The kernel of the reciprocity map of simple normal crossing varieties over finite fields (Q1932215)

Summary: For a smooth and proper variety \(Y\) over a finite field \(k\) the reciprocity map \(\rho^Y: \text{CH}_0(Y) \to \pi_1^{ab}(Y)\) 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.