Lemma 10.98.1 (0G1Q)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.98: Taking limits of modules
Lemma 10.98.1 (cite)

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Lemma 10.98.1. Let $I \subset A$ be a finitely generated ideal of a ring. Let $(M_ n)$ be an inverse system of $A$ -modules with $I^ n M_ n = 0$ . Then $M = \mathop{\mathrm{lim}}\nolimits M_ n$ is $I$ -adically complete.

Proof. We have $M \to M/I^ nM \to M_ n$ . Taking the limit we get $M \to M^\wedge \to M$ . Hence $M$ is a direct summand of $M^\wedge$ . Since $M^\wedge$ is $I$ -adically complete by Lemma 10.96.3, so is $M$ . $\square$

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