On the weight zero motivic cohomology

TL;DR

This paper establishes an isomorphism between singular cohomology of Berkovich analytifications, cdh-cohomology, and weight zero motivic cohomology for schemes over trivially-valued fields, revealing new structural insights.

Contribution

It proves the equivalence of singular cohomology, cdh-cohomology, and weight zero motivic cohomology, and explores implications for sheaves with transfers and morphisms between algebraic groups.

Findings

Singular cohomology is isomorphic to cdh-cohomology for schemes over trivially-valued fields.

Vanishing of certain RHom groups involving sheaves with transfers associated to algebraic groups.

Explicit description of RHom groups between sheaves associated to abelian varieties.

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 $Hom_{\mathbf{Ab_k}}(A,B)$.