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Let $X$ be an integral noetherian scheme, let $\xi$ be its generic point. Then it is not so hard to show that $\{ \xi\}$ is open in $X$ if and only if $X$ is a finite set. In termes of algebra, it says the following:

Let $A$ be a noetherian integral domain. Suppose there exists $f\in A$ non-zero such that the localization $A_f$ is a field. Then $A$ is semi-local of Krull dimension at most $1$.

Question: What is the name of this result ? I am sure I saw it once somewhere.

The motivation comes from a construction of non-discrete zero-dimensional schemes here.

Cantlog
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  • @YACP: thanks ! Do you mind to convert your comment into an answer ? Meanwhile I found the place where I saw this statement attributed to Artin-Tate, it is in Görtz and Wedhorn, Appendix B.62. – Cantlog Aug 29 '13 at 21:29

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This result is attributed to Artin and Tate: see Theorem 2.13, pg. 281.