# 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.