  For a certain class of complexes of pre-Hilbert $A$-modules, we prove that
their cohomology groups equipped with a canonical quotient structure are again
pre-Hilbert $A$-modules and derive the Hodge decomposition for them. We call
these complexes self-adjoint parametrix possessing. We show that $A$-elliptic
complexes of differential operator acting on sections of finitely generated
projective $A$-Hilbert bundles over compact manifolds have this property if the
images of certain extensions of their Laplacians are closed.
