Example 35.29.1. 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 35.13.1). 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 $Z$.

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