Lemma 38.13.4 (05II)—The Stacks project

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Part 2: Schemes
Chapter 38: More on Flatness
Section 38.13: Flat finite type modules, Part II
Lemma 38.13.4 (cite)

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Lemma 38.13.4. Let $R \to S$ be a ring map which is essentially of finite type. Let $N$ be a localization of a finite $S$ -module flat over $R$ . Let $M$ be an $R$ -module. Then

$\text{WeakAss}_ S(M \otimes _ R N) = \bigcup \nolimits _{\mathfrak p \in \text{WeakAss}_ R(M)} \text{Ass}_{S \otimes _ R \kappa (\mathfrak p)}(N \otimes _ R \kappa (\mathfrak p))$

Proof. This lemma is a translation of Lemma 38.13.3 into algebra. Details of translation omitted. $\square$

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