The Stacks project

Lemma 10.65.5. Let $R \to S$ be a ring map. Let $M$ be an $R$-module, and let $N$ be an $S$-module. Assume $N$ is flat as $R$-module. Then

\[ \text{Ass}_ S(M \otimes _ R N) \supset \bigcup \nolimits _{\mathfrak p \in \text{Ass}_ R(M)} \text{Ass}_{S \otimes _ R \kappa (\mathfrak p)}(N \otimes _ R \kappa (\mathfrak p)) \]

where we use Remark 10.17.8 to think of the spectra of fibre rings as subsets of $\mathop{\mathrm{Spec}}(S)$. If $R$ is Noetherian then this inclusion is an equality.

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