Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory

Inspired by Brown’s collapsing method (or discrete Morse theory) to obtain a free resolution of $\text{bbZ}$ over the monoid ring $\text{bbZ}M$, we apply algebraic discrete Morse theory to compute the homology groups of Lawvere theories, which is defined as Tor of a certain module. We reinterpret known partial free resolutions arising from complete term rewriting systems in terms of collapsing of the normalized bar resolution. This perspective yields homological inequalities that bound the number of equational axioms in presentations and recovers classical results, such as lower bounds for group axiomatizations. Our main contribution is to extend these resolutions to higher dimensions.