Section 87.38 (0GXT): Completion along a closed subspace

87.38 Completion along a closed subspace

This section is the analgue of Section 87.14 for completions with respect to a closed subspace.

Definition 87.38.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $Z \subset X$ be a closed subspace and denote $Z_ n \subset X$ the $n$ th order infinitesimal neighbourhood. The formal algebraic space

$X^\wedge _ Z = \mathop{\mathrm{colim}}\nolimits Z_ n$

(see Lemma 87.36.2) is called the completion of $X$ along $Z$ .

Observe that if $T = |Z|$ then there is a canonical morphism $X^\wedge _ Z \to X_{/T}$ comparing the completions along $Z$ and $T$ (Section 87.14) which need not be an isomorphism.

Let $f : X \to X'$ be a morphism of algebraic spaces over a scheme $S$ . Suppose that $Z \subset X$ and $Z' \subset X'$ are closed subspaces such that $f|_ Z$ maps $Z$ into $Z'$ inducing a morphism $Z \to Z'$ . Then it is clear that $f$ defines a morphism of formal algebraic spaces

$X^\wedge _ Z \longrightarrow (X')^\wedge _{Z'}$

between the completions.

Lemma 87.38.2. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$ . Let $Z \subset X$ be a closed subspace and let $Z' = f^{-1}(Z) = X' \times _ X Z$ . Then

$\xymatrix{ (X')^\wedge _{Z'} \ar[r] \ar[d] & X' \ar[d]^ f \\ X^\wedge _ Z \ar[r] & X }$

is a cartesian diagram of sheaves. In particular, the morphism $(X')^\wedge _{Z'} \to X^\wedge _ Z$ is representable by algebraic spaces.

Proof. Namely, suppose that $Y \to X$ is a morphism from a scheme into $X$ such that $Y \to X$ factors through $Z$ . Then $Y \times _ X X' \to X$ is a morphism of algebraic spaces such that $Y \times _ X X' \to X'$ factors through $Z'$ . Since $Z'_ n = X' \times _ X Z_ n$ for all $n \geq 1$ the same is true for the infinitesimal neighbourhoods. Hence the cartesian square of functors follows from the formulas $X^\wedge _ Z = \mathop{\mathrm{colim}}\nolimits Z_ n$ and $(X')^\wedge _{Z'} = \mathop{\mathrm{colim}}\nolimits Z'_ n$ . $\square$

Lemma 87.38.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $Z \subset X$ be a closed subspace. The reduction $(X^\wedge _ Z)_{red}$ of the completion $X^\wedge _ Z$ of $X$ along $Z$ is $Z_{red}$ .

Proof. Omitted. $\square$

Lemma 87.38.4. Let $S$ be a scheme. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$ . Let $Z \subset X$ be a closed subscheme. Let $X^\wedge _ Z$ be the formal completion of $X$ along $Z$ .

The affine formal algebraic space $X^\wedge _ Z$ is weakly adic.
If $Z \to X$ is of finite presentation, then $X^\wedge _ Z$ is adic*.
If $Z = V(I)$ for some finitely generated ideal $I \subset A$ , then $X^\wedge _ Z = \text{Spf}(A^\wedge )$ where $A^\wedge$ is the $I$ -adic completion of $A$ .
If $X$ is Noetherian, then $X^\wedge _ Z$ is Noetherian.

Proof. Omitted. $\square$

Lemma 87.38.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $Z \subset X$ be a closed subspace. Let $X^\wedge _ Z$ be the formal completion of $X$ along $Z$ .

The formal algebraic space $X^\wedge _ Z$ is locally weakly adic.
If $Z \to X$ is of finite presentation, then $X^\wedge _ Z$ is locally adic*.
If $X$ is locally Noetherian, then $X_ Z$ is locally Noetherian.

Proof. Omitted. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

The Stacks project

87.38 Completion along a closed subspace

Comments (0)

Post a comment