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

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 $X$, $H_0(X,\mathbb{Z}{)}_{\tau }$ is the zeroth group of truncated homology ${\mathrm{Hom}}_{D(k_{\mathrm{sm}})}({\tau }_{\le 1}R{\varphi }_{\ast }{\mathbb{G}}_{m,X},{\mathbb{G}}_{m,k})$, $\varphi$ is the structure morphism of $X$, and $(-{)}^{\wedge }$ is the profinite completion functor.

arXiv comment39 pages