Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) is not isomorphic to $\mathbb Z$?

## 1 Answer

This is just to flesh out @Tom Goodwillie example.

For any reasonable scheme $X$ and an open set $U$, one has a natural exact sequence,

$$K_0(X-U)\to K_0(X)\to K_0(U)\to 0.$$

Taking $X=\operatorname{Spec} (\mathbb{Q}[x,y]/xy)_{(x,y)}$ (or a number of similar examples) and $U$ the punctured spectrum, we note that the punctured spectrum is two points and thus $K_0(U)=\mathbb{Z}^2$. The kernel is generated by the closed point, but going mod $x+y, x+y^2$, one can easily see that 2 and 3 times the closed point is zero in $K_0(X)$. So, we get $K_0(X)=\mathbb{Z}^2$.

8more comments