An affineness criterion for algebraic groups and applications
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2018
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866916442886111232 |
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| author | de Salas, C. Sancho de Salas, F. Sancho de Salas, J. B. Sancho |
| author_facet | de Salas, C. Sancho de Salas, F. Sancho de Salas, J. B. Sancho |
| contents | We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth algebraic group and $\mathcal L$ is a $G$-invariant invertible sheaf on a $G$-variety $X$, then the action of $G$ on $X$ extends to a projective action on the complete linear ${\mathbb P}(H^0(X,{\mathcal L})$. As an application of the affineness criterion, we give a new and simple proof of Chevalley-Barsotti Theorem on the structure of algebraic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_05901 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | An affineness criterion for algebraic groups and applications de Salas, C. Sancho de Salas, F. Sancho de Salas, J. B. Sancho Algebraic Geometry 14L17, 14L30, 14L15 We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth algebraic group and $\mathcal L$ is a $G$-invariant invertible sheaf on a $G$-variety $X$, then the action of $G$ on $X$ extends to a projective action on the complete linear ${\mathbb P}(H^0(X,{\mathcal L})$. As an application of the affineness criterion, we give a new and simple proof of Chevalley-Barsotti Theorem on the structure of algebraic groups. |
| title | An affineness criterion for algebraic groups and applications |
| topic | Algebraic Geometry 14L17, 14L30, 14L15 |
| url | https://arxiv.org/abs/1802.05901 |