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 isomorphism^{1}.

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