Lichtenbaum-van Hamel duality for singular varieties over $p$-adic fields

TL;DR

This paper generalizes the Lichtenbaum-van Hamel duality theorem to include singular, proper, and geometrically integral varieties over p-adic fields, establishing a natural perfect pairing involving the algebraic Brauer group.

Contribution

It extends duality results to singular varieties over p-adic fields, providing a new perfect pairing between the algebraic Brauer group and a truncated homology group.

Findings

Established a natural continuous perfect pairing for singular varieties over p-adic fields.

Extended the duality theorem to not necessarily smooth, proper, geometrically integral varieties.

Connected the algebraic Brauer group with truncated homology via a duality pairing.

Abstract

In this article, we extend the van Hamel-Lichtenbaum duality theorem to (not necessarily smooth) proper and geometrically integral varieties defined over a $p$-adic field $k$. More precisely, we prove that for such variety $X$ there exists a natural continuous perfect pairing \[ \mathrm{Br}_1(X)\times H_0(X,\mathbb{Z})_\tau^{\wedge} \to \mathbb{Q}/\mathbb{Z}, \] where $\mathrm{Br}_1(X):=\ker(\mathrm{Br}(X)\to\mathrm{Br}(\overline{X}))$ is the algebraic Brauer group of $X$, $H_0(X,\mathbb{Z})_\tau$ is the zeroth group of truncated homology $\mathrm{Hom}_{D(k_{\mathrm{sm}})}(\tau_{\leq 1}R\phi_*\mathbb{G}_{m,X},\mathbb{G}_{m,k})$, $\phi$ is the structure morphism of $X$, and $(-)^{\wedge}$ is the profinite completion functor.