van Hamel-Lichtenbaum duality for singular varieties over $p$-adic fields
Felipe Rivera-Mesas
TL;DR
The paper generalizes van Hamel's duality by extending Lichtenbaum's Brauer–Picard framework to possibly singular proper varieties over a $p$-adic field. It introduces truncated homology $H_i(X,\mathbb{Z})_{\tau}$ and proves a natural continuous perfect pairing $H_0(X,\mathbb{Z})_{\tau}^{\wedge} \times \mathrm{Br}_1(X) \to \mathbb{Q}/\mathbb{Z}$, recovering the classical curve case $\mathrm{Br}(X)^* \simeq H_0(X,\mathbb{Z})_{\tau}^{\wedge}$. The approach rests on a detailed construction of a filtration of $\tau_{\le 1}R\phi_*\mathbb{G}_{m,X}$, a duality theory for generalized $1$-motives, and a robust topology on cohomology groups to obtain Tate duality for each graded piece. The results unify and extend Tate duality for singular geometries, with explicit applications to pinched varieties and normalization maps, providing new arithmetic control over $\mathrm{Br}_1(X)$ and the $k$-rational structure of $0$-cycles via $H_0(X,\mathbb{Z})_{\tau}$.
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 \[ \mathrm{Br}_1(X)\times H_0(X,\mathbb{Z})_τ^{\wedge} \to \mathbb{Q}/\mathbb{Z}, \] where $\mathrm{Br}_1(X):=\ker(\mathrm{Br}(X)\to\mathrm{Br}(\overline{X}))$ is the algebraic Brauer group of $X$, $H_0(X,\mathbb{Z})_τ$ is the zeroth group of truncated homology $\mathrm{Hom}_{D(k_{\mathrm{sm}})}(τ_{\leq 1}Rφ_*\mathbb{G}_{m,X},\mathbb{G}_{m,k})$, $φ$ is the structure morphism of $X$, and $(-)^{\wedge}$ is the profinite completion functor.