Definition 10.160.1. Let $(R, \mathfrak m)$ be a local ring. We say $R$ is a complete local ring if the canonical map

$R \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n R/\mathfrak m^ n$

to the completion of $R$ with respect to $\mathfrak m$ is an isomorphism1.

[1] This includes the condition that $\bigcap \mathfrak m^ n = (0)$; in some texts this may be indicated by saying that $R$ is complete and separated. Warning: It can happen that the completion $\mathop{\mathrm{lim}}\nolimits _ n R/\mathfrak m^ n$ of a local ring is non-complete, see Examples, Lemma 108.7.1. This does not happen when $\mathfrak m$ is finitely generated, see Lemma 10.96.3 in which case the completion is Noetherian, see Lemma 10.97.5.

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