On Frobenius rigidity for motivic cohomology

Abstract

We study how motivic and étale motivic cohomology of a smooth and proper variety $X$ changes under extensions of algebraically closed base fields $k$. We show that with finite coefficients, they are independent of $k$ away from the characteristic. At the characteristic, étale cohomology does depend on $k$, whereas for motivic cohomology this is only known in weights $0,1,\dim X$. We then consider the fiber of Frobenius on motivic cohomology and étale motivic cohomology for varieties defined over finite fields, i.e., Weil-étale cohomology. Combining the results of the first part with the structure theory of perfect unipotent group schemes, we show that this fiber is independent of $k$ with finite coefficients. Finally, we give some results and examples with integral coefficients.