Two Approaches to Supermanifolds

Abstract

The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways.Either one extends the sheaf of functions formally, as in the category of graded manifolds [3], [8], or one mimicks the usual definition of a manifold, having replaced Euclidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds [6].This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds.Introduction.Supermanifolds or graded manifolds were defined to provide a "space" whose "functions" would include anticommuting elements.There have been two approaches to the definition of these objects, emerging from two approaches to the study of the geometry of a manifold.Traditionally, the differential geometer regards the space itself as the primary object, but it is also possible to take the algebraic geometer's point of view and study the geometry of the space through the algebraic structure of its sheaf of functions.Supermanifolds as defined by de Witt [6] for example, follows the first approach, while graded manifolds, as defined by Kostant [8] are inspired by the algebraic geometer's approach.The purpose of this paper is to establish the equivalence of the category of supermanifolds with the category of graded manifolds.In the first section, basic definitions are given and the main theorem is stated.Smooth maps on "super-Euclidean space" are constructed in 2, and the proof of the equivalence theorem is given in 3.I would like to thank Professor S. Steinberg for bringing de Witt's work to my attention, and for subsequent discussions, and also Carolyn Schroeder for her help in finding the "right" definition for smooth maps on super-Euclidean space.1. Definitions and results.All algebras and vector spaces are over the real numbers, although similar constructions could be carried out using complex numbers.All (ordinary) manifolds are considered to be real, smooth, Hausdorff paracompact manifolds.1.1.Definition.An algebra A is called a Z2-graded algebra (or simply a graded algebra) if A can be written as a direct sum of linear subspaces A = A0 A, such that AjAj C ^4,-+;(mod2).