Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}\_q$.


The Grothendieck group of the category of finitely-generated $\mathbf{F}\_q[G]$-modules is naturally identified with a group of $\mathbf{C}$-valued functions on the set of conjugacy classes of elements of $G$ of order prime to $p$, via Brauer characters.


What is the Grothendieck group of finitely generated $\mathcal{O}[G]$-modules?



What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?
If I remember correctly, it is isomorphic to the group of ordinary (virtual) characters over $K=\mbox{Frac}(\mathcal{O})$, the field of fractions of $\mathcal{O}$. That is, the Grothendieck group of $KG$. An isomorphism is given by simply tensoring with $K$, so that $\mathcal{O}G$-modules $M,N$ are identified in $K\_0(\mathcal{O}G)$ if and only if $K\otimes\_{\mathcal{O}}M$ and $K\otimes\_{\mathcal{O}}N$ afford the same character. In particular, all torsion modules are zero in the Grothendieck group. 


This was proved by Swan in 'The Grothendieck ring of a finite group', Topology 2, 85-110, 1963. He proves a more general result over an arbitrary integral domain there, and Theorem 3 is the one you're after.


