Lemma 47.7.1. Let $R \to S$ be a surjective map of local rings with kernel $I$. Let $E$ be the injective hull of the residue field of $R$ over $R$. Then $E[I]$ is the injective hull of the residue field of $S$ over $S$.
Proof. Observe that $E[I] = \mathop{\mathrm{Hom}}\nolimits _ R(S, E)$ as $S = R/I$. Hence $E[I]$ is an injective $S$-module by Lemma 47.3.4. Since $E$ is an essential extension of $\kappa = R/\mathfrak m_ R$ it follows that $E[I]$ is an essential extension of $\kappa $ as well. The result follows. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)