The Stacks project

[Proposition 7.5.5, EGA1]

Lemma 86.2.2. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Then

  1. every object of the category $\mathcal{C}'$, in particular the completion $A[x_1, \ldots , x_ r]^\wedge $, is Noetherian,

  2. if $B$ is an object of $\mathcal{C}'$ and $J \subset B$ is an ideal, then $B/J$ is an object of $\mathcal{C}'$.

Proof. To see (1) by Lemma 86.2.1 we reduce to the case of the completion of the polynomial ring. This case follows from Algebra, Lemma 10.96.6 as $A[x_1, \ldots , x_ r]$ is Noetherian (Algebra, Lemma 10.30.1). Part (2) follows from Algebra, Lemma 10.96.1 which tells us that ever finite $B$-module is $IB$-adically complete. $\square$

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