Étale degree map and 0-cycles

Abstract

By using the triangulated category of étale motives over a field $k$, for a smooth projective variety $X$ over $k$, we define the group ${\text{CH}}_0^{\text{A˜c◯t}}(X)$ as an étale analogue of 0-cycles. We study the properties of ${\text{CH}}_0^{\text{A˜c◯t}}(X)$, giving a description about the birational invariance of such group. We define and present the étale degree map by 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, however, this property is not always true as we present examples where the étale degree map is not surjective.