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In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g.

For some properties specified on stalks this does not happen, e.g.

  • factoriality
  • Noetherianity (to clarify, an example of a reduced affine scheme with Noetherian topological space and Noetherian stalks that is not Noetherian is given).

In my experience, the latter are less common than the former. What are other examples of properties specified on stalks that do not lift to properties of the values of the structure sheaf on affine opens?

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    $\begingroup$ so this guy only has 9 alter egos ? $\endgroup$
    – meh
    Commented Jun 20, 2019 at 12:47
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    $\begingroup$ I stand corrected 10. $\endgroup$
    – meh
    Commented Jun 20, 2019 at 13:16
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    $\begingroup$ @aginensky Please feel free to flag others that you see. $\endgroup$
    – Todd Trimble
    Commented Jun 20, 2019 at 14:45

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