  The nilpotent BRST, anti-BRST, dual-BRST and anti-dual-BRST symmetry
transformations are constructed in the context of noncommutative (NC) 1-form as
well as 2-form gauge theories. The corresponding Noether's charges for these
symmetries on the Moyal plane are shown to satisfy the same algebra as by the
de Rham cohomological operators of differential geometry. The Hodge
decomposition theorem on compact manifold is also studied. We show that
noncommutative gauge theories are field theoretic models for Hodge theory.
