K_0-motives and the Chow weight-heart

Abstract

Bondarko and Luzgarev constructed the Chow weight structure on the homotopy category of $KGL$-modules; we show that the heart of this weight structure is equivalent to the category of $K_0$-motives introduced by Gillet and Soulé, which is the $K$-theoretic counterpart of the category of relative Chow motives defined by Corti and Hanamura. The proof is achieved via a detailed study on the Borel-Moore theory in $KGL$-modules, as well as explicit formulas for products in algebraic $G$-theory in terms of $\mathrm{Tor}$-operations.