Let $A$ be a Dedekind domain, and define $K_0(A)$ using isomorphism classes of finitely generated projective $A$-modules. For every finitely generated projective $A$-module $P$, let $\operatorname{rank}_A(P)\in\mathbb Z_{\ge 0}$ denote its constant rank and let $\det_A(P)$ denote its determinant invertible $A$-module, with the rank-zero convention $\det_A(P)=A$. There is an isomorphism of abelian groups

\begin{align*} \Phi_A:K_0(A)\to \mathbb Z\oplus \operatorname{Pic}(A) \end{align*}

characterized on classes of finitely generated projective $A$-modules by

\begin{align*} \Phi_A([P])=(\operatorname{rank}_A(P),[\det_A(P)]). \end{align*}

Equivalently, for a virtual class $[P]-[Q]\in K_0(A)$,

\begin{align*} \Phi_A([P]-[Q])=(\operatorname{rank}_A(P)-\operatorname{rank}_A(Q),[\det_A(P)\otimes_A \det_A(Q)^{-1}]). \end{align*}

Moreover, if $f:A\to B$ is a unital homomorphism of Dedekind domains and $B\otimes_A P$ is finitely generated projective over $B$ for every finitely generated projective $A$-module $P$, then under the identifications $\Phi_A$ and $\Phi_B$, the induced homomorphism

\begin{align*} K_0(f):K_0(A)\to K_0(B) \end{align*}

is given by

\begin{align*} (n,[L])\mapsto (n,[B\otimes_A L]) \end{align*}

for every $n\in\mathbb Z$ and every invertible $A$-module $L$.