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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.09969 |
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| _version_ | 1866915233725939712 |
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| author | Yang, Yuan |
| author_facet | Yang, Yuan |
| contents | For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent. The latter is an extension of a finite group $J$ by the group of $k$-points of a connected commutative unipotent algebraic group $U$. In this paper we show that (1) if $X$ is ordinary, then $U = 0$; (2) if $X$ is a surface, then $J$ is the Pontryagin dual of $NS(X)[p^\infty]$; (3) if $X$ is an abelian variety, then $J = 0$. Using Crew's formula, we compute the dimension of $U$ for surfaces and abelian $3$-folds. We show that, if $X$ is ordinary, then the unipotent subgroup of $Br(X\times Y)$ is isomorphic to the unipotent subgroup of $Br(Y)$. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on $p$-primary torsion of the Brauer group Yang, Yuan Algebraic Geometry Number Theory For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent. The latter is an extension of a finite group $J$ by the group of $k$-points of a connected commutative unipotent algebraic group $U$. In this paper we show that (1) if $X$ is ordinary, then $U = 0$; (2) if $X$ is a surface, then $J$ is the Pontryagin dual of $NS(X)[p^\infty]$; (3) if $X$ is an abelian variety, then $J = 0$. Using Crew's formula, we compute the dimension of $U$ for surfaces and abelian $3$-folds. We show that, if $X$ is ordinary, then the unipotent subgroup of $Br(X\times Y)$ is isomorphic to the unipotent subgroup of $Br(Y)$. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree $2$. |
| title | Remarks on $p$-primary torsion of the Brauer group |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2410.09969 |