I am trying to compute the $G$-theory groups of the ring $k[x,y]/(xy)$ for any field $k$. What What I have tried so far are two approaches. Approach
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
$$\dots\rightarrow G_{n+1}(A_x)\rightarrow G_n(A/x)\rightarrow G_n(A)\rightarrow G_n(A_x)\rightarrow …\rightarrow G_0(A_x)\rightarrow 0.$$
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 So by exactness of the localization sequence, I got $G_0(A)\cong G_0(A_x)\cong \mathbb{Z}$. I I know that $G_n(A_x)\cong G_n(k)\oplus G_{n-1}(k)$ for any $n\geq 1$. And And $G_n(A/x)\cong G_n(k)$ for all $n$. But But I don’t know how to compute the boundary map $\partial:G_{n+1}(A_x)\rightarrow G_n(A/x)$.This This is where I am stuck. Approach
Approach 2:Use Use the coniveau spectral sequence. I 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 Coniveau spectral sequence. Now 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 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 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
Any help will be greatly appreciated.
