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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.

  1. 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 $S$.
  2. 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 effective if 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}

    Comments (3)

    Comment #1589 by Ariyan on July 30, 2015 a 10:42 am UTC

    Typo: In (1) of Definition 78.9.1 the $f_n$ are missing. (It should be "and morphisms $f_n:\xi_n \to \xi_{n + 1}$ of $\mathcal{X}$ lying")

    Comment #1590 by Ariyan on July 30, 2015 a 10:51 am UTC

    Typo: After Definition 9.1, an s is missing: "Thus we obtain a category of formal object(s)".

    Comment #2781 by Ariyan Javanpeykar on August 20, 2017 a 5:19 pm UTC

    You refer to "Formal spaces" several times. Should that be "Formal algebraic spaces"?

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