The Stacks project

Lemma 47.6.2. Let $(R, \mathfrak m, \kappa )$ be an artinian local ring. Let $E$ be an injective hull of $\kappa $. For any finite $R$-module $M$ the evaluation map

\[ M \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, E), E) \]

is an isomorphism. In particular $R = \mathop{\mathrm{Hom}}\nolimits _ R(E, E)$.

Proof. Observe that the displayed arrow is injective. Namely, if $x \in M$ is a nonzero element, then there is a nonzero map $Rx \to \kappa $ which we can extend to a map $\varphi : M \to E$ that doesn't vanish on $x$. Since the source and target of the arrow have the same length by Lemma 47.6.1 we conclude it is an isomorphism. The final statement follows on taking $M = R$. $\square$

Comments (0)

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Your email address will not be published.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