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

Abstract

Let $K$ be a finite extension of the $p$-adic numbers ${\mathbb{Q}}_p$ with ring of integers ${\mathcal{O}}_K$, $\mathcal{X}$ a regular scheme, proper, flat, and geometrically irreducible over ${\mathcal{O}}_K$ of dimension $d$, and ${\mathcal{X}}_K$ its generic fiber. We show, under some assumptions on ${\mathcal{X}}_K$, that there is a reciprocity isomorphism of locally compact groups $H_{ar}^{2d-1}({\mathcal{X}}_K,\mathbb{Z}(d))\simeq {\pi }_1^{ab}({\mathcal{X}}_K{)}_W$ from a new cohomology theory to an integral model ${\pi }_1^{ab}({\mathcal{X}}_K{)}_W$ of the abelianized geometric fundamental groups ${\pi }_1^{ab}({\mathcal{X}}_K{)}^{geo}$. After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups.