Lemma 47.7.4. Let $R \to S$ be a flat local homomorphism of local Noetherian rings such that $R/\mathfrak m_ R \cong S/\mathfrak m_ R S$. Then the injective hull of the residue field of $R$ is the injective hull of the residue field of $S$.

Proof. Note that $\mathfrak m_ RS = \mathfrak m_ S$ as the quotient by the former is a field. Set $\kappa = R/\mathfrak m_ R = S/\mathfrak m_ S$. Let $E_ R$ be the injective hull of $\kappa$ over $R$. Let $E_ S$ be the injective hull of $\kappa$ over $S$. Observe that $E_ S$ is an injective $R$-module by Lemma 47.3.2. Choose an extension $E_ R \to E_ S$ of the identification of residue fields. This map is an isomorphism by Lemma 47.7.3 because $R \to S$ induces an isomorphism $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$ for all $n$. $\square$

## 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Your email address will not be published.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 08Z5. Beware of the difference between the letter 'O' and the digit '0'.