The Stacks project

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

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

  2. $N$ is flat over $R$,

  3. $S \to S'$ is flat, and

  4. 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.

Proof. By Algebra, Lemma 10.82.12 it suffices to show that $N_{\mathfrak q} \to (N \otimes _ R S')_{\mathfrak q}$ is a $R_{\mathfrak p}$-universally injective for any prime $\mathfrak q$ of $S$ lying over $\mathfrak p$ in $R$. Thus we may apply Lemma 38.7.3 to the ring maps $R_{\mathfrak p} \to S_{\mathfrak q} \to S'_{\mathfrak q}$ and the module $N_{\mathfrak q}$. $\square$

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