Weird Example: Pull-back of Noetherian Isn’t Noetherian

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

\begin{matrix}\text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}) & \to & \text{Spec}(\overline{\mathbb{Q}})\\ \downarrow & & \downarrow\\ \text{Spec}(\overline{\mathbb{Q}}) & \to & \text{Spec}(\mathbb{Q})\end{matrix}

Certainly \text{Spec}(\overline{\mathbb{Q}}) is Noetherian, but we claim that \text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}) is not.

Indeed, suppose that it were. Note that \overline{\mathbb{Q}}\to\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}} is an integral extension, and so \dim \overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}=0. So, if \text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}) 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, \text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}) is not finite. Indeed, note that for each finite extension L/K we have the following fibered diagram(s):

\begin{matrix}\text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}) & \to & \text{Spec}(\overline{\mathbb{Q}})\\ \downarrow & & \downarrow\\ \text{Spec}(L\otimes_\mathbb{Q}\overline{\mathbb{Q}}) & \to & \text{Spec}(L)\\ \downarrow & & \downarrow \\ \text{Spec}(\overline{\mathbb{Q}}) & \to & \text{Spec}(\mathbb{Q}) \end{matrix}

and since \text{Spec}(\overline{\mathbb{Q}})\to\text{Spec}(L) is surjective, so is \text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}})\to\text{Spec}(L\otimes_\mathbb{Q}\overline{\mathbb{Q}}). But, note that if L=\mathbb{Q}[x]/(f(x)), with \deg f=n (i.e. [L:K]=n) then L\otimes_\mathbb{Q}\overline{\mathbb{Q}} is isomorphic to \overline{\mathbb{Q}}[x]/(f(x)), which is in turn isomorphic to \overline{\mathbb{Q}}^n. So, \text{Spec}(L\otimes_\mathbb{Q}\overline{\mathbb{Q}}) has n-points. Since L was arbitrary, we see that \text{Spec}(\overline{\mathbb{Q}}\otimes_\mathbb{Q}\overline{\mathbb{Q}}) is infinite, contradicting that it is Noetherian.

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