Lemma 10.65.4 (05C1)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.65: Relative assassin
Lemma 10.65.4 (cite)

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Lemma 10.65.4. Let $R \to S$ be a ring map. Let $N$ be an $S$ -module. Assume $N$ is flat as an $R$ -module and $R$ is a domain with fraction field $K$ . Then

$\text{Ass}_ S(N) = \text{Ass}_ S(N \otimes _ R K) = \text{Ass}_{S \otimes _ R K}(N \otimes _ R K)$

via the canonical inclusion $\mathop{\mathrm{Spec}}(S \otimes _ R K) \subset \mathop{\mathrm{Spec}}(S)$ .

Proof. Note that $S \otimes _ R K = (R \setminus \{ 0\} )^{-1}S$ and $N \otimes _ R K = (R \setminus \{ 0\} )^{-1}N$ . For any nonzero $x \in R$ multiplication by $x$ on $N$ is injective as $N$ is flat over $R$ . Hence the lemma follows from Lemma 10.63.17 combined with Lemma 10.63.16 part (1). $\square$

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