Let $R$ be a unital ring, and interpret $K_0$ and $K_1$ as the low Quillen algebraic $K$-groups of unital rings, with $K_0(R)$ defined from finitely generated projective left $R$-modules and $K_1(R)$ in the standard stable automorphism model. Let $R[t]$ be the [polynomial ring](/page/Polynomial%20Ring) in one central indeterminate $t$, let $R[t,t^{-1}]$ be its Laurent localization obtained by adjoining an inverse to $t$, and let

\begin{align*} \operatorname{ev}_0:R[t]\to R \end{align*}

be the unital evaluation homomorphism determined by $\operatorname{ev}_0(t)=0$ and $\operatorname{ev}_0(r)=r$ for $r\in R$. Define

\begin{align*} NK_1(R):=\ker\bigl(K_1(\operatorname{ev}_0):K_1(R[t])\to K_1(R)\bigr). \end{align*}

Let $i:R\to R[t,t^{-1}]$ be the constant Laurent polynomial inclusion. For a finitely generated projective left $R$-module $P$, let $\tau_P$ denote multiplication by $t$ on $R[t,t^{-1}]\otimes_R P$; the induced Bass suspension homomorphism is

\begin{align*} \beta_R:K_0(R)\to K_1(R[t,t^{-1}]),\qquad [P]\mapsto [\tau_P]. \end{align*}

Let $j_+:R[t]\to R[t,t^{-1}]$ send $t$ to $t$, and let $j_-:R[s]\to R[t,t^{-1}]$ send $s$ to $t^{-1}$, where $s$ is an independent central polynomial variable. Then the homomorphism

\begin{align*} K_1(R)\oplus K_0(R)\oplus NK_1(R)\oplus NK_1(R)\to K_1(R[t,t^{-1}]) \end{align*}

whose four summands are induced respectively by $K_1(i)$, $\beta_R$, $K_1(j_+)$ on the evaluation kernel of $R[t]\to R$, and $K_1(j_-)$ on the evaluation kernel of $R[s]\to R$, is a natural isomorphism of abelian groups.