The $n$-th $\mathcal{K}$-homology sheaf on $X$ is defined as the sheafification of the presheaf $U\mapsto K_n(U)$. Here $K_n(U)$ is the $n$-th algebraic $K$-group. I was just wondering whether it is possible to give a description of this sheaf in the simplest possible cases i.e. $\mathbb{A}^m$ and $\mathbb{P}^m$. My guess is that the sheaf is probably a constant sheaf just because of how symmetric these spaces are!
I think my idea of being a locally constant sheaf cannot be true, sicne $H^i(X,\mathcal{K}_i)\cong CH^i(X)$ and sheaf cohomology of locally constant sheaves are trivial but the group of $i$-cycles in the projective space is not trivial.