TITLE: What sort of ring is the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$?
QUESTION [20 upvotes]: I begin losing intuition when I start dealing with infinitely generated fields over $\mathbb{Q}$...
The naive guess is that it is noetherian of krull dimension 1. Is this correct?
A related question is what sort of morphism is $Spec($the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}) \rightarrow Spec(O_K)$ for some number field $K$? For example, is it a flat morphism?

REPLY [6 votes]: To your related question "For example is flat?" the answer is yes. The point is that over a Dedekind domain, more generally over a Prüfer ring, a module is flat if and only if it is torsion free( see here). On the other hand I'm not so sure if I can say more about of the morphism. It is not going to be locally finite, locally of finite type, locally Noetherian etc...