  A Lagrangian theory giving rise to a version of the Dirac-Kahler equations on
curved backgrounds is considered. The principal pieces are the general fields
which have values in the algebra of the Dirac matrices and satisfy a Dirac-type
equation. Their components are scalar, pseudo-scalar, vector, axial-vector
fields and fields strength which satisfy an irreducible systems of first-order
Dirac-Kahler equations having remarkable gauge and duality properties similar
to those of the flat case. The vector and axial-vector fields are the physical
potentials giving rise to the field strength while the scalar fields play an
auxiliary role and can be eliminated by fixing a suitable gauge. The chiral
components of the field strength are either self-dual or anti self-dual with
respect to the Hodge duality.
