Definition 27.5.1. Let $X$ be a scheme.

We say $X$ is

*locally Noetherian*if every $x \in X$ has an affine open neighbourhood $\mathop{\mathrm{Spec}}(R) = U \subset X$ such that the ring $R$ is Noetherian.We say $X$ is

*Noetherian*if $X$ is locally Noetherian and quasi-compact.

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