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arXiv · 2605.07169

The Structure of $C^\infty$-Superschemes

Cristian Danilo Olarte·Pedro Rizzo·Alexander Torres-Gomez
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Abstract

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.

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Cristian Danilo Olarte, Pedro Rizzo, Alexander Torres-Gomez. 2026-05-08. The Structure of $C^\infty$-Superschemes. https://arxiv.org/abs/2605.07169

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