that sends $(g,m)$ to $g t^m$. Let $U\le K_1(R[t,t^{-1}])$ be the subgroup generated by the unit classes $[\varepsilon g t^m]$ with $\varepsilon\in\{\pm1\}$, $g\in G$, and $m\in\mathbb Z$. Assume the normalized Bass-Heller-Swan isomorphism is the natural isomorphism
for which the inclusion $R\hookrightarrow R[t,t^{-1}]$ gives the $K_1(R)$ summand, the Laurent unit class $[t]$ maps to $[R]\in K_0(R)$, and the two copies of $NK_1(R)$ are the positive and negative Bass nil summands. Then $\beta$ descends through the quotient by $U$ to a natural isomorphism