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. Let $X^\wedge _ Z$ be the formal completion of $X$ along $Z$.

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

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

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

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

Proof. Omitted. $\square$

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