The Stacks project

Lemma 47.26.4. Let $\varphi : A \to B$ be a flat homorphism of Noetherian rings such that for all primes $\mathfrak q \subset B$ we have $\mathfrak p B_\mathfrak q = \mathfrak qB_\mathfrak q$ where $\mathfrak p = \varphi ^{-1}(\mathfrak q)$, for example if $\varphi $ is étale. If $I$ is an injective $A$-module, then $I \otimes _ A B$ is an injective $B$-module.

Proof. Étale maps satisfy the assumption by Algebra, Lemma 10.143.5. By Lemma 47.3.7 and Proposition 47.5.9 we may assume $I$ is the injective hull of $\kappa (\mathfrak p)$ for some prime $\mathfrak p \subset A$. Then $I$ is a module over $A_\mathfrak p$. It suffices to prove $I \otimes _ A B = I \otimes _{A_\mathfrak p} B_\mathfrak p$ is injective as a $B_\mathfrak p$-module, see Lemma 47.3.2. Thus we may assume $(A, \mathfrak m, \kappa )$ is local Noetherian and $I = E$ is the injective hull of the residue field $\kappa $. Our assumption implies that the Noetherian ring $B/\mathfrak m B$ is a product of fields (details omitted). Thus there are finitely many prime ideals $\mathfrak m_1, \ldots , \mathfrak m_ n$ in $B$ lying over $\mathfrak m$ and they are all maximal ideals. Write $E = \bigcup E_ n$ as in Lemma 47.7.3. Then $E \otimes _ A B = \bigcup E_ n \otimes _ A B$ and $E_ n \otimes _ A B$ is a finite $B$-module with support $\{ \mathfrak m_1, \ldots , \mathfrak m_ n\} $ hence decomposes as a product over the localizations at $\mathfrak m_ i$. Thus $E \otimes _ A B = \prod (E \otimes _ A B)_{\mathfrak m_ i}$. Since $(E \otimes _ A B)_{\mathfrak m_ i} = E \otimes _ A B_{\mathfrak m_ i}$ is the injective hull of the residue field of $\mathfrak m_ i$ by Lemma 47.26.3 we conclude. $\square$


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Your email address will not be published.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 0E4F. Beware of the difference between the letter 'O' and the digit '0'.