$C^\infty$-superrings and $C^\infty$-superschemes

TL;DR

This paper develops a theory of $C^inity$-superrings and superschemes, establishing an equivalence between certain categories and focusing on split structures that generalize supermanifold function algebras.

Contribution

It introduces the concept of $C^inity$-superrings and superschemes, proving an equivalence between categories and analyzing split structures as foundational elements.

Findings

Established an equivalence between fair affine $C^inity$-superschemes and $C^inity$-superrings.

Characterized split $C^inity$-superrings as generalizations of supermanifold function algebras.

Provided a framework for constructing complex superschemes from split structures.

Abstract

This paper develops a theory of $C^\infty$-superrings and their associated $C^\infty$-superschemes. We prove a key equivalence between the category of fair affine $C^\infty$-superschemes and the category of fair $C^\infty$-superrings. We place special emphasis on split $C^\infty$-superrings, which generalize the function algebras of supermanifolds and serve as building blocks for more complex, non-split structures.