Resolution of the residue class field via algebraic discrete Morse theory

Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules over a ring. We provide three applications of this theory: We construct new resolutions of the residue class field $k$ over $A$, where $A$ is the quotient of a (i) commutative polynomial ring or (ii) non-commutaitve polynomial ring by a (twosided) ideal and (iii) we construct a new resolution of $A$ as an $A \otimes A^{op}$-module in the situation (ii). In either case we prove minimality of the resolution for certain classes of algebras $A$.