The Stacks project

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

I feel like the conclusion should be either

is an essential injection.


is an essential extension of .

There are also:

  • 4 comment(s) on Section 47.2: Essential surjections and injections

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08XL. Beware of the difference between the letter 'O' and the digit '0'.