Grothendieck Groups of Invariant Rings: Filtrations

Abstract

We investigate the Grothendieck group G0(R) of finitely generated modules over the ring of invariants R = SG of the action of a finite group G on an FBN ring S under the assumption that the trace map from S to R is surjective. Using a certain filtration of G0(R) that is defined in terms of (Gabriel-Rentschler) Krull dimension, properties of G0(R) are derived to a large extent from the connections between the sets of prime ideals of S and R. A crucial ingredient is an equivalence relation ∼ on Spec R that was introduced by Montgomery [25]. For example, we show that rank G0(R) ⩽ rank G0(S)G + ∑ Ω ( # Ω - 1 ) where Ω runs over the ∼-equivalence classes in Spec R and (·)G denotes G-coinvariants. The torsion subgroup of G0(R) is also considered. We apply our results to group actions on the Weyl algebra in positive characteristics, the quantum plane, and the localized quantum plane for roots of unity.