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The Stacks project

Lemma 71.4.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let y \in |Y|. The following are equivalent

  1. for some scheme V, point v \in V, and étale morphism V \to Y mapping v to y, the algebraic space X_ v is locally Noetherian,

  2. for every scheme V, point v \in V, and étale morphism V \to Y mapping v to y, the algebraic space X_ v is locally Noetherian, and

  3. there exists a field k and a morphism \mathop{\mathrm{Spec}}(k) \to Y representing y such that X_ k is locally Noetherian.

If there exists a field k_0 and a monomorphism \mathop{\mathrm{Spec}}(k_0) \to Y representing y, then these are also equivalent to

  1. the algebraic space X_{k_0} is locally Noetherian.

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

X_ v \longleftarrow X_ w \longrightarrow X_ k

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


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