Algebraic Morse theory via Homological Perturbation Lemma

As a generalization of the classical killing-contractible-complexes lemma, we present algebraic Morse theory via homological perturbation lemma, in a form more general than existing presentations in the literature. Two-sided Anick resolutions due to E.~Sk\"{o}ldberg are generalised to algebras given by quivers with relations and a minimality criterion is provided as well. Two applications of algebraic Morse theory are presented. It is shown that the Chinese algebra of rank $n\geq 1$ is homologically smooth and of global dimension $\frac{n(n+1)}{2}$, and the minimal two-sided projective resolution of a Koszul algebra is constructed.