Lemma 10.66.11. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Then we have $\mathop{\mathrm{Spec}}(\varphi )(\text{WeakAss}_ S(M)) \supset \text{WeakAss}_ R(M)$.

Proof. Let $\mathfrak p$ be an element of $\text{WeakAss}_ R(M)$. Then there exists an $m \in M_{\mathfrak p}$ whose annihilator $I = \{ x \in R_{\mathfrak p} \mid xm = 0\}$ has radical $\mathfrak pR_{\mathfrak p}$. Consider the annihilator $J = \{ x \in S_{\mathfrak p} \mid xm = 0 \}$ of $m$ in $S_{\mathfrak p}$. As $IS_{\mathfrak p} \subset J$ we see that any minimal prime $\mathfrak q \subset S_{\mathfrak p}$ over $J$ lies over $\mathfrak p$. Moreover such a $\mathfrak q$ corresponds to a weakly associated prime of $M$ for example by Lemma 10.66.2. $\square$

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