\rm Spec}(R), X) \longrightarrow \mathop{\rm lim}\nolimits \mathop{\rm Mor}\nolimits_S(\mathop{\rm Spec}(R/\mathfrak m^n), X) $$ is bijective for $R$ as above. This is Formal Spaces, Lemma 77.26.3.

Case III: General case of an algebraic stack. A general remark is that the left and right hand side of (88.9.3.1) 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 $\mathop{\rm Spec}(R)$, see Algebraic Stacks, Lemma 84.10.11. Applying Case II to $I$ over $\mathop{\rm Spec}(R)$ implies immediately that (88.9.3.1) is fully faithful on fibre categories over $\mathop{\rm Spec}(R)$. Hence the functor is fully faithful by Categories, Lemma 4.34.8.

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 $f : (\textit{Sch}/U)_{fppf} \to \mathcal{X}$. For every $n$ consider the fibre product $$ (\textit{Sch}/\mathop{\rm Spec}(R/\mathfrak m^n))_{fppf} \times_{\xi_n, \mathcal{X}, f} (\textit{Sch}/U)_{fppf} $$ By assumption this is representable by an algebraic space $V_n$ surjective and smooth over $\mathop{\rm Spec}(R/\mathfrak m^n)$. The morphisms $f_n : \xi_n \to \xi_{n + 1}$ induce cartesian squares $$ \xymatrix{ V_{n + 1} \ar[d] & V_n \ar[d] \ar[l] \\ \mathop{\rm Spec}(R/\mathfrak m^{n + 1}) & \mathop{\rm Spec}(R/\mathfrak m^n) \ar[l] } $$ of algebraic spaces. By Spaces over Fields, Lemma 63.12.2 we can find a finite separable extension $k \subset k'$ and a point $v'_1 : \mathop{\rm Spec}(k') \to V_1$ over $k$. Let $R \subset R'$ be the finite étale extension whose residue field extension is $k \subset k'$ (exists and is unique by Algebra, Lemmas 10.148.7 and 10.148.9). By the infinitesimal lifting criterion of smoothness (see More on Morphisms of Spaces, Lemma 67.19.6) applied to $V_n \to \mathop{\rm Spec}(R/\mathfrak m^n)$ for $n = 2, 3, 4, \ldots$ we can successively find morphisms $v'_n : \mathop{\rm Spec}(R'/(\mathfrak m')^n) \to V_n$ over $\mathop{\rm Spec}(R/\mathfrak m^n)$ fitting into commutative diagrams $$ \xymatrix{ \mathop{\rm Spec}(R'/(\mathfrak m')^{n + 1}) \ar[d]_{v'_{n + 1}} & \mathop{\rm Spec}(R'/(\mathfrak m')^n) \ar[d]^{v'_n} \ar[l] \\ V_{n + 1} & V_n \ar[l] } $$ Composing with the projection morphisms $V_n \to U$ we obtain a compatible system of morphisms $u'_n : \mathop{\rm Spec}(R'/(\mathfrak m')^n) \to U$. By Case I the family $(u'_n)$ comes from a unique morphism $u' : \mathop{\rm Spec}(R') \to U$. Denote $x'$ the object of $\mathcal{X}$ over $\mathop{\rm Spec}(R')$ we get by applying the $1$-morphism $f$ to $u'$. By construction, there exists a morphism of formal objects $$ (88.9.3.1)(x') = (R', x'|_{\mathop{\rm Spec}(R'/(\mathfrak m')^n)}, \ldots) \longrightarrow (R, \xi_n, f_n) $$ lying over $\mathop{\rm Spec}(R') \to \mathop{\rm Spec}(R)$. Note that $R' \otimes_R R'$ is a finite product of spectra of Noetherian complete local rings to which our current discussion applies. Denote $p_0, p_1 : \mathop{\rm Spec}(R' \otimes_R R') \to \mathop{\rm Spec}(R')$ 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 $\mathop{\rm Spec}((R' \otimes_R R')/\mathfrak m^n(R' \otimes_R R'))$. We omit the proof that the isomorphism $\varphi$ satisfies the cocycle condition (see Stacks, Definition 8.3.1). Since $\{\mathop{\rm Spec}(R') \to \mathop{\rm Spec}(R)\}$ is an fppf covering we conclude that $x'$ descends to an object $x$ of $\mathcal{X}$ over $\mathop{\rm Spec}(R)$. We omit the proof that $x_n$ is the restriction of $x$ to $\mathop{\rm Spec}(R/\mathfrak m^n)$. $\square$

Lemma 88.9.6. 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 $(\textit{Sch}/S)_{fppf}$. If the functor (88.9.3.1) is an equivalence for $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, then it is and equivalence for $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.

Proof.The left and the right hand side of (88.9.3.1) for $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ are simply the $2$-fibre products of the left and the right hand side of (88.9.3.1) 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 4.30.7. $\square$

The code snippet corresponding to this tag is a part of the file `artin.tex` and is located in lines 836–1103 (see updates for more information).

```
\section{Formal objects}
\label{section-formal-objects}
\noindent
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 $\Spec(R) \to S$.
\begin{definition}
\label{definition-formal-objects}
Let $S$ be a locally Noetherian scheme. Let
$p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids.
\begin{enumerate}
\item A {\it 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 $\Spec(R/\mathfrak m_R^n)$, and morphisms
$f_n : \xi_n \to \xi_{n + 1}$ of $\mathcal{X}$ lying over
$\Spec(R/\mathfrak m^n) \to \Spec(R/\mathfrak m^{n + 1})$
such that $R/\mathfrak m$ is a field of finite type over $S$.
\item A {\it 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
$$
\xymatrix{
\xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\
\eta_{n + 1} \ar[r]^{g_n} & \eta_n
}
$$
is commutative. Applying the functor $p$ we obtain a compatible collection
of morphisms $\Spec(R/\mathfrak m_R^n) \to \Spec(T/\mathfrak m_T^n)$ and
hence a morphism $a_0 : \Spec(R) \to \Spec(T)$ over $S$. We say that
$a$ {\it lies over} $a_0$.
\end{enumerate}
\end{definition}
\noindent
Thus we obtain a category of formal objects of $\mathcal{X}$.
\begin{remark}
\label{remark-formal-objects-match}
Let $S$ be a locally Noetherian scheme. Let
$p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids.
Let $\xi = (R, \xi_n, f_n)$ be a formal object. Set $k = R/\mathfrak m$ and
$x_0 = \xi_1$. The formal object $\xi$ defines a formal object
$\xi$ of the predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$.
This follows immediately from
Definition \ref{definition-formal-objects} above,
Formal Deformation Theory, Definition
\ref{formal-defos-definition-formal-objects},
and our construction of the predeformation category
$\mathcal{F}_{\mathcal{X}, k, x_0}$ in
Section \ref{section-predeformation-categories}.
\end{remark}
\noindent
If $F : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred
in groupoids over $(\Sch/S)_{fppf}$, then $F$ induces a functor between
categories of formal objects as well.
\begin{lemma}
\label{lemma-smooth-lift-formal}
Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$
be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
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 \ref{criteria-section-formally-smooth}),
then there exists a formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{X}$
such that $F(\xi) \cong \eta$.
\end{lemma}
\begin{proof}
Note that each of the morphisms
$\Spec(R/\mathfrak m^n) \to \Spec(R/\mathfrak m^{n + 1})$ 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
$\eta_n|_{\Spec(R/\mathfrak m^{n - 1})} \cong \eta_{n - 1}$
and $F(\eta_n) \cong \xi_n$.
\end{proof}
\noindent
Let $S$ be a locally Noetherian scheme. Let
$p : \mathcal{X} \to (\Sch/S)_{fppf}$ 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
$\xi_n = x|_{\Spec(R/\mathfrak m^n)}$ 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
\begin{equation}
\label{equation-approximation}
\left\{
\begin{matrix}
\text{objects }x\text{ of }\mathcal{X} \text{ such that }p(x) = \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 \ref{definition-formal-objects}.
\begin{definition}
\label{definition-effective}
Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category
fibred in groupoids over $(\Sch/S)_{fppf}$. A formal object
$\xi = (R, \xi_n, f_n)$ of $\mathcal{X}$ is called {\it effective}
if it is in the essential image of the functor
(\ref{equation-approximation}).
\end{definition}
\noindent
If the category fibred in groupoids is an algebraic stack, then every
formal object is effective as follows from the next lemma.
\begin{lemma}
\label{lemma-effective}
Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be an algebraic
stack over $S$. The functor (\ref{equation-approximation}) is an equivalence.
\end{lemma}
\begin{proof}
Case I: $\mathcal{X}$ is representable (by a scheme). Say
$\mathcal{X} = (\Sch/X)_{fppf}$ 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
$$
\Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X)
$$
is bijective. This follows from Formal Spaces, Lemma
\ref{formal-spaces-lemma-map-into-scheme}.
\medskip\noindent
Case II. $\mathcal{X}$ is representable by an algebraic space. Say
$\mathcal{X}$ is representable by $X$. Again we have to show that
$$
\Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X)
$$
is bijective for $R$ as above. This is Formal Spaces, Lemma
\ref{formal-spaces-lemma-map-into-algebraic-space}.
\medskip\noindent
Case III: General case of an algebraic stack. A general remark is that
the left and right hand side of (\ref{equation-approximation}) 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.
\medskip\noindent
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 $\Spec(R)$, see
Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}.
Applying Case II to $I$ over $\Spec(R)$ implies immediately that
(\ref{equation-approximation}) is fully faithful on fibre categories over
$\Spec(R)$. Hence the functor is fully faithful by
Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories}.
\medskip\noindent
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
$f : (\Sch/U)_{fppf} \to \mathcal{X}$. For every $n$ consider the fibre product
$$
(\Sch/\Spec(R/\mathfrak m^n))_{fppf}
\times_{\xi_n, \mathcal{X}, f}
(\Sch/U)_{fppf}
$$
By assumption this is representable by an algebraic space $V_n$ surjective and
smooth over $\Spec(R/\mathfrak m^n)$. The morphisms
$f_n : \xi_n \to \xi_{n + 1}$ induce cartesian squares
$$
\xymatrix{
V_{n + 1} \ar[d] & V_n \ar[d] \ar[l] \\
\Spec(R/\mathfrak m^{n + 1}) & \Spec(R/\mathfrak m^n) \ar[l]
}
$$
of algebraic spaces. By Spaces over Fields, Lemma
\ref{spaces-over-fields-lemma-smooth-separable-closed-points-dense}
we can find a finite separable extension $k \subset k'$ and a point
$v'_1 : \Spec(k') \to V_1$ over $k$. Let $R \subset R'$ be the finite \'etale
extension whose residue field extension is $k \subset k'$ (exists and
is unique by
Algebra, Lemmas \ref{algebra-lemma-henselian-cat-finite-etale} and
\ref{algebra-lemma-complete-henselian}).
By the infinitesimal lifting criterion of smoothness (see
More on Morphisms of Spaces, Lemma
\ref{spaces-more-morphisms-lemma-smooth-formally-smooth})
applied to $V_n \to \Spec(R/\mathfrak m^n)$ for $n = 2, 3, 4, \ldots$
we can successively find morphisms
$v'_n : \Spec(R'/(\mathfrak m')^n) \to V_n$ over $\Spec(R/\mathfrak m^n)$
fitting into commutative diagrams
$$
\xymatrix{
\Spec(R'/(\mathfrak m')^{n + 1}) \ar[d]_{v'_{n + 1}} &
\Spec(R'/(\mathfrak m')^n) \ar[d]^{v'_n} \ar[l] \\
V_{n + 1} & V_n \ar[l]
}
$$
Composing with the projection morphisms $V_n \to U$ we obtain a compatible
system of morphisms $u'_n : \Spec(R'/(\mathfrak m')^n) \to U$.
By Case I the family $(u'_n)$ comes from a unique
morphism $u' : \Spec(R') \to U$. Denote $x'$ the object of $\mathcal{X}$
over $\Spec(R')$ we get by applying the $1$-morphism $f$ to $u'$.
By construction, there exists a morphism of formal objects
$$
(\ref{equation-approximation})(x') =
(R', x'|_{\Spec(R'/(\mathfrak m')^n)}, \ldots)
\longrightarrow
(R, \xi_n, f_n)
$$
lying over $\Spec(R') \to \Spec(R)$. Note that $R' \otimes_R R'$ is a finite
product of spectra of Noetherian complete local rings to which our current
discussion applies. Denote $p_0, p_1 : \Spec(R' \otimes_R R') \to \Spec(R')$
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
$\Spec((R' \otimes_R R')/\mathfrak m^n(R' \otimes_R R'))$.
We omit the proof that the isomorphism $\varphi$ satisfies the cocycle
condition (see Stacks, Definition \ref{stacks-definition-descent-data}).
Since $\{\Spec(R') \to \Spec(R)\}$ is an fppf covering we conclude
that $x'$ descends to an object $x$ of $\mathcal{X}$ over $\Spec(R)$.
We omit the proof that $x_n$ is the restriction of $x$ to
$\Spec(R/\mathfrak m^n)$.
\end{proof}
\begin{lemma}
\label{lemma-fibre-product-effective}
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 $(\Sch/S)_{fppf}$. If the functor
(\ref{equation-approximation}) is an equivalence for
$\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, then it is
and equivalence for $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.
\end{lemma}
\begin{proof}
The left and the right hand side of (\ref{equation-approximation})
for $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ are simply the $2$-fibre
products of the left and the right hand side of (\ref{equation-approximation})
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 \ref{categories-lemma-equivalence-2-fibre-product}.
\end{proof}
```

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