Lemma 88.2.2. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Then

every object of the category $\mathcal{C}'$ (88.2.0.2) is Noetherian,

if $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ and $J \subset B$ is an ideal, then $B/J$ is an object of $\mathcal{C}'$,

for a finite type $A$-algebra $C$ the $I$-adic completion $C^\wedge $ is in $\mathcal{C}'$,

in particular the completion $A[x_1, \ldots , x_ r]^\wedge $ is in $\mathcal{C}'$.

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