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

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.

40 pages. Revised version. Includes significant updates to Sections 3.1 and 3.2. Specifically corrects inaccuracies regarding $n$-ary operations on split $C^{\infty }$-superrings, including the Weil superalgebra example in Section 3.4