Grothendieck groups of invariant rings and of group rings (Q1330062)

Let \(G\) be a finite group acting as automorphisms of a (right) Noetherian ring \(S\), and \(R = S^ G\) be the fixed ring under this action. There is a Morita context linking the skew group ring \(T = S * G\) with \(R\), via the bimodules \(tT\) and \(Tt\) where \(t = \sum_{g \in G} g\). Suppose that the trace map \(\text{tr} : S \to R\) is surjective. Then \(-\otimes_ T Tt\) is a localization functor from the category of finitely generated (right) \(T\)-modules to that of finitely generated \(R\)-modules, which yields a long exact sequence of Grothendieck groups. The purpose of this paper is to study this sequence and to derive information about the Grothendieck groups \(G_ 0(R)\) and \(G_ 0(T)\). Some examples are really interesting. When \(S = kA\) is the group algebra of the free abelian group \(A\) of rank \(d\) over a field \(k\), and \(\alpha : G \to \text{GL}(A)\) is a homomorphism which induces a fixed-point-free action of \(G\) on \(S\), the authors obtain a short exact sequence \(0 \to G_ 0 (\overline{T}) \to G_ 0(T) \to G_ 0(R) \to 0\) where \(\overline{T} = T/TtT\), by counting torsion free ranks. In case \(S\) is a commutative Noetherian domain and \(G_ 0 (S)\) is generated by \([S]\), the authors show that \(G_ 0 (R) = \langle [R]\rangle \oplus F \cong \mathbb{Z} \oplus F\), where \(F\) is a \(| G|\)- torsion group.