The axioms of supermanifolds and a new structure arising from them

Abstract

An analysis of supermanifolds over an arbitrary graded-commmutative algebra is given, proceeding from a set of axioms the first of which is that the derivations of the structure sheaf of a supermanifold are locally free. These axioms are satisfied not by the sheaf of G ∞ {G^\infty } functions, as has been asserted elsewhere, but by an extension of this sheaf. A given G ∞ {G^\infty } manifold may admit many supermanifold extensions, and it is unknown at present whether there are G ∞ {G^\infty } manifolds that admit no such extension. When the underlying graded-commutative algebra is commutative, the axioms reduce to the Berezin-Kostant supermanifold theory.