The Stacks project

Remark 12.29.2. Let $\mathcal{A}$, $\mathcal{B}$, $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be as in Lemma 12.29.1. In the presence of assumption (1) assumption (2) is equivalent to requiring that $v$ is exact. Moreover, condition (2) is necessary. Here is an example. Let $A \to B$ be a ring map. Let $u : \text{Mod}_ B \to \text{Mod}_ A$ be $u(N) = N_ A$ and let $v : \text{Mod}_ A \to \text{Mod}_ B$ be $v(M) = M \otimes _ A B$. Then $u$ is right adjoint to $v$, and $u$ is exact and $v$ is right exact, but $v$ does not transform injective maps into injective maps in general (i.e., $v$ is not left exact). Moreover, it is not the case that $u$ transforms injective $B$-modules into injective $A$-modules. For example, if $A = \mathbf{Z}$ and $B = \mathbf{Z}/p\mathbf{Z}$, then the injective $B$-module $\mathbf{Z}/p\mathbf{Z}$ is not an injective $\mathbf{Z}$-module. In fact, the lemma applies to this example if and only if the ring map $A \to B$ is flat.