  We develop a theory of parafree augmented algebras similar to the theory of
parafree groups and explore some questions related to the Parafree Conjecture.
We provide an example of finitely generated parafree augmented algebra of
infinite cohomological dimension. Motivated by this example, we prove a version
of the Composition-Diamond lemma for complete augmented algebras and provide a
sufficient condition for augmented algebra to be residually nilpotent on the
language of its relations.
