Orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for a number field $K$.

Question

I was reading some notes of Keith Conrad where he proves that the number of orbits of the $\text{SL}(2,\mathcal{O}_K)$-action on $\mathbb{P}^{1}(K)$ for a number field $K$ is precisely the class number of $K$.

I am wondering if there is any kind of "higher-order" arithmetic information found in looking at the number of orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for $n>2$ (or even on higher Grassmannians $\text{Gr}(r,K^{n})$, but let's not get too crazy for now). As a first question, will these numbers even be finite for all $n$?

For $K=\mathbb{Q}$, I believe one can use a generalized Euclidean algorithm to show that the action above is transitive for all $n$, at least for the projective spaces $\mathbb{P}^{n-1}(\mathbb{Q})$, but I'm not sure if one can adapt this argument to work even for $\mathcal{O}_K$ that are UFDs but not Euclidean.

Does anyone know of any references on this question?

Answer 1

First, I'm a little nervous about that assertion being exactly right, due to distinctions about "narrow class group", and so on, but I don't want to think about it...

The relevant factoid is "Steinitz' Theorem", generalizing the structure theorem for finitely-generated modules over principal ideal domains to finitely-generated modules over Dedekind domains $\mathfrak o$: among other things, the torsion-free ones are of the form $\mathfrak o\oplus\ldots\oplus \mathfrak o\oplus \mathfrak a$ for some ideal $\mathfrak a$ of $\mathfrak o$. And subsidiary points like $\mathfrak a\oplus \mathfrak b\approx \mathfrak o\oplus \mathfrak a\mathfrak b$.

The $SL_n(\mathfrak o)$ or $GL_n(\mathfrak o)$ action is then just about change-of-basis.