Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such classes is like a class group of a number field, except it is just a pointed set. It has an adelic description which you can find in Proposition 13 here. On the other hand, we can form the Grothendieck group of the category of projective left modules over $\mathcal{O}$. This maps to $\mathbb{Z}$ by the "rank" map. The kernel is a group $G$. My questions are:

1) Does $G$ have the same size as $S$?

2) Is $S$ a torsor for $G$ in a natural way?

3) Is there a simple description of $G$ (adelic or otherwise) in terms of $\mathcal{O}$ that doesn't reference the Grothendieck group construction?


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