The axioms of supermanifolds and a new structure arising from them

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 ${G^\infty }$ functions, as has been asserted elsewhere, but by an extension of this sheaf. A given ${G^\infty }$ manifold may admit many supermanifold extensions, and it is unknown at present whether there are ${G^\infty }$ manifolds that admit no such extension. When the underlying graded-commutative algebra is commutative, the axioms reduce to the Berezin-Kostant supermanifold theory.