Let $k$ be an algebraically closed field and let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$.Let $X$ be the affine toric variety over $k$ associated to the cone $\sigma$, i.e. set $X$ to be $\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^3])$. I am trying to compute the $G$-theory groups of $X$ using the localization sequence. I am trying to start by applying some coordinate transformations in $GL(3,\mathbb{Z})$ to the cone $\sigma$, so that the cone $\sigma$ can be simplified without modifying the isomorphism type of the associated affine toric variety. Suppose $v_1,v_2,v_3$ are the three minimal generators of the cone $\sigma$. Since the vector $v_1\in\mathbb{Z}^3$ can be seen as a unimodular column, it can be completed to a matrix $A\in GL(3,\mathbb{Z})$. Let $B=A^{-1}$. Then $B\in GL(3,\mathbb{Z})$ and left multiplication by $B$ transforms the matrix taking $v_i$ as its $i$-th column to a 3 by 3 matrix taking $e_1$ as its first column. So there exists a coordinate transformation in $GL(3,\mathbb{Z})$ such that the minimal generator of the cone $\sigma$ becomes the first standard basis vector $e_1$ of $\mathbb{R}^3$. Now I am thinking of a further simplification of the cone $\sigma$. Is it possible to apply some transformation in $GL(3,\mathbb{Z})$ such that the resulting cone from $\sigma$ has one of its two-dimensional faces lying on the $xy$-plane?
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