The Stacks project

Example 35.32.1. Consider the property $\mathcal{P}$ of morphisms of schemes defined by the rule $\mathcal{P}(X \to Y) = $“$Y$ is locally Noetherian”. The reader can verify that this is étale local on the source and étale local on the target (omitted, see Lemma 35.16.1). But it is not true that if $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is étale, then $g \circ f$ has $\mathcal{P}$. Namely, $f$ could be the identity on $Y$ and $g$ could be an open immersion of a locally Noetherian scheme $Y$ into a non locally Noetherian scheme $Z$.