The Stacks project

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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