On some negative motivic homology groups

Abstract: For an arbitrary separated scheme $X$ of finite type over a finite field $\mathbb F_q$ and an integer $j=-1,-2,$ we prove under the assumption of resolution of singularities, that the two groups $H_{-1}(X,\mathbb Z(j))$ and $H_{-1}(\pi_0(X),\mathbb Z(j))$ are canonically isomorphic. This gives an explicit computation of $H_{-1}(X,\mathbb Z(j)).$