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
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
follows from Algebra, Lemma 10.63.13. We have the inclusion
by Lemma 31.5.3. We have the inclusion
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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