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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.07169 |
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| _version_ | 1866911661364871168 |
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| author | Olarte, Cristian Danilo Rizzo, Pedro Torres-Gomez, Alexander |
| author_facet | Olarte, Cristian Danilo Rizzo, Pedro Torres-Gomez, Alexander |
| contents | This paper establishes a structural generalization of Batchelor's theorem within the framework of $C^\infty$-superschemes. Our main result proves that any Batchelor space satisfies a global splitness condition, establishing an isomorphism between the structure sheaf and its associated graded sheaf. Although this isomorphism is non-canonical, the existence of a splitting endows the structure sheaf with a natural $\mathbb{Z}_{\geq 0}$-grading. This grading is shown to be equivalent to the data of an even superderivation, which we term an Euler vector field. Consequently, global splittings of $C^\infty$-superspaces can be characterized in terms of Euler vector fields, providing a differential-geometric formulation of the splitting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07169 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Structure of $C^\infty$-Superschemes Olarte, Cristian Danilo Rizzo, Pedro Torres-Gomez, Alexander Algebraic Geometry Differential Geometry This paper establishes a structural generalization of Batchelor's theorem within the framework of $C^\infty$-superschemes. Our main result proves that any Batchelor space satisfies a global splitness condition, establishing an isomorphism between the structure sheaf and its associated graded sheaf. Although this isomorphism is non-canonical, the existence of a splitting endows the structure sheaf with a natural $\mathbb{Z}_{\geq 0}$-grading. This grading is shown to be equivalent to the data of an even superderivation, which we term an Euler vector field. Consequently, global splittings of $C^\infty$-superspaces can be characterized in terms of Euler vector fields, providing a differential-geometric formulation of the splitting. |
| title | The Structure of $C^\infty$-Superschemes |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2605.07169 |