  We show the existence of a co(dual)-BRST symmetry for the usual BRST
invariant Lagrangian density of an Abelian gauge theory in two dimensions of
space-time where a U(1) gauge field is coupled to the Noether conserved current
(constructed by the Dirac fields). Under this new symmetry, it is the
gauge-fixing term that remains invariant and the symmetry transformations on
the Dirac fields are analogous to the chiral transformations. This interacting
theory is shown to provide a tractable field theoretical model for the Hodge
theory. The Hodge dual operation is shown to correspond to a discrete symmetry
in the theory and the extended BRST algebra for the generators of the
underlying symmetries turns out to be reminiscent of the algebra obeyed by the
de Rham cohomology operators of differential geometry.
