Lemma 76.8.3. Let $A$ be a ring. Let $u : M \to N$ be a map of $A$-modules. If $N$ is projective as an $A$-module, then there exists an ideal $I \subset A$ such that for any ring map $\varphi : A \to B$ the following are equivalent
$u \otimes 1 : M \otimes _ A B \to N \otimes _ A B$ is zero, and
$\varphi (I) = 0$.