On the weight zero motivic cohomology

Abstract

We prove that singular cohomology of the underlying space of Berkovich's analytification of a scheme $X$ locally of finite type over a trivially-valued field $k$ of characteristic $0$ is isomorphic to cdh-cohomology with integer coefficients which is also isomorphic to the weight zero motivic cohomology $H^*(X, \mathbb{Z})$. Using this isomorphism, we demonstrate the vanishing of $RHom_{Sh_{Nis}(cor_k)}(\underline{G},\mathbb{Z})$, where $\underline{G}$ denotes the Nisnevich sheaf with transfers associated with a commutative algebraic group $G$ over $k$. For abelian $k$-varieties $A$ and $B$, we prove that $RHom_{\mathcal{PS}h_{tr}}(\underline{A},\underline{B})$ is isomorphic to $Hom_{\mathbf{Ab_k}}(A,B)$.