On integral class field theory for varieties over p-adic fields

Abstract

Let K be a finite extension of the p-adic numbers Qp with ring of integers OK and residue field κ. Let X a regular scheme, proper, flat, and geometrically irreducible over OK of dimension d, and XK its generic fiber. We show, under some assumptions on X, that there is a reciprocity isomorphism of locally compact groups Har2d−1(XK,Z(d))≃π1ab(XK)W from the cohomology theory defined in [10] to an integral model π1ab(XK)W of the abelianized fundamental group π1ab(XK). After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups. The key ingredient is the duality result in [10].