The Stacks project

Lemma 35.13.1. The property $\mathcal{P}(S) =$“$S$ is locally Noetherian” is local in the fppf topology.

Proof. We will use Lemma 35.12.2. First we note that “being locally Noetherian” is local in the Zariski topology. This is clear from the definition, see Properties, Definition 28.5.1. Next, we show that if $S' \to S$ is a flat, finitely presented morphism of affines and $S$ is locally Noetherian, then $S'$ is locally Noetherian. This is Morphisms, Lemma 29.15.6. Finally, we have to show that if $S' \to S$ is a surjective flat, finitely presented morphism of affines and $S'$ is locally Noetherian, then $S$ is locally Noetherian. This follows from Algebra, Lemma 10.164.1. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$


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