Lemma 86.14.6. Let $S$ be a scheme. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$. Let $T \subset X$ be a closed subset. Let $X_{/T}$ be the formal completion of $X$ along $T$.

If $X \setminus T$ is quasi-compact, i.e., $T$ is constructible, then $X_{/T}$ is adic*.

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

If $X$ is Noetherian, then $X_{/T}$ is Noetherian.

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