I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true.
This leads me to the following question: what are some properties of the underlying space of a Noetherian affine scheme that do not hold for arbitrary Noetherian spectral spaces?
To clarify, consider an almost complex structure as an additional structure one puts on a smooth manifold. The existence of an almost complex structure implies that the smooth manifold is orientable. Is there something analogous if we think of a sheaf of rings making a spectral space into a Noetherian affine scheme as the additional structure?