Chow Künneth decomposition for étale motives

Abstract: In the present article we define an integral analogue of Chow-K\"unneth decomposition for \'etale motives. By using families of conservative functors we are able to establish a decomposition of the \'etale motive of commutative group schemes over a base and we relate to an integral \'etale Chow-K\"unneth decomposition of abelian varieties. For a projective variety $X$ of dimension $d$ over an algebraically closed field, we construct integral sub-motives $h^1_{\text{\'et}}(X)$ and $h^{2d-1}_{\text{\'et}}(X)$ of the motive $h_{\text{\'et}}(X)$.