The Stacks project

Lemma 31.6.2. Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $X$ is locally Noetherian, then we have

\[ f(\text{Ass}_ X(\mathcal{F})) = \text{Ass}_ S(f_*\mathcal{F}) = \text{WeakAss}_ S(f_*\mathcal{F}) = f(\text{WeakAss}_ X(\mathcal{F})) \]

Proof. We may assume $X$ and $S$ affine, so $X \to S$ comes from a ring map $A \to B$. As $X$ is locally Noetherian the ring $B$ is Noetherian, see Properties, Lemma 28.5.2. Write $\mathcal{F} = \widetilde M$ for some $B$-module $M$. By Lemma 31.2.2 the associated points of $\mathcal{F}$ correspond exactly to the associated primes of $M$, and any associated prime of $M$ as an $A$-module is an associated points of $f_*\mathcal{F}$. Hence the inclusion

\[ f(\text{Ass}_ X(\mathcal{F})) \subset \text{Ass}_ S(f_*\mathcal{F}) \]

follows from Algebra, Lemma 10.63.13. We have the inclusion

\[ \text{Ass}_ S(f_*\mathcal{F}) \subset \text{WeakAss}_ S(f_*\mathcal{F}) \]

by Lemma 31.5.3. We have the inclusion

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

by Lemma 31.6.1. The outer sets are equal by Lemma 31.5.8 hence we have equality everywhere. $\square$

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