The Stacks project

Lemma 47.7.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $E$ be the injective hull of $\kappa $. Let $M$ be a $\mathfrak m$-power torsion $R$-module with $n = \dim _\kappa (M[\mathfrak m]) < \infty $. Then $M$ is isomorphic to a submodule of $E^{\oplus n}$.

Proof. Observe that $E^{\oplus n}$ is the injective hull of $\kappa ^{\oplus n} = M[\mathfrak m]$. Thus there is an $R$-module map $M \to E^{\oplus n}$ which is injective on $M[\mathfrak m]$. Since $M$ is $\mathfrak m$-power torsion the inclusion $M[\mathfrak m] \subset M$ is an essential extension (for example by Lemma 47.2.4) we conclude that the kernel of $M \to E^{\oplus n}$ is zero. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 47.7: Injective hull of the residue field

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Your email address will not be published.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 08Z3. Beware of the difference between the letter 'O' and the digit '0'.