On Brauer groups of tame stacks
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909695208325120 |
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| author | Achenjang, Niven |
| author_facet | Achenjang, Niven |
| contents | We develop some general tools for computing the Brauer group of a tame algebraic stack $\mathscr X$ by studying the difference between it and the Brauer group of the coarse space $X$ of $\mathscr X$. It is our hope that these tools will be used to simplify future computations of Brauer groups of stacks. Informally, we show that $\operatorname{Br}\mathscr X$ is often built from $\operatorname{Br} X$ and "information about the Picard groups of the fibers of $\mathscr X\to X$''. Along these lines, we compute, for example, the Brauer group of the moduli stack $\mathscr Y(1)_S$ of elliptic curves, over any regular noetherian $\mathbb Z[1/2]$-scheme $S$ as well as the Brauer groups of (many) stacky curves (allowing generic stabilizers) over algebraically closed fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06217 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Brauer groups of tame stacks Achenjang, Niven Algebraic Geometry Number Theory 14F22 (Primary) 14D23, 14A20 (Secondary) We develop some general tools for computing the Brauer group of a tame algebraic stack $\mathscr X$ by studying the difference between it and the Brauer group of the coarse space $X$ of $\mathscr X$. It is our hope that these tools will be used to simplify future computations of Brauer groups of stacks. Informally, we show that $\operatorname{Br}\mathscr X$ is often built from $\operatorname{Br} X$ and "information about the Picard groups of the fibers of $\mathscr X\to X$''. Along these lines, we compute, for example, the Brauer group of the moduli stack $\mathscr Y(1)_S$ of elliptic curves, over any regular noetherian $\mathbb Z[1/2]$-scheme $S$ as well as the Brauer groups of (many) stacky curves (allowing generic stabilizers) over algebraically closed fields. |
| title | On Brauer groups of tame stacks |
| topic | Algebraic Geometry Number Theory 14F22 (Primary) 14D23, 14A20 (Secondary) |
| url | https://arxiv.org/abs/2410.06217 |