Equivariant K-theory for curves (Q1063076)

Let X be a smooth curve over an algebraically closed field, k, and let \(G\subseteq Aut_ k(X)\) be a finite group such that \((char(k),| G|)=1\). Let \(Y=X/G\) then, in equivariant algebraic K-theory (defined by locally free sheaves) we have \(\pi^*: K^{\bullet}(Y)\to K_ G^{\bullet}(X)\), a split injection. The authors prove that \(K_ G^{\bullet}(X)\cong K^{\bullet}(Y)\oplus (\oplus^{r}_{i=1}R_ k(H_ i))\) where \(H_ i\) runs through the ramification groups of \(\pi\) : \(X\to Y\). Here \(R_ k(H_ i)\) is the k-representation ring of \(H_ i\). The \(R_ k(G)\)-module structure of \(K_ G^{\bullet}(X)\) is also described.