By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, consider the complex (of functors) $X\mapsto (k(X)^*\to Div(X))$. As far as I understand, this is a complex of sheaves with transfers; its cohomology sheaves are $G_m$ and $Pic$. So, as an object of $DM^{eff}_-$ it is isomorphic to a cone of a homomorphism from $\mathbb{Z}(1)[2]$ (or is it $\mathbb{Z}(1)[2]\oplus \mathbb{Z}$? sorry, I am somewhat confused here) to $G_m[2]=\mathbb{Z}(1)[3]$. It seems that there are no non-trivial homomorphisms here; so this complex should be quasi-isomorphic to the direct sum of its cohomology sheaves. Is there an explicit quasi-isomorphism here? Is my reasoning correct?