Quasi-compact, quasi-separated

Asked Dec 05 '17 at 06:07

Active Dec 05 '17 at 06:07

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I am now learning about quasi-compact and quasi-separated morphism of schemes. I know that the examples for schemes, but I am having trouble finding out the examples for morphism of schemes:

Non-quasi-compact morphism of scheme
Non-quasi-separated morphism of scheme
Quasi-compact but non-quasi-separated morphism of scheme

Thanks!

asked Dec 05 '17 at 06:07

Alahoi

You realize how 3. would give you examples of 1. and 2. automatically? – MooS Dec 05 '17 at 08:16
It is technically not a duplicate of the question, however you find an example of 3. here: https://math.stackexchange.com/a/256244/211913 – MooS Dec 05 '17 at 08:26
I know 3 can give 2 automatically. But how can 3 give 1? – Alahoi Dec 05 '17 at 14:02
By definition, if $X \to Y$ is not qs, then $X \to X \times_Y X$ is not qc. – MooS Dec 05 '17 at 14:33
Furthermore, the linked answer explicitly constructs an open set, that is not quasi-compact. – MooS Dec 05 '17 at 14:35
@MooS Thank you! – Alahoi Dec 06 '17 at 07:02

Quasi-compact, quasi-separated

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