  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.
