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 79.4.6. Let $R$ be a topological ring. Let $M$ be a topological $R$-module. Let $I \subset R$ be a finitely generated ideal. Assume $M$ has an open submodule whose topology is $I$-adic. Then $M^\wedge $ has an open submodule whose topology is $I$-adic and we have $M^\wedge /I^ n M^\wedge = M/I^ nM$ for all $n \geq 1$.

Proof. Let $M' \subset M$ be an open submodule whose topology is $I$-adic. Then $\{ I^ nM'\} _{n \geq 1}$ is a fundamental system of open submodules of $M$. Thus $M^\wedge = \mathop{\mathrm{lim}}\nolimits M/I^ nM'$ contains $(M')^\wedge = \mathop{\mathrm{lim}}\nolimits M'/I^ nM'$ as an open submodule and the topology on $(M')^\wedge $ is $I$-adic by Algebra, Lemma 10.95.3. Since $I$ is finitely generated, $I^ n$ is finitely generated, say by $f_1, \ldots , f_ r$. Observe that the surjection $(f_1, \ldots , f_ r) : M^{\oplus r} \to I^ n M$ is continuous and open by our description of the topology on $M$ above. By Lemma 79.4.5 applied to this surjection and to the short exact sequence $0 \to I^ nM \to M \to M/I^ nM \to 0$ we conclude that

\[ (f_1, \ldots , f_ r) : (M^\wedge )^{\oplus r} \longrightarrow M^\wedge \]

surjects onto the kernel of the surjection $M^\wedge \to M/I^ nM$. Since $f_1, \ldots , f_ r$ generate $I^ n$ we conclude. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F1S. Beware of the difference between the letter 'O' and the digit '0'.