Question

Let $G$ be a connected linear algebraic group over $\mathbb{C}$ and $X$ a $G$-variety. Let $K_G^0(X)$ be the Grothedieck group of coherent $G$-equivariant sheaves on $X$, and $K_G^i(X)$ for $i>0$ the higher analogues of $K_G^0(X)$ (Quillen, Thomason). My question is that when $X=\text{pt}$ is a single point, do we have $K_G^i(\text{pt})=0$ for $i>0$?

Answer 1

If your point is $Spec({\mathbb C})$ and $G$ is the trivial group, then $K^i_G(pt)$ contains a rational vector space of uncountable dimension for every $i>0$.