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The Stacks project

Lemma 87.20.5. Let S be a scheme. Let X \to Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and (locally) of finite type. Then X is Noetherian if and only if Y is Noetherian.

Proof. Observe that a Noetherian affine formal algebraic space is adic*, see Lemma 87.10.3. Thus by Lemma 87.20.4 we may assume that both X and Y are adic*. We will use the criterion of Lemma 87.10.5 to see that the lemma holds. Namely, write Y = \mathop{\mathrm{colim}}\nolimits Y_ n as in Lemma 87.10.1. For each n set X_ n = Y_ n \times _ Y X. Then X_ n is an affine scheme (Lemma 87.19.7) and X = \mathop{\mathrm{colim}}\nolimits X_ n. Each of the morphisms X_ n \to Y_ n is faithfully flat and of finite type. Thus the lemma follows from the fact that in this situation X_ n is Noetherian if and only if Y_ n is Noetherian, see Algebra, Lemma 10.164.1 (to go down) and Algebra, Lemma 10.31.1 (to go up). \square


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