The Stacks project

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. $\square$