The Cartan algebra of exterior differential forms as a supermanifold: morphisms and manifolds associated with them

A supermanifold, Mm/n, can be caracterired by its smooth superfunctions which constitute an algebra A (Leites, Kostant). We associate canonically ~a Ia Gelfand, certain fibred manifolds on which the automorphisms (the Jordan automorphisms) of A actas diffeomorphisms. For example, the kernelsof all homomorphisms from the algebra of superfunctions onto the Grassmann algebra of dimension n form naturally a manifold of dimension m 2~1 if n is even. To be more specific we explain this and similar constructions in the case of the algebra of smooth exterior differential forms defined on a smooth manifold. This algebra defines a particular supermanifold Mm/rn.