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 10.97.1. Let $A$ be a ring. Let $I \subset A$ be an ideal. Let $(M_ n)$ be an inverse system of $A$-modules. Set $M = \mathop{\mathrm{lim}}\nolimits M_ n$. If $M_ n = M_{n + 1}/I^ nM_{n + 1}$ and $I$ is finitely generated then $M/I^ nM = M_ n$ and $M$ is $I$-adically complete.

Proof. As $M_{n + 1} \to M_ n$ is surjective, the map $M \to M_1$ is surjective. Pick $x_ t \in M$, $t \in T$ mapping to generators of $M_1$. This gives a map $\bigoplus _{t \in T} A \to M$. Note that the images of $x_ t$ in $M_ n$ generate $M_ n$ for all $n$ too. Consider the exact sequences

\[ 0 \to K_ n \to \bigoplus \nolimits _{t \in T} A/I^ n \to M_ n \to 0 \]

We claim the map $K_{n + 1} \to K_ n$ is surjective. Namely, if $y \in K_ n$ choose a lift $y' \in \bigoplus _{t \in T} A/I^{n + 1}$. Then $y'$ maps to an element of $I^ n M_{n + 1}$ by our assumption $M_ n = M_{n + 1}/I^ nM_{n + 1}$. Hence we can modify our choice of $y'$ by an element of $\bigoplus _{t \in T} I^ n/I^{n + 1}$ so that $y'$ maps to zero in $M_{n + 1}$. Then $y' \in K_{n +1}$ maps to $y$. Hence $(K_ n)$ is a sequence of modules with surjective transition maps and we obtain an exact sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits K_ n \to \left(\bigoplus \nolimits _{t \in T} A\right)^\wedge \to M \to 0 \]

by Lemma 10.86.1. Fix an integer $m$. As $I$ is finitely generated, the completion with respect to $I$ is complete and $(\bigoplus _{t \in T} A)^\wedge / I^ m(\bigoplus _{t \in T} A)^\wedge = \bigoplus _{t \in T} A/I^ m$ (Lemma 10.95.3). We obtain a short exact sequence

\[ (\mathop{\mathrm{lim}}\nolimits K_ n)/I^ m(\mathop{\mathrm{lim}}\nolimits K_ n) \to \bigoplus \nolimits _{t \in T} A/I^ m \to M/I^ mM \to 0 \]

Since $\mathop{\mathrm{lim}}\nolimits K_ n \to K_ m$ is surjective we conclude that $M/I^ mM = M_ m$. It follows in particular that $M$ is $I$-adically complete. $\square$

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