The Structure of $C^{\infty }$-Superschemes

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 ${\mathbb{Z}}_{\ge 0}$-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.

25 pages. Revised version. Includes significant structural changes and corrects inaccuracies from v1 regarding $n$-ary operations defined on split $C^{\infty }$-superrings