The Stacks project

Lemma 85.16.3. 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 85.6.3. Thus by Lemma 85.16.2 we may assume that both $X$ and $Y$ are adic*. We will use the criterion of Lemma 85.6.5 to see that the lemma holds. Namely, write $Y = \mathop{\mathrm{colim}}\nolimits Y_ n$ as in Lemma 85.6.1. For each $n$ set $X_ n = Y_ n \times _ Y X$. Then $X_ n$ is an affine scheme (Lemma 85.15.7) 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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