Lemma 10.97.6 (0316)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.97: Completion for Noetherian rings
Lemma 10.97.6 (cite)

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Lemma 10.97.6. Let $R$ be a Noetherian ring. Let $I$ be an ideal of $R$ . The completion $R^\wedge$ of $R$ with respect to $I$ is Noetherian.

Proof. This is a consequence of Lemma 10.97.5. It can also be seen directly as follows. Choose generators $f_1, \ldots , f_ n$ of $I$ . Consider the map

$R[[x_1, \ldots , x_ n]] \longrightarrow R^\wedge , \quad x_ i \longmapsto f_ i.$

This is a well defined and surjective ring map (details omitted). Since $R[[x_1, \ldots , x_ n]]$ is Noetherian (see Lemma 10.31.2) we win. $\square$

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