The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to $Hom_{Chow\otimes \mathbb Q}(M(X),M(P^n))$ for $n\ge \dim X$. Is there a conceptual explanation for this?