For a smooth and proper variety over a finite field the reciprocity map 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.
Cite this article
Patrick Forré, The Kernel of the Reciprocity Map of Simple Normal Crossing Varieties over Finite Fields. Publ. Res. Inst. Math. Sci. 48 (2012), no. 4, pp. 919–936