On automorphisms of $p$-torsion $\mathbf{G}_m$-gerbes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917189505777664 |
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| author | Olander, Noah |
| author_facet | Olander, Noah |
| contents | Olsson showed in [Ols25] that if $\mathcal{X} \to X$ is a $\mathbf{G}_m$-gerbe over a smooth projective variety over an algebraically closed field $k$ such that the Brauer class of $\mathcal{X}$ has order prime to the characteristic of $k$, then the homomorphism of $k$-group algebraic spaces $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ is surjective. We provide an example to show that this need not be the case when the Brauer class of $\mathcal{X}$ has order equal to the characteristic. Our main tools are deformation theory of the fppf sheafified Artin--Mazur formal groups and nice properties of the flat cohomology of ordinary varieties in positive characteristic which are presumably well-known, but which we collect and give an exposition of here. We additionally prove some sufficient conditions for surjectivity of $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ using representability results of Bragg and Olsson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04427 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On automorphisms of $p$-torsion $\mathbf{G}_m$-gerbes Olander, Noah Algebraic Geometry Olsson showed in [Ols25] that if $\mathcal{X} \to X$ is a $\mathbf{G}_m$-gerbe over a smooth projective variety over an algebraically closed field $k$ such that the Brauer class of $\mathcal{X}$ has order prime to the characteristic of $k$, then the homomorphism of $k$-group algebraic spaces $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ is surjective. We provide an example to show that this need not be the case when the Brauer class of $\mathcal{X}$ has order equal to the characteristic. Our main tools are deformation theory of the fppf sheafified Artin--Mazur formal groups and nice properties of the flat cohomology of ordinary varieties in positive characteristic which are presumably well-known, but which we collect and give an exposition of here. We additionally prove some sufficient conditions for surjectivity of $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ using representability results of Bragg and Olsson. |
| title | On automorphisms of $p$-torsion $\mathbf{G}_m$-gerbes |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2601.04427 |