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

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 $${\mathrm{B}\mathrm{r}}_1(X)\times H_0(X,\mathbb{Z}{)}_{\tau }^{\wedge }\to \mathbb{Q}\mathrm{/}\mathbb{Z},$$ where ${\mathrm{B}\mathrm{r}}_1(X):=\ker (\mathrm{B}\mathrm{r}(X)\to \mathrm{B}\mathrm{r}(\bar{X}))$ is the algebraic Brauer group of $X$, $H_0(X,\mathbb{Z}{)}_{\tau }$ is the zeroth group of truncated homology ${\mathrm{H}\mathrm{o}\mathrm{m}}_{D(k_{\mathrm{s}\mathrm{m}})}({\tau }_{\le 1}R{\phi }_{\ast }{\mathbb{G}}_{m,X},{\mathbb{G}}_{m,k})$, $\phi$ is the structure morphism of $X$, and $(-{)}^{\wedge }$ is the profinite completion functor.