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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.15353 |
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| _version_ | 1866912131050373120 |
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| author | Petrov, Alexander Skorobogatov, Alexei |
| author_facet | Petrov, Alexander Skorobogatov, Alexei |
| contents | We give a new, short proof of the formula for the first potentially non-zero differential of the Hochschild--Serre spectral sequence for semiabelian varieties over non-closed fields. We show that this differential is non-zero for the Jacobian of a curve when the image of the torsor of theta-characteristics under the Bockstein map is non-zero. An explicit example is a curve of genus 2 whose Albanese torsor is not divisible by 2. When the Albanese torsor is trivial, we show that the Hochschild--Serre spectral sequence for the Jacobian degenerates at the second page. We give a formula for the differential of the Hochschild--Serre spectral sequence for a torus which computes its Brauer group. Finally, we describe the differentials of the Hochschild--Serre spectral sequence for a smooth projective curve, generalising a lemma of Suslin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On spectral sequences for semiabelian varieties over non-closed fields Petrov, Alexander Skorobogatov, Alexei Algebraic Geometry 18G40, 11G10 We give a new, short proof of the formula for the first potentially non-zero differential of the Hochschild--Serre spectral sequence for semiabelian varieties over non-closed fields. We show that this differential is non-zero for the Jacobian of a curve when the image of the torsor of theta-characteristics under the Bockstein map is non-zero. An explicit example is a curve of genus 2 whose Albanese torsor is not divisible by 2. When the Albanese torsor is trivial, we show that the Hochschild--Serre spectral sequence for the Jacobian degenerates at the second page. We give a formula for the differential of the Hochschild--Serre spectral sequence for a torus which computes its Brauer group. Finally, we describe the differentials of the Hochschild--Serre spectral sequence for a smooth projective curve, generalising a lemma of Suslin. |
| title | On spectral sequences for semiabelian varieties over non-closed fields |
| topic | Algebraic Geometry 18G40, 11G10 |
| url | https://arxiv.org/abs/2411.15353 |