A $p$-adic local invariant cycle theorem with applications to Brauer groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912194758705152 |
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| author | Qin, Yanshuai |
| author_facet | Qin, Yanshuai |
| contents | In this article, we prove a $p$-adic analogue of the local invariant cycle theorem for $H^2$ in mixed characteristics. As a result, for a smooth projective variety $X$ over a $p$-adic local field $K$ with a proper flat regular model $\mathcal{X}$ over $O_K$, we show that the natural map $Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}$ has a finite kernel and a finite cokernel. And we prove that the natural map $Hom(Br(X)/Br(K)+Br(\mathcal{X}), \mathbb{Q}/\mathbb{Z}) \rightarrow Alb_X(K)$ has a finite kernel and a finite cokernel, generalizing Lichtenbaum's duality between Brauer groups and Jacobians for curves to arbitrary dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_04945 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A $p$-adic local invariant cycle theorem with applications to Brauer groups Qin, Yanshuai Algebraic Geometry Number Theory 14F22 In this article, we prove a $p$-adic analogue of the local invariant cycle theorem for $H^2$ in mixed characteristics. As a result, for a smooth projective variety $X$ over a $p$-adic local field $K$ with a proper flat regular model $\mathcal{X}$ over $O_K$, we show that the natural map $Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}$ has a finite kernel and a finite cokernel. And we prove that the natural map $Hom(Br(X)/Br(K)+Br(\mathcal{X}), \mathbb{Q}/\mathbb{Z}) \rightarrow Alb_X(K)$ has a finite kernel and a finite cokernel, generalizing Lichtenbaum's duality between Brauer groups and Jacobians for curves to arbitrary dimensions. |
| title | A $p$-adic local invariant cycle theorem with applications to Brauer groups |
| topic | Algebraic Geometry Number Theory 14F22 |
| url | https://arxiv.org/abs/2103.04945 |