Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ fails to be isomorphic to $K_0(R[s])$. Thus my interest in computing $K_0(k[t^2,t^3])$ and $K_0(k[t^2,t^3,s])$ to see if they are isomorphic or not.

For $K_0(k[t^2,t^3])$ I have used the fact that since it is Noetheiran of dimension $1$, it will be isomorphic to $\mathbb{Z} \oplus Pic(k[t^2,t^3])$. To compute the Picard group I took the help of the conductor ideals to show that it is $k^{\times}\times k.$ But for $K_0(k[t^2,t^3,s])$ I can understand that it will be isomorphic to a quotient of $\mathbb{Z} \oplus Pic(k[t^2,t^3,s])$ but I am unable to compute it explicitely. I would be grateful for any help or suggestions in this direction. Thank you.

Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ fails to be isomorphic to $K_0(R[s])$. Thus my interest in computing $K_0(k[t^2,t^3])$ and $K_0(k[t^2,t^3,s])$ to see if they are isomorphic or not.

For $K_0(k[t^2,t^3])$ I have used the fact that since it is Noetheiran of dimension $1$, it will be isomorphic to $\mathbb{Z} \oplus Pic(k[t^2,t^3])$. To compute the Picard group I took the help of the conductor ideals to show that it is $ k.$ But for $K_0(k[t^2,t^3,s])$ I can understand that it will be isomorphic to a quotient of $\mathbb{Z} \oplus Pic(k[t^2,t^3,s])$ but I am unable to compute it explicitly. I would be grateful for any help or suggestions in this direction. Thank you.