Remark 10.63.12. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Then it is not always the case that $\mathop{\mathrm{Spec}}(\varphi )(\text{Ass}_ S(M)) \supset \text{Ass}_ R(M)$. For example, consider the ring map $R = k \to S = k[x_1, x_2, x_3, \ldots ]/(x_ i^2)$ and $M = S$. Then $\text{Ass}_ R(M)$ is not empty, but $\text{Ass}_ S(S)$ is empty.

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