The Stacks project

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 corresponding to the ideal $I \subset A$. Then

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

2. If $I$ is finitely generated, then $X^\wedge _ Z = \text{Spf}(A^\wedge )$ where $A^\wedge $ is the $I$-adic completion of $A$.

3. If $Z \to X$ is of finite presentation, i.e., $I$ is finitely generated, then $X^\wedge _ Z$ is adic*.

4. If $X$ is Noetherian, then $X^\wedge _ Z$ is Noetherian.