The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Definition 10.154.1. 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 102.6.1. This does not happen when $\mathfrak m$ is finitely generated, see Lemma 10.95.3 in which case the completion is Noetherian, see Lemma 10.96.5.

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