In this post we just give a simple, but nice example, to show that ‘Noetherian’ is not a property preserved under pull-back.
Noetherian is Not Stable
The stability of a property of schemes, its preservation under arbitrary pull-backs, is a very desirable property of schemes. That said, while most natural examples of relative properties are preserved under pull-backs (e.g. finite type, integral, etc.) , some nice global properties are not. In particular, we show in this example that Noetherianess is not preserved under pull-backs.
In particular, we consider the following fibered diagram
Certainly is Noetherian, but we claim that is not.
Indeed, suppose that it were. Note that is an integral extension, and so . So, if were Noetherian, it would have to be a finite set of points. Indeed, every Noetherian space is a union of finitely many irreducible components. And, the only dimension 0 irreducible space is a point.
But, is not finite. Indeed, note that for each finite extension we have the following fibered diagram(s):
and since is surjective, so is . But, note that if , with (i.e. ) then is isomorphic to , which is in turn isomorphic to . So, has -points. Since was arbitrary, we see that is infinite, contradicting that it is Noetherian.