Let $A$ be a Dedekind domain with fraction field $F$. Let $S=A\setminus\{0_A\}$, and let $\mathcal T_S(A)$ be the exact category of finitely generated $A$-modules $M$ such that $S^{-1}M=0$. Let $\operatorname{Div}(A)$ denote the free abelian group on the nonzero prime ideals of $A$, so $\operatorname{Div}(A)=\bigoplus_{\mathfrak p \subset A,\ \mathfrak p\ne 0}\mathbb Z[\mathfrak p]$. For each nonzero prime ideal $\mathfrak p\subset A$, let $v_{\mathfrak p}:F^\times\to \mathbb Z$ be the normalized valuation at the discrete valuation ring $A_{\mathfrak p}$, and define the divisor map

\begin{align*} \operatorname{div}:F^\times \to \operatorname{Div}(A) \end{align*}

by $\operatorname{div}(x)=\sum_{\mathfrak p\ne 0} v_{\mathfrak p}(x)[\mathfrak p]$. Let

\begin{align*} \lambda:K_0(\mathcal T_S(A))\to \operatorname{Div}(A) \end{align*}

be the length-devissage isomorphism $[M]\mapsto \sum_{\mathfrak p\ne0}\operatorname{length}_{A_{\mathfrak p}}(M_{\mathfrak p})[\mathfrak p]$. Let $\operatorname{Div}(A)\to K_0(A)$ be the composite $\operatorname{Div}(A)\xrightarrow{\lambda^{-1}}K_0(\mathcal T_S(A))\to K_0(A)$, where the second map is the localization-sequence map. Let $K_0(A)\to K_0(F)$ be induced by extension of scalars along $A\hookrightarrow F$. Then there is an exact sequence of abelian groups

\begin{align*} A^\times \longrightarrow F^\times \xrightarrow{\operatorname{div}} \operatorname{Div}(A) \longrightarrow K_0(A) \longrightarrow K_0(F) \longrightarrow 0. \end{align*}

Here the first map is the inclusion $A^\times\hookrightarrow F^\times$. Under the dimension isomorphism $K_0(F)\cong \mathbb Z$, the map $K_0(A)\to K_0(F)$ is the rank map. Consequently, the kernel of the rank map $K_0(A)\to \mathbb Z$ is naturally isomorphic to the ideal class group $\operatorname{Cl}(A)=\operatorname{Div}(A)/\operatorname{div}(F^\times)$.