Proposition 15.91.5 (091T)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.91: Derived Completion
Proposition 15.91.5 (cite)

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Proposition 15.91.5. Let $I \subset A$ be a finitely generated ideal of a ring $A$ . Let $M$ be an $A$ -module. The following are equivalent

$M$ is $I$ -adically complete, and
$M$ is derived complete with respect to $I$ and $\bigcap I^ nM = 0$ .

Proof. This is clear from the results of Lemma 15.91.3. $\square$

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