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*}

Lemma 10.95.6. Let $R$ be a ring, let $I \subset R$ be an ideal, and let $R^\wedge = \mathop{\mathrm{lim}}\nolimits R/I^ n$.

  1. any element of $R^\wedge $ which maps to a unit of $R/I$ is a unit,

  2. any element of $1 + I$ maps to an invertible element of $R^\wedge $,

  3. any element of $1 + IR^\wedge $ is invertible in $R^\wedge $, and

  4. the ideals $IR^\wedge $ and $\mathop{\mathrm{Ker}}(R^\wedge \to R/I)$ are contained in the Jacobson radical of $R^\wedge $.

Proof. Let $x \in R^\wedge $ map to a unit $x_1$ in $R/I$. Then $x$ maps to a unit $x_ n$ in $R/I^ n$ for every $n$ by Lemma 10.31.4. Hence $y = (x_ n^{-1}) \in \mathop{\mathrm{lim}}\nolimits R/I^ n = R^\wedge $ is an inverse to $x$. Parts (2) and (3) follow immediately from (1). Part (4) follows from (1) and Lemma 10.18.1. $\square$


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