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| Main Author: | |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2009.14128 |
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| _version_ | 1866929495064182784 |
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| author | Hübner, Katharina |
| author_facet | Hübner, Katharina |
| contents | On a smooth discretely ringed adic space $\mathcal{X}$ over a field $k$ we define a subsheaf $Ω_{\mathcal{X}}^+$ of the sheaf of differentials $Ω_{\mathcal{X}}$. It is defined in a similar way as the subsheaf $\mathcal{O}^+_{\mathcal{X}}$ of $\mathcal{O}_{\mathcal{X}}$ using Kähler seminorms on $Ω_{\mathcal{X}}$. We give a description of $Ω^+_{\mathcal{X}}$ in terms of logarithmic differentials. If $\mathcal{X}$ is of the form $\mathrm{Spa}(X,\bar{X})$ for a scheme $\bar{X}$ and an open subscheme $X$ such that the corresponding log structure on $\bar{X}$ is smooth, we show that $Ω^+_{\mathcal{X}}(\mathcal{X})$ is isomorphic to the logarithmic differentials of $(X,\bar{X})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_14128 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Logarithmic differentials on discretely ringed adic spaces Hübner, Katharina Algebraic Geometry 14F10, 14G17, 14F20 On a smooth discretely ringed adic space $\mathcal{X}$ over a field $k$ we define a subsheaf $Ω_{\mathcal{X}}^+$ of the sheaf of differentials $Ω_{\mathcal{X}}$. It is defined in a similar way as the subsheaf $\mathcal{O}^+_{\mathcal{X}}$ of $\mathcal{O}_{\mathcal{X}}$ using Kähler seminorms on $Ω_{\mathcal{X}}$. We give a description of $Ω^+_{\mathcal{X}}$ in terms of logarithmic differentials. If $\mathcal{X}$ is of the form $\mathrm{Spa}(X,\bar{X})$ for a scheme $\bar{X}$ and an open subscheme $X$ such that the corresponding log structure on $\bar{X}$ is smooth, we show that $Ω^+_{\mathcal{X}}(\mathcal{X})$ is isomorphic to the logarithmic differentials of $(X,\bar{X})$. |
| title | Logarithmic differentials on discretely ringed adic spaces |
| topic | Algebraic Geometry 14F10, 14G17, 14F20 |
| url | https://arxiv.org/abs/2009.14128 |