Parafree augmented algebras and Gröbner-Shirshov bases for complete augmented algebras

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 a 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 give a sufficient condition for an augmented algebra to be residually nilpotent in terms of its relations.