Notes on Super Projective Modules

TL;DR

This paper introduces the concept of super projective modules, extending classical projective modules to supergeometry, and demonstrates their properties using the supersphere as a key example.

Contribution

It defines super projective modules and explores their geometric and algebraic properties, particularly in the context of superspheres and supersmooth functions.

Findings

The module of vector fields over a supersphere is super projective.

Super projective modules can be constructed from projection maps on supersmooth function modules.

The paper establishes a link between supergeometry and algebra via super projective modules.

Abstract

Projective modules are a link between geometry and algebra as established by the theorem of Serre-Swan. In this paper, we define the super analog of projective modules and explore this link in the case of some particular super geometric objects. We consider the tangent bundle over the supersphere and show that the module of vector field over a supersphere is a super projective module over the ring of supersmooth functions. Also, we discuss a class of super projective modules that can be constructed from a projection map on modules defined over the ring of supersmooth functions over superspheres.