I am trying to compute the $G$-theory groups of the ring $k[x,y]/(xy)$ for any field $k$.
What I have tried so far are two approaches.
Approach 1: Use the $G$-theory localization sequence for $k[x,y]/(xy)$.
Set $A=k[x,y]/(xy)$ and consider $A/x\cong k[y]$ and $A_x\cong k[x,x^{-1}]$.
There is the $G$-theory localization sequence 
$…\rightarrow G_{n+1}(A_x)\rightarrow G_n(A/x)\rightarrow G_n(A)\rightarrow G_n(A_x)\rightarrow …\rightarrow G_0(A_x)\rightarrow 0$.
Since $A_x\cong k[x,x^{-1}]$, I obtained $G_0(A_x)\cong G_0(k)\cong \mathbb{Z}$.
And the boundary map $G_1(A_x)\rightarrow G_0(A/x)$ is surjective, since the class $[A/x]\in G_0(A/x)$ is in the image of this boundary map.
So by exactness of the localization sequence, I got $G_0(A)\cong G_0(A_x)\cong \mathbb{Z}$.
I know that $G_n(A_x)\cong G_n(k)\oplus G_{n-1}(k)$ for any $n\geq 1$.
And $G_n(A/x)\cong G_n(k)$ for all $n$.
But I don’t know how to compute the boundary map $\partial:G_{n+1}(A_x)\rightarrow G_n(A/x)$.This is where I am stuck.
Approach 2:Use the coniveau spectral sequence.
I know that $E_{\infty}^{0,-n}\cong F^0G_n(A)/F^1G_n(A)$, where $E_{\infty}^{0,-n}$ is the stable value at the $(0,-n)$ spot of the 
Coniveau spectral sequence. 
Now we have $F^0G_n(A)=G_n(A)$ and $F^1G_n(A)$ is defined to be the image of the map $K_nM^1(A)\rightarrow K_nM(A)$.
Since the coniveau spectral sequence is a bounded fourth quadrant cohomological spectral sequence, I calculated that the $E_2$ page gives the stable values.
So there are two difficulties here:how to compute $E_{\infty}^{0,-n}$ and how to compute $G_n(A)$ from the fact that $G_n(A)/F^1G_n(A)\cong E_{\infty}^{0,-n}$.
Any help will be greatly appreciated.