\'Etale degree map and 0-cycles

Abstract

By using the triangulated category of \'etale motives over a field $k$, for a smooth projective variety $X$ over $k$, we define the group $\text{CH}^\text{\'et}_0(X)$ as an \'etale analogue of 0-cycles. We study the properties of $\text{CH}^\text{\'et}_0(X)$, giving a description about the birational invariance of such group. We define and present the \'etale degree map by using Gysin morphisms in \'etale motivic cohomology and the \'etale 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 \'etale zero cycles of degree one, however, this property is not always true as we present examples where the \'etale degree map is not surjective.