The Stacks project

Proof. Observe that $X_ v = v \times _ Y X = \mathop{\mathrm{Spec}}(\kappa (v)) \times _ Y X$. Hence the implications (2) $\Rightarrow $ (1) $\Rightarrow $ (3) are clear. Assume that $\mathop{\mathrm{Spec}}(k) \to Y$ is a morphism from the spectrum of a field such that $X_ k$ is locally Noetherian. Let $V \to Y$ be an étale morphism from a scheme $V$ and let $v \in V$ a point mapping to $y$. Then the scheme $v \times _ Y \mathop{\mathrm{Spec}}(k)$ is nonempty. Choose a point $w \in v \times _ Y \mathop{\mathrm{Spec}}(k)$. Consider the morphisms

Since $V \to Y$ is étale and since $w$ may be viewed as a point of $V \times _ Y \mathop{\mathrm{Spec}}(k)$, we see that $\kappa (w)/k$ is a finite separable extension of fields (Morphisms, Lemma 29.36.7). Thus $X_ w \to X_ k$ is a finite étale morphism as a base change of $w \to \mathop{\mathrm{Spec}}(k)$. Hence $X_ w$ is locally Noetherian (Morphisms of Spaces, Lemma 67.23.5). The morphism $X_ w \to X_ v$ is a surjective, affine, flat morphism as a base change of the surjective, affine, flat morphism $w \to v$. Then the fact that $X_ w$ is locally Noetherian implies that $X_ v$ is locally Noetherian. This can be seen by picking a surjective étale morphism $U \to X$ and then using that $U_ w \to U_ v$ is surjective, affine, and flat. Working affine locally on the scheme $U_ v$ we conclude that $U_ w$ is locally Noetherian by Algebra, Lemma 10.164.1.

Finally, it suffices to prove that (3) implies (4) in case we have a monomorphism $\mathop{\mathrm{Spec}}(k_0) \to Y$ in the class of $y$. Then $\mathop{\mathrm{Spec}}(k) \to Y$ factors as $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k_0) \to Y$. The argument given above then shows that $X_ k$ being locally Noetherian impies that $X_{k_0}$ is locally Noetherian. $\square$