I was wondering it while studying Hartshorne.

In II.4.8 proof of the corollary (This part is about valuative criterion of properness.), it is written that a morphism from noetherian scheme is automatically quasi-compact.

Precisely, for a noetherian scheme $X$, any morphism $f:X\rightarrow Y$ of schemes is quasi-compact.

But I cannot find why.

Could you give me any idea?

  • 4
    $\begingroup$ Every subset of a Noetherian space is quasi-compact. $\endgroup$ – Alex Youcis Oct 13 '16 at 16:04
  • $\begingroup$ Oh, even for not open subset? $\endgroup$ – wooa0923 Oct 13 '16 at 16:12
  • 2
    $\begingroup$ Yes. This is actually a characterization of noetherian spaces i.e. noetherian iff every subspace is quasi-compact $\endgroup$ – basket Oct 13 '16 at 18:27

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