The Structure of $C^\infty$-Superschemes

TL;DR

This paper generalizes Batchelor's theorem for $C^inity$-superschemes, showing that global splittings relate to Euler vector fields, thus providing a differential-geometric perspective on superspace structures.

Contribution

It establishes a non-canonical isomorphism between the structure sheaf and its graded version, linking splittings to Euler vector fields in $C^inity$-superspaces.

Findings

Any Batchelor space satisfies a global splitness condition.

The structure sheaf admits a natural $Z_{ e 0}$-grading.

Global splittings are characterized by Euler vector fields.

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.