The Kernel of the Reciprocity Map of Simple Normal Crossing Varieties over Finite Fields

Abstract

For a smooth and proper variety $Y$ over a finite field $k$ the reciprocity map ${\rho }^Y:{\mathrm{CH}}_0(Y)\to {\pi }_1^{\mathrm{ab}}(Y)$ is injective with dense image. For a proper simple normal crossing variety this is no longer true in general. In this paper we give a discription of the kernel and cokernel of the reciprocity map in terms of homology groups of a complex filled with descent data using an algebraic Seifert–van Kampen theorem. Furthermore, we give a new criterion for the injectivity of the reciprocity map for proper simple normal crossing varieties over finite fields.