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An integral scheme of locally finite type over a field is always of finite type of the field?

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up vote 3 down vote accepted

No, take an infinite disjoint union of $\mathrm{Spec}(k)$. if it is not supposed to be separated. Just glue infinitely many copies of affines lines $\mathbb A^1$ along $\mathbb A^1\setminus \{ 0\}$.

Now even if $X$ is integral and separated, this is not necessarily true. See this example of BCnrd at mathoverflow.

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Dear QiL, I think it is not integral. – Makoto Kato Dec 19 '12 at 16:23
Dear @MakotoKato: you are absolutely right. I will edit. – user18119 Dec 19 '12 at 18:55
Dear @QiL and MakotoKato Thank you very much. I understood. – Tom Dec 20 '12 at 5:02

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