Ordinary Grothendieck groups of a Frobenius P-category

TL;DR

This paper introduces a new inverse limit of Grothendieck groups for Frobenius categories over a finite p-group, linking characteristic zero and p characteristic cases through canonical isomorphisms.

Contribution

It extends previous work by defining an analogous inverse limit in characteristic zero and establishing its relation to centralizers in Frobenius categories.

Findings

The inverse limit's rank is determined.

Extension to a field is canonically isomorphic to a direct sum.

Connections between characteristic zero and p characteristic Grothendieck groups.

Abstract

In "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, we have introduced the Frobenius categories F over a finite p-group P, and we have associated to F - 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 F, 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 F, 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 F of a suitable set of representatives of the F-classes of elements of P.