K0 of a local ring

Example

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$ and residue field $k = R/\mathfrak{m}$. Then $K_0(R) \cong \Z$, generated by $[R]$, and the isomorphism sends $[P] - [Q]$ to $\dim_k P/\mathfrak{m}P - \dim_k Q/\mathfrak{m}Q$.

Check: every finitely generated projective $R$-module $P$ is free, so $P \cong R^n$ for some $n \ge 0$. The number $n$ is determined by $P$, because $R^n/\mathfrak{m}R^n \cong k^n$ has dimension $n$ over $k$. So the monoid of isomorphism classes is $\N$, with $[R^n] \mapsto n$, and its Grothendieck group is $\Z$. In particular the reduced group $\tilde{K}_0(R)$ is $0$.

For instance, $K_0(\Z_{(p)}) \cong \Z$ for the localization of $\Z$ at a prime $p$. Also $K_0(A) \cong \Z$ for the ring $A$ of formal power series in one variable over a field.