On some negative motivic homology groups

Abstract

For an arbitrary separated scheme $X$ of finite type over a finite field ${\mathbb{F}}_q$ and a negative integer $j$, we prove, under the assumption of resolution of singularities, that $H_{-1}\left(X,\mathbb{Z}\left(j\right)\right)$ is canonically isomorphic to $H_{-1}\left({\pi }_0\left(X\right),\mathbb{Z}\left(j\right)\right)$ if $j=-1$ or $-2$, and $H_i\left(X,\mathbb{Z}\left(j\right)\right)$ vanishes if $i\le -2$ and $i-j\le 1$. As the group $H_{-1}\left({\pi }_0\left(X\right),\mathbb{Z}\left(j\right)\right)$ is explicitly known, this gives a explicit calculation of motivic homology of degree $-1$ and weight $-1$ or $-2$ of an arbitrary scheme over a finite field.