Supervector Bundles, general ground rings formulation

A well-behaved theory of supervector bundles within the general category of supermanifolds is described in [3] and a more detailed treatment in [1].

Let be two supermanifolds based on a ground graded Banach algebra Λ satisfying appropriate (but quite loose) conditions, cf. [3]. The product supermanifold is constructed by taking Z = X × Y as topological spaces while the structure sheaf is the sheaf associated to the presheaf defined by

where U, V are open sets in X, Y, and ⊗_Λ,π denotes the topological completion of the graded tensor product over Λ in the Grothendieck π topology [4]. The evaluation morphism , where is the sheaf of Λ-valued continuous functions on Z, is induced by continuity by the morphism .

Let be the free graded (m, n)-dimensional Λ-module Λ^m|n with its natural supermanifold structure [3], that we shall denote as ). Then the sheaf of sections of the projection map is a free graded module of rank (m, n) over the sheafof...