Lemma 38.23.2. Let $A$ be a ring. Let $u : M \to N$ be a surjective map of $A$-modules. If $M$ 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 an isomorphism, and

$\varphi (I) = 0$.

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