The Stacks project

Remark 10.66.10. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Then it is not always the case that $\mathop{\mathrm{Spec}}(\varphi )(\text{WeakAss}_ S(M)) \subset \text{WeakAss}_ R(M)$ contrary to the case of associated primes (see Lemma 10.63.11). An example is to consider the ring map

\[ R = k[x_1, x_2, x_3, \ldots ] \to S = k[x_1, x_2, x_3, \ldots , y_1, y_2, y_3, \ldots ]/ (x_1y_1, x_2y_2, x_3y_3, \ldots ) \]

and $M = S$. In this case $\mathfrak q = \sum x_ iS$ is a minimal prime of $S$, hence a weakly associated prime of $M = S$ (see Lemma 10.66.8). But on the other hand, for any nonzero element of $S$ the annihilator in $R$ is finitely generated, and hence does not have radical equal to $R \cap \mathfrak q = (x_1, x_2, x_3, \ldots )$ (details omitted).

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