The Stacks project

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 $T$.

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

  2. 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

Maybe it is worthwhile to include a Remark warning that if is affine and is the zero locus of a finitely generated ideal (so is quasi-compact) with radical then from the definitions for the -adic completion of whereas the natural map from to the -adic completion of can fail to be a ring isomorphism.

As an example, to illustrate the issues, one can mention the old standby for an algebraically closed field equipped with a rank-1 valuation, for a nonzero nonunit , and the maximal ideal. Since the -adic completion of is the residue field, whereas the -adic completion of is the valuation ring of the completion of for the given rank-1 valuation).

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AQ1. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 87.20.8 as pdf