Weight homology of motives

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.