Lemma 10.97.1 (00MA)—The Stacks project

Lemma 10.97.1. Let $I$ be an ideal of a Noetherian ring $R$. Denote ${}^\wedge $ completion with respect to $I$.

If $K \to N$ is an injective map of finite $R$-modules, then the map on completions $K^\wedge \to N^\wedge $ is injective.
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.
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 10.87.1 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 10.51.2 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 10.39. The bottom row is exact by part (2). By Lemma 10.96.1 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 (6)

Comment #2979 by Kestutis Cesnavicius on October 30, 2017 at 10:46

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 #3103 by Johan on February 01, 2018 at 00:10

Thanks, fixed here.

Comment #3278 by Nicolas on May 02, 2018 at 22:19

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

Comment #3370 by Johan on May 19, 2018 at 17:24

Thanks, fixed here.

Comment #8753 by Maozhou Huang on December 24, 2023 at 17:32

In the proof of (3), to apply 0315, we need to show that $K$ is finite. This seems true because $R^{\oplus n}$ is a Noetherian module ( $R$ is Noetherian). Am I correct?

Comment #9324 by Stacks project on May 31, 2024 at 14:42

Yes, but we don't even need it for the argument to work.

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5 comment(s) on Section 10.97: Completion for Noetherian rings

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