On the Brauer groups of fibrations II
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910646550921216 |
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| author | Qin, Yanshuai |
| author_facet | Qin, Yanshuai |
| contents | Let $K$ be a number field, and let $\mathcal{X}$ be a proper regular flat scheme over $\mathcal{O}_{K}$ with a generic fiber $X$ geometrically connected over $K$. We prove that there is an exact sequence up to finite groups $0\rightarrow Sha(Pic_{X/K}^0)\rightarrow Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}\rightarrow 0$, which generalizes a theorem of Artin and Grothendieck for arithmetic surfaces to arbitrary dimensions. Consequently, we reduce Artin's question regarding the finiteness of $Br(\mathcal{X})$ for proper regular flat schemes $\mathcal{X}$ over $\mathbb{Z}$ to $3$-dimensional arithmetic schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_06910 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Brauer groups of fibrations II Qin, Yanshuai Algebraic Geometry Number Theory 14F22 Let $K$ be a number field, and let $\mathcal{X}$ be a proper regular flat scheme over $\mathcal{O}_{K}$ with a generic fiber $X$ geometrically connected over $K$. We prove that there is an exact sequence up to finite groups $0\rightarrow Sha(Pic_{X/K}^0)\rightarrow Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}\rightarrow 0$, which generalizes a theorem of Artin and Grothendieck for arithmetic surfaces to arbitrary dimensions. Consequently, we reduce Artin's question regarding the finiteness of $Br(\mathcal{X})$ for proper regular flat schemes $\mathcal{X}$ over $\mathbb{Z}$ to $3$-dimensional arithmetic schemes. |
| title | On the Brauer groups of fibrations II |
| topic | Algebraic Geometry Number Theory 14F22 |
| url | https://arxiv.org/abs/2103.06910 |