Étale degree map and 0-cycles

Abstract

Using the triangulated category of étale motives over a field k, for a smooth projective variety X over k, we define the group ${\text{CH}}_0^{\text{ét}}\left(X\right)$ as an étale analogue of 0-cycles. We study the properties of ${\text{CH}}_0^{\text{ét}}\left(X\right)$ and give a description of the birational invariance of such a group. We define and present the étale degree map using Gysin morphisms in étale motivic cohomology and the étale index as an analogue to the classical case. We give examples of smooth projective varieties over a field k without zero cycles of degree one but with étale zero cycles of degree one, but this property is not always true as we give examples where the étale degree map is not surjective.