Ordinary Grothendieck Groups of a Frobenius P-Category

In [7] we have introduced the Frobenius categories [Formula: see text] over a finite p-group P, and we have associated to [Formula: see text] — suitably endowed with some central k*-extensions — a “Grothendieck group” as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from [Formula: see text], determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from [Formula: see text], determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the “Grothendieck groups” above associated with the centralizers in [Formula: see text] of a suitable set of representatives of the [Formula: see text]-classes of elements of P.