Mayer–Vietoris sequence in K-theory

Remarks

The idea of the proof is Milnor's patching construction. Given finitely generated projective modules $P_1$ over $R_1$ and $P_2$ over $R_2$ and an isomorphism $h \colon R' \otimes P_1 \to R' \otimes P_2$, the set of pairs $(p_1, p_2)$ with $h(1 \otimes p_1) = 1 \otimes p_2$ is a finitely generated projective $R$-module. The boundary $\partial$ sends the class of a matrix $h \in GL_n(R')$ to $[M(h)] - [R^n]$, where $M(h)$ is the module patched from $R_1^n$ and $R_2^n$ along $h$. Surjectivity of $g_2$ is used to lift elementary matrices from $R'$ to $R_2$, which makes $\partial$ well defined on $K_1$. Example: for $R = k[x, y]/(xy)$ with $R_1 = k[x]$, $R_2 = k[y]$ and $R' = k$, all four rings have $K_0 \cong \Z$ and the sequence gives $K_0(R) \cong \Z$, since $K_1(k[x]) \oplus K_1(k[y]) \to K_1(k) = k^\times$ is onto.

Going deeper

For a student

The sequence is the main tool for computing $K_0$ of non-normal rings and of group rings such as $\Z[C_p]$, which sit in Milnor squares with simpler rings. It continues to the left with negative K-groups, which is one way to motivate them. It does not continue to the right in general, since excision fails for $K_1$. For the square built from an ideal $I$ common to $R \subseteq S$, it is closely related to excision for $K_0$.