Lemma 38.7.4. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Let $S \to S'$ be a ring map. Assume

$R \to S$ is of finite presentation and $N$ is of finite presentation over $S$,

$N$ is flat over $R$,

$S \to S'$ is flat, and

the image of $\mathop{\mathrm{Spec}}(S') \to \mathop{\mathrm{Spec}}(S)$ contains all primes $\mathfrak q$ such that $\mathfrak q$ is an associated prime of $N \otimes _ R \kappa (\mathfrak p)$ where $\mathfrak p$ is the inverse image of $\mathfrak q$ in $R$.

Then $N \to N \otimes _ S S'$ is $R$-universally injective.

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