In this section we define the completion of a formal algebraic space along a closed subset of its reduction. It is the natural generalization of Section 87.14.
Lemma 87.37.1. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Let $T \subset |X_{red}|$ be a closed subset. Then the functor
is an affine formal algebraic space.
Proof. Write $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 87.9.1. Then $X_{\lambda , red} = X_{red}$ and we may and do view $T$ as a closed subset of $|X_\lambda | = |X_{\lambda , red}|$. By Lemma 87.14.1 for each $\lambda $ the completion $(X_\lambda )_{/T}$ is an affine formal algebraic space. The transition morphisms $(X_\lambda )_{/T} \to (X_\mu )_{/T}$ are closed immersions as base changes of the transition morphisms $X_\lambda \to X_\mu $, see Lemma 87.14.4. Also the morphisms $((X_\lambda )_{/T})_{red} \to ((X_\mu )_/T)_{red}$ are isomorphisms by Lemma 87.14.5. Since $X_{/T} = \mathop{\mathrm{colim}}\nolimits (X_\lambda )_{/T}$ we conclude by Lemma 87.36.1. $\square$
Lemma 87.37.2. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $T \subset |X_{red}|$ be a closed subset. Then the functor
is a formal algebraic space.
Proof. The functor $X_{/T}$ is an fppf sheaf since if $\{ U_ i \to U\} $ is an fppf covering, then $\coprod |U_ i| \to |U|$ is surjective.
Choose a covering $\{ g_ i : X_ i \to X\} _{i \in I}$ as in Definition 87.11.1. The morphisms $X_ i \times _ X X_{/T} \to X_{/T}$ are étale (see Spaces, Lemma 65.5.5) and the map $\coprod X_ i \times _ X X_{/T} \to X_{/T}$ is a surjection of sheaves. Thus it suffices to prove that $X_{/T} \times _ X X_ i$ is an affine formal algebraic space. A $U$-valued point of $X_ i \times _ X X_{/T}$ is a morphism $U \to X_ i$ whose image is contained in the closed subset $|g_{i, red}|^{-1}(T) \subset |X_{i, red}|$. Thus this follows from Lemma 87.37.1. $\square$
Definition 87.37.3. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $T \subset |X_{red}|$ be a closed subset. The formal algebraic space $X_{/T}$ of Lemma 87.14.2 is called the completion of $X$ along $T$.
Let $f : X \to X'$ be a morphism of formal algebraic spaces over a scheme $S$. Suppose that $T \subset |X_{red}|$ and $T' \subset |X'_{red}|$ are closed subsets such that $|f_{red}|(T) \subset T'$. Then it is clear that $f$ defines a morphism of formal algebraic spaces
between the completions.
Lemma 87.37.4. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of formal algebraic spaces over $S$. Let $T \subset |X_{red}|$ be a closed subset and let $T' = |f_{red}|^{-1}(T) \subset |X'_{red}|$. Then
is a cartesian diagram of formal algebraic spaces over $S$.
Proof. Namely, observe that the horizontal arrows are monomorphisms by construction. Thus it suffices to show that a morphism $g : U \to X'$ from a scheme $U$ defines a point of $X'_{/T}$ if and only if $f \circ g$ defines a point of $X_{/T}$. In other words, we have to show that $g(U)$ is contained in $T' \subset |X'_{red}|$ if and only if $(f \circ g)(U)$ is contained in $T \subset |X_{red}|$. This follows immediately from our choice of $T'$ as the inverse image of $T$. $\square$
Lemma 87.37.5. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $T \subset |X_{red}|$ be a closed subset. The reduction $(X_{/T})_{red}$ of the completion $X_{/T}$ of $X$ along $T$ is the reduced induced closed subspace $Z$ of $X_{red}$ corresponding to $T$.
Proof. It follows from Lemma 87.12.1, Properties of Spaces, Definition 66.12.5 (which uses Properties of Spaces, Lemma 66.12.3 to construct $Z$), and the definition of $X_{/T}$ that $Z$ and $(X_{/T})_{red}$ are reduced algebraic spaces characterized the same mapping property: a morphism $g : Y \to X$ whose source is a reduced algebraic space factors through them if and only if $|Y|$ maps into $T \subset |X|$. $\square$
Lemma 87.37.6. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Let $T \subset X_{red}$ be a closed subset and let $X_{/T}$ be the formal completion of $X$ along $T$. Then
$X_{/T}$ is an affine formal algebraic space,
if $X$ is McQuillan, then $X_{/T}$ is McQuillan,
if $|X_{red}| \setminus T$ is quasi-compact and $X$ is countably indexed, then $X_{/T}$ is countably indexed,
if $|X_{red}| \setminus T$ is quasi-compact and $X$ is adic*, then $X_{/T}$ is adic*,
if $X$ is Noetherian, then $X_{/T}$ is Noetherian.
Proof. Part (1) is Lemma 87.37.1. If $X$ is McQuillan, then $X = \text{Spf}(A)$ for some weakly admissible topological ring $A$. Then $X_{/T} \to X \to \mathop{\mathrm{Spec}}(A)$ satisfies property (2) of Lemma 87.9.6 and hence $X_{/T}$ is McQuillan, see Definition 87.9.7.
Assume $X$ and $T$ are as in (3). Then $X = \text{Spf}(A)$ where $A$ has a fundamental system $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots $ of weak ideals of definition, see Lemma 87.10.4. By Algebra, Lemma 10.29.1 we can find a finitely generated ideal $\overline{J} = (\overline{f}_1, \ldots , \overline{f}_ r) \subset A/I_1$ such that $T$ is cut out by $\overline{J}$ inside $\mathop{\mathrm{Spec}}(A/I_1) = |X_{red}|$. Choose $f_ i \in A$ lifting $\overline{f}_ i$. If $Z = \mathop{\mathrm{Spec}}(B)$ is an affine scheme and $g : Z \to X$ is a morphism with $g(Z) \subset T$ (set theoretically), then $g^\sharp : A \to B$ factors through $A/I_ n$ for some $n$ and $g^\sharp (f_ i)$ is nilpotent in $B$ for each $i$. Thus $J_{m, n} = (f_1, \ldots , f_ r)^ m + I_ n$ maps to zero in $B$ for some $n, m \geq 1$. It follows that $X_{/T}$ is the formal spectrum of $\mathop{\mathrm{lim}}\nolimits _{n, m} A/J_{m, n}$ and hence countably indexed. This proves (3).
Proof of (4). Here the argument is the same as in (3). However, here we may choose $I_ n = I^ n$ for some finitely generated ideal $I \subset A$. Then it is clear that $X_{/T}$ is the formal spectrum of $\mathop{\mathrm{lim}}\nolimits A/J^ n$ where $J = (f_1, \ldots , f_ r) + I$. Some details omitted.
Proof of (5). In this case $X_{red}$ is the spectrum of a Noetherian ring and hence the assumption that $|X_{red}| \setminus T$ is quasi-compact is satisfied. Thus as in the proof of (4) we see that $X_{/T}$ is the spectrum of $\mathop{\mathrm{lim}}\nolimits A/J^ n$ which is a Noetherian adic topological ring, see Algebra, Lemma 10.97.6. $\square$
Lemma 87.37.7. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $T \subset X_{red}$ be a closed subset and let $X_{/T}$ be the formal completion of $X$ along $T$. Then
if $X_{red} \setminus T \to X_{red}$ is quasi-compact and $X$ is locally countably indexed, then $X_{/T}$ is locally countably indexed,
if $X_{red} \setminus T \to X_{red}$ is quasi-compact and $X$ is locally adic*, then $X_{/T}$ is locally adic*, and
if $X$ is locally Noetherian, then $X_{/T}$ is locally Noetherian.
Proof. Choose a covering $\{ X_ i \to X\} $ as in Definition 87.11.1. Let $T_ i \subset X_{i, red}$ be the inverse image of $T$. We have $X_ i \times _ X X_{/T} = (X_ i)_{/T_ i}$ (Lemma 87.37.4). Hence $\{ (X_ i)_{/T_ i} \to X_{/T}\} $ is a covering as in Definition 87.11.1. Moreover, if $X_{red} \setminus T \to X_{red}$ is quasi-compact, so is $X_{i, red} \setminus T_ i \to X_{i, red}$ and if $X$ is locally countably indexed, or locally adic*, pr locally Noetherian, the is $X_ i$ is countably index, or adic*, or Noetherian. Thus the lemma follows from the affine case which is Lemma 87.37.6. $\square$
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