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

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