The Stacks project

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

  1. if $X_{red} \setminus T \to X_{red}$ is quasi-compact and $X$ is locally countably indexed, then $X_{/T}$ is locally countably indexed,

  2. if $X_{red} \setminus T \to X_{red}$ is quasi-compact and $X$ is locally adic*, then $X_{/T}$ is locally adic*, and

  3. 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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