Lemma 27.5.6. Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian.
Proof. Omitted. Hint: Any quotient, and any localization of a Noetherian ring is Noetherian. For the Noetherian case use again that any subset of a Noetherian space is a Noetherian space (with induced topology). $\square$
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