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 $\mathcal{C}'$ (86.2.0.2) is Noetherian,

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

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

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

Proof. Part (4) follows from Algebra, Lemma 10.97.6 as $A[x_1, \ldots , x_ r]$ is Noetherian (Algebra, Lemma 10.31.1). To see (1) by Lemma 86.2.1 we reduce to the case of the completion of the polynomial ring which we just proved. Part (2) follows from Algebra, Lemma 10.97.1 which tells us that ever finite $B$-module is $IB$-adically complete. Part (3) follows in the same manner as part (4). $\square$


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