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.
- 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?