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\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 10.96.1. Let $I$ be an ideal of a Noetherian ring $R$. Denote ${}^\wedge $ completion with respect to $I$.

  1. If $K \to N$ is an injective map of finite $R$-modules, then the map on completions $K^\wedge \to N^\wedge $ is injective.

  2. If $0 \to K \to N \to M \to 0$ is a short exact sequence of finite $R$-modules, then $0 \to K^\wedge \to N^\wedge \to M^\wedge \to 0$ is a short exact sequence.

  3. If $M$ is a finite $R$-module, then $M^\wedge = M \otimes _ R R^\wedge $.

Proof. Setting $M = N/K$ we find that part (1) follows from part (2). Let $0 \to K \to N \to M \to 0$ be as in (2). For each $n$ we get the short exact sequence

\[ 0 \to K/(I^ nN \cap K) \to N/I^ nN \to M/I^ nM \to 0. \]

By Lemma 10.86.1 we obtain the exact sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits K/(I^ nN \cap K) \to N^\wedge \to M^\wedge \to 0. \]

By the Artin-Rees Lemma 10.50.2 we may choose $c$ such that $I^ nK \subset I^ n N \cap K \subset I^{n-c} K$ for $n \geq c$. Hence $K^\wedge = \mathop{\mathrm{lim}}\nolimits K/I^ nK = \mathop{\mathrm{lim}}\nolimits K/(I^ nN \cap K)$ and we conclude that (2) is true.

Let $M$ be as in (3) and let $0 \to K \to R^{\oplus t} \to M \to 0$ be a presentation of $M$. We get a commutative diagram

\[ \xymatrix{ & K \otimes _ R R^\wedge \ar[r] \ar[d] & R^{\oplus t} \otimes _ R R^\wedge \ar[r] \ar[d] & M \otimes _ R R^\wedge \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^\wedge \ar[r] & (R^{\oplus t})^\wedge \ar[r] & M^\wedge \ar[r] & 0 } \]

The top row is exact, see Section 10.38. The bottom row is exact by part (2). By Lemma 10.95.1 the vertical arrows are surjective. The middle vertical arrow is an isomorphism. We conclude (3) holds by the Snake Lemma 10.4.1. $\square$


Comments (4)

Comment #2979 by Kestutis Cesnavicius on

The statement of this lemma could be improved: completions are taken with respect to what? Similar remarks apply to the statements of the next several lemmas in this section.

Comment #3278 by Nicolas on

There's a typo in the statement of the lemma: it should be "Let I be an ideal".

There are also:

  • 3 comment(s) on Section 10.96: Completion for Noetherian rings

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