On Integral Class field theory for varieties over $p$-adic fields

. Let K be a finite extension of the p -adic numbers Q p with ring of integers O K , X a regular scheme, proper, flat, and geometrically irreducible over O K of dimension d , and X K its generic fiber. We show, under some assumptions on X K , that there is a reciprocity isomorphism of locally compact groups H 2 d − 1 ar ab W from the cohomology theory defined in [9] to an integral model π ab 1 ( X K ) W of the abelianized geometric fundamental groups π ab 1 ( X K ) geo . 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 [9].