Lemma 10.66.18 (05CC)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.66: Weakly associated primes
Lemma 10.66.18 (cite)

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Lemma 10.66.18. 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

\[ \text{WeakAss}_ S(N) = \text{WeakAss}_{S \otimes _ R K}(N \otimes _ R K) \]

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 10.66.16. $\square$

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