super vector bundle
An ordinary smooth vector bundle on a manifold may be identified with its sheaf of sections, which is a locally free sheaf of modules over its structure sheaf and all such locally free module sheaves arise this way.
The definition of vector bundles in terms of sheaves of sections therefore immediately generalizes to every ringed space and in particular to supermanifolds.
A super vector bundle over a supermanifold is a locally free sheaf on the category of open subsets of of modules for the structure sheaf .
Super tangent bundles
The tangent bundle of an ordinary manifold has the sheaf of sections given by the derivations of the structure sheaf. The same definition works here:
the super tangent bundle of a supermanifold is given by the sheaf .
so a super tangent vector field is a global section of this sheaf of derivations.
On the supermanifold with its canonical coordinates
there is the odd vector field
whose super Lie bracket with itself vanishes
This odd vector field is left invariant with respect to the super translation group structure on .
This means that is free on one odd generator.