The Stacks project

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 $Z$.

  1. The formal algebraic space $X^\wedge _ Z$ is locally weakly adic.

  2. If $Z \to X$ is of finite presentation, then $X^\wedge _ Z$ is locally adic*.

  3. If $X$ is locally Noetherian, then $X_ Z$ is locally Noetherian.

Proof. Omitted. $\square$

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