Lichtenbaum-van Hamel duality for singular varieties over $p$-adic fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918432830652416 |
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| author | Rivera-Mesas, Felipe |
| author_facet | Rivera-Mesas, Felipe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22614 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lichtenbaum-van Hamel duality for singular varieties over $p$-adic fields Rivera-Mesas, Felipe Number Theory Algebraic Geometry 14G20, 14F22, 11S25 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. |
| title | Lichtenbaum-van Hamel duality for singular varieties over $p$-adic fields |
| topic | Number Theory Algebraic Geometry 14G20, 14F22, 11S25 |
| url | https://arxiv.org/abs/2512.22614 |