Let $A$ be a Dedekind domain, and define $K_0(A)$ as the Grothendieck group of the commutative monoid of isomorphism classes of finitely generated projective $A$-modules under direct sum. For every finitely generated projective $A$-module $P$, let $\operatorname{rank}(P)\in \mathbb Z_{\ge 0}$ denote its constant rank and let $\det(P)$ denote its determinant invertible $A$-module, with the convention $\det(0)=A$. Then the assignment $[P]\longmapsto \bigl(\operatorname{rank}(P),[\det(P)]\bigr)$ induces an isomorphism of abelian groups $K_0(A)\cong \mathbb Z\oplus \operatorname{Pic}(A)$. Moreover, if $f:A\to B$ is a unital ring homomorphism between Dedekind domains, then extension of scalars along $f$ sends finitely generated projective $A$-modules to finitely generated projective $B$-modules, induces a homomorphism $K_0(f):K_0(A)\to K_0(B)$, and, under the above isomorphisms for $A$ and $B$, this homomorphism is the map $(n,[L])\longmapsto (n,[L\otimes_A B])$ from $\mathbb Z\oplus \operatorname{Pic}(A)$ to $\mathbb Z\oplus \operatorname{Pic}(B)$, where $L$ is any invertible $A$-module representing the class $[L]$.