The Stacks project

Lemma 28.5.2. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is locally Noetherian.

  2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is Noetherian.

  3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is Noetherian.

  4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is locally Noetherian.

Moreover, if $X$ is locally Noetherian then every open subscheme is locally Noetherian.

Proof. To show this it suffices to show that being Noetherian is a local property of rings, see Lemma 28.4.3. Any localization of a Noetherian ring is Noetherian, see Algebra, Lemma 10.31.1. By Algebra, Lemma 10.23.2 we see the second property to Definition 28.4.1. $\square$

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