Algebraic Morse Theory

In this section we give a version of discrete Morse theory that is adapted to the setting of arbitrary free chain complexes. 

---

In order to introduce a combinatorial element into this setting, we need to choose a basis (i.e., a set of free generators) for each C . When this is done, we say that we have chosen a basis i = (J S7 for the entire chain complex C. We write (C, 1 ) to denote a chain complex with a basis. A free chain complex with a basis is the main object of study of algebraic Morse theory. 

The main theorem of algebraic Morse theory brings to light a certain structure in (T. Namely, by choosing a different basis, one can represent C as a direct sum of two chain complexes, of which one is a direct sum of atom chain complexes, in particular acyclic, and the other one is isomorphic to C. For convenience, the choice of basis can be performed in several steps, one step for each matched pair of the basis elements. 

JW05] M. JoUenbeck, V. Welker, Resolution of the residue class field via algebraic discrete Morse theory, preprint 2005. arXiv math.AC/0501179... 

---