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Is an irreducible separated scheme locally of finite type necessarily quasi-compact?

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    $\begingroup$ No. Let $(S_0,p_0)$ be $(\mathbb{A}^2,(0,0))$. Iteratively define $S_{i+1}$ to be the blow up of $S_i$ along $p_i$ and $p_{i+1}$ to be an arbitrary point on the new exceptional divisor. You can glue all the $S_i-p_i$ into one scheme $S_{\infty}.$ $\endgroup$ – dhy Dec 26 '18 at 15:52
  • $\begingroup$ @dhy what if we require the scheme to be one-dimensional? $\endgroup$ – geometer Dec 26 '18 at 16:05
  • $\begingroup$ If you require the scheme to be one-dimensional, then I think the answer should be yes, essentially because there will be a scheme parametrizing valuations on the underlying field of fractions... but I need to think a bit to make sure this can actually be made rigorous when you are not necessarily over a field. $\endgroup$ – dhy Dec 27 '18 at 18:46

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