On some negative motivic homology groups

TL;DR

This paper proves an isomorphism between certain negative motivic homology groups for schemes over finite fields, providing explicit computations under the assumption of resolution of singularities.

Contribution

It establishes a canonical isomorphism between negative motivic homology groups of schemes and their connected components, assuming resolution of singularities.

Findings

Proves isomorphism of $H_{-1}(X,Z(j))$ and $H_{-1}(pi_0(X),Z(j))$

Provides explicit computation of negative motivic homology groups

Assumes resolution of singularities for the proof

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 $H_{-1}(X,\mathbb Z(j)).$