Lemma 38.7.4 (05DH)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.7: Localization and universally injective maps
Lemma 38.7.4 (cite)

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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.

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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