Lemma 87.20.8 (0AQ1)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.20: Types of formal algebraic spaces
Lemma 87.20.8 (cite)

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Lemma 87.20.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $T \subset |X|$ be a closed subset. Let $X_{/T}$ be the formal completion of $X$ along $T$ .

If $X \setminus T \to X$ is quasi-compact, then $X_{/T}$ is locally adic*.
If $X$ is locally Noetherian, then $X_{/T}$ is locally Noetherian.

Proof. Choose a surjective étale morphism $U \to X$ with $U = \coprod U_ i$ a disjoint union of affine schemes, see Properties of Spaces, Lemma 66.6.1. Let $T_ i \subset U_ i$ be the inverse image of $T$ . We have $X_{/T} \times _ X U_ i = (U_ i)_{/T_ i}$ (Lemma 87.14.4). Hence $\{ (U_ i)_{/T_ i} \to X_{/T}\}$ is a covering as in Definition 87.11.1. Moreover, if $X \setminus T \to X$ is quasi-compact, so is $U_ i \setminus T_ i \to U_ i$ and if $X$ is locally Noetherian, so is $U_ i$ . Thus the lemma follows from the affine case which is Lemma 87.14.6. $\square$

Comments (2)

Comment #2029 by Brian Conrad on May 05, 2016 at 21:47

Maybe it is worthwhile to include a Remark warning that if $X = {\rm{Spec}}(A)$ is affine and $T$ is the zero locus of a finitely generated ideal $I$ (so $X-T$ is quasi-compact) with radical $J$ then from the definitions $X_{/T} = {\rm{Spf}}(A^{\wedge})$ for the $I$ -adic completion $A^{\wedge}$ of $A$ whereas the natural map from $A^{\wedge}$ to the $J$ -adic completion of $A$ can fail to be a ring isomorphism.

As an example, to illustrate the issues, one can mention the old standby $A = O_K$ for an algebraically closed field $K$ equipped with a rank-1 valuation, $I = (\pi)$ for a nonzero nonunit $\pi \in A$ , and $J = \mathfrak{m}$ the maximal ideal. Since $J^2=J$ the $J$ -adic completion of $A$ is the residue field, whereas the $I$ -adic completion of $A$ is the valuation ring of the completion of $K$ for the given rank-1 valuation).

Comment #2068 by Johan on June 16, 2016 at 12:39

Thanks very much. See changes here.

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