Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be the cone in $\mathbb{R}^3$ generated by $e_1,2e_1+e_2,e_1+2e_2+3e_3$. Then it is easy to check that $\sigma$ is a 3-dimensional simplicial cone. Let $X=\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^3])$ and let $R= k[\sigma^{\vee}\cap\mathbb{Z}^3]$. Then $X$ is a 3-dimensional affine simplicial toric variety over $k$. I am trying to compute all the $G$-theory groups of $X$. I have shown that the dual cone $\sigma^{\vee}$ is the cone in $\mathbb{R}^3$ generated by $e_3,e_1-2e_2+e_3,3e_2-2e_3$. My strategy is to use the $G$-theory localization sequence for the Noetherian ring $R$:$…G_1(R_z)\xrightarrow{\partial}G_0(R/zR)\rightarrow G_0(R)\rightarrow G_0(R_z)\rightarrow 0$. I used the fact that $\operatorname{Spec}(k[\tau^{\vee}\cap\mathbb{Z}^3])$ is the localization $R_z$, where $\tau$ is the face of the cone $\sigma$ determined by $\tau=\sigma\cap H_{e_3}$, and $H_{e_3}$ is the $xy$-plane in $\mathbb{R}^3$. So I managed to compute $R_z\cong k[z,z^{-1},x,x^2y^{-1},y]$ as rings. Now I am stuck on the problem of computing the quotient ring $R/zR$.
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