Let $A$ be a Dedekind domain with fraction field $F$, and let $S=A\setminus\{0\}$. Let

\begin{align*} \partial:K_1(F)\longrightarrow K_0(\mathcal T_S(A)) \end{align*}

be the boundary homomorphism in the low-degree localization sequence, using the boundary convention that an injective lattice map whose cokernel is a torsion module $M$ contributes $[M]$ in $K_0(\mathcal T_S(A))$. Here $\mathcal T_S(A)$ is the exact category of finitely generated torsion $A$-modules. Identify

\begin{align*} K_0(\mathcal T_S(A))\cong \bigoplus_{\mathfrak p\ne 0}K_0(A/\mathfrak p) \end{align*}

by primary decomposition and devissage over the nonzero prime ideals of $A$, and normalize this identification so that the class of $A/\mathfrak p$ maps to the positive generator of $K_0(A/\mathfrak p)\cong \mathbb Z$. Under the identifications $K_1(F)\cong F^\times$ and $K_0(A/\mathfrak p)\cong \mathbb Z$ for every nonzero prime ideal $\mathfrak p\subset A$, the boundary map is the divisor map

\begin{align*} F^\times\longrightarrow \bigoplus_{\mathfrak p\ne 0}\mathbb Z, \end{align*}

which sends $x\in F^\times$ to $(v_{\mathfrak p}(x))_{\mathfrak p}$, where $v_{\mathfrak p}:F^\times\to\mathbb Z$ is the normalized discrete valuation associated to the discrete valuation ring $A_{\mathfrak p}$.