Nisnevich Descent Fails for 0-cycles With Modulus

Abstract We show that, contrary to the ordinary Chow groups, the canonical map from the Chow group of 0-cycles with modulus on a $d$-dimensional smooth variety $X$ with an effective divisor $D$ over a field to the $(2d,d)$-th Nisnevich motivic cohomology of $X$ relative to $D$ is not always an isomorphism. This is achieved by showing that the degree zero part of the motivic cohomology with modulus over a finite field is isomorphic to the geometric part of an abelian étale fundamental group with modulus.