Lemma 47.2.3 (08XL)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.2: Essential surjections and injections
Lemma 47.2.3 (cite)

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Lemma 47.2.3. Let $R$ be a ring. Let $M$ be an $R$ -module. Let $E = \mathop{\mathrm{colim}}\nolimits E_ i$ be a filtered colimit of $R$ -modules. Suppose given a compatible system of essential injections $M \to E_ i$ of $R$ -modules. Then $M \to E$ is an essential injection.

Proof. Immediate from the definitions and the fact that filtered colimits are exact (Algebra, Lemma 10.8.8). $\square$

Comments (2)

Comment #1365 by jojo on March 19, 2015 at 11:48

I feel like the conclusion should be either

$M \to E$ is an essential injection.

or

$E$ is an essential extension of $M$ .

Comment #1384 by Johan on April 01, 2015 at 19:24

Yes that is better. See here.

There are also:

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