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). $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)