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Ravi Vakil definition 5.3.4.

Suppose $X$ is a scheme. If $X$ can be covered by affine open sets Spec($A$) where $A$ is Noetherian, we say that $X$ is a locally Noetherian scheme.

The definition only says "If $X$ can be covered by affine open sets Spec($A$) where $A$ is Noetherian". I wonder if this can imply that every affine open set is Noetherian?

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1 Answer 1

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Yes, because an affine scheme is quasicompact. So, if $U\subseteq X$ is an affine open subscheme, then $U$ is covered by finitely many spectra of Noetherian rings and so is Noetherian.

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