I'm reading about orders and invertible ideals, but I'm having a little trouble connecting a few different concepts.
In a Dedekind domain, all ideals are invertible, and they correspond to divisors in algebraic geometry, which in turn are formal sums of codimension 1 irreducible subvarieties. But in an order, an ideal $\mathfrak{a}$ is invertible iff all the localisations $\mathfrak{a}_{\mathcal{p}}$ are principal.
My two questions are:
- What is the significance of an invertible ideal of an order. It seems that even non-invertible ideals would define a subvariety, so why exclude them? I see that it is necessary to do this in order to form the class group, but there must be some more intuitive reason...
- Do principal ideals correspond in some way to smooth points? I understand that a DVR defines a smooth local neighborhood of a point, and a singularity in an order is a point where the local ring is not a PID, but at these points, what is the difference between the principal and non-principal ideals?
Thanks in advance!