Geometry of Supermanifolds through Sheaf and Ringed Space Methods

TL;DR

This paper develops an algebraic-geometric framework for supermanifolds using sheaf and ringed space methods, providing a rigorous foundation for their structure and morphisms.

Contribution

It introduces a sheaf-theoretic approach to supermanifolds, characterizing morphisms and establishing categorical properties, advancing the mathematical understanding of supergeometry.

Findings

Superfunctions exhibit richer structures than ordinary functions.

A key theorem characterizes morphisms between supermanifolds.

The category of supermanifolds admits finite products.

Abstract

This paper introduces the concept of supermanifolds, viewed as the super-analogues of classical manifolds. Instead of treating supermanifolds as sets of points, we adopt an algebraic-geometric perspective, emphasizing the algebra of functions and utilizing the framework of ringed spaces and sheaf theory. We begin by constructing presheaves and sheaves to define locally ringed spaces, which model the local structure of supermanifolds. Superfunctions, a key element, are shown to differ significantly from ordinary functions, leading to a richer structure. We also prove a key characterization theorem for morphisms between supermanifolds and demonstrate that the category of supermanifolds admits finite products. This approach provides a solid foundation for further studies in mathematics and theoretical physics.