Lemma 31.6.1. Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then we have

$\text{WeakAss}_ S(f_*\mathcal{F}) \subset f(\text{WeakAss}_ X(\mathcal{F}))$

Proof. We may assume $X$ and $S$ affine, so $X \to S$ comes from a ring map $A \to B$. Then $\mathcal{F} = \widetilde M$ for some $B$-module $M$. By Lemma 31.5.2 the weakly associated points of $\mathcal{F}$ correspond exactly to the weakly associated primes of $M$. Similarly, the weakly associated points of $f_*\mathcal{F}$ correspond exactly to the weakly associated primes of $M$ as an $A$-module. Hence the lemma follows from Algebra, Lemma 10.66.11. $\square$

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