Bass–Heller–Swan theorem

Remarks

The proof studies the projective $R[t]$-modules that become free after inverting $t$, and the localization sequence for $R[t] \to R[t, t^{-1}]$. The cokernel of $K_1(R[t]) \oplus K_1(R[t^{-1}]) \to K_1(R[t, t^{-1}])$ is identified with $K_0(R)$ through the multiplication by $t$.

Example: let $F$ be a field. The ring $F[t, t^{-1}]$ is Euclidean, so its $K_1$ is its group of units, which is $\set{c t^m : c \in F^\times, m \in \Z} \cong F^\times \times \Z$. This agrees with $K_1(F) \oplus K_0(F) = F^\times \oplus \Z$, and the factor $\Z$ is generated by the class of $t$.

For a group $G$, applied to the group ring $R = \Z G$, the theorem gives a decomposition of the Whitehead group of $G \times \Z$ that contains the reduced group $\tilde K_0(\Z G)$ as a summand.

Going deeper

For a student

This is the degree one case of the fundamental theorem of algebraic K-theory, which gives the same pattern for $K_n$ in every degree. Bass used the cokernel above to define negative K-groups, so that the pattern continues below degree zero. In topology the formula for the Whitehead group of $G \times \Z$ controls the torsion of h-cobordisms over manifolds with fundamental group $G \times \Z$.