Weight homology of motives

TL;DR

This paper introduces a new weight homology functor for effective motives, explores its properties, and compares motivic homology with étale motivic homology, extending previous theories to more general schemes.

Contribution

It defines a novel weight homology functor on Voevodsky's motives and establishes a comparison theorem between motivic and étale motivic homology without smoothness restrictions.

Findings

Recovery of Gillet-Soulé's weight homology in special cases

Recovery of Geisser's Kato-Suslin homology in special cases

Proved a comparison theorem between motivic and étale motivic homology

Abstract

In the first half of this article we define a new weight homology functor on Voevodsky's category of effective motives, and investigate some of its properties. In special cases we recover Gillet-Soul\'e's weight homology, and Geisser's Kato-Suslin homology. In the second half, we consider the notions of "co-\'etale" and "reduced" motives, and use the notions to a prove a theorem comparing motivic homology to \'etale motivic homology. Due to the first author's Ph.D. thesis arXiv:1305.5349 we do not have to restrict to smooth schemes.