Acyclic Matchings on Free Chain Complexes and the Morse Complex

1 Acyclic Matchings on Free Chain Complexes and the Morse Complex 

Let TZ be an arbitrary commutative ring with unit. Recall that a chain complex C consisting of 7 -modules, 

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. 

Note that a free chain complex with a basis (C, 1 ) can be represented as a ranked poset P(C, 17), with P-weights on the order relations. The elements of rank n correspond to the elements of 17 , and the weight of the covering relation b y a, for b e fin, a G I7 i, is simply defined by wo b y a) = lin db,a). In other words, 

It is important to note that Definition 11.22 is different from the topological one, which was used in Theorem 11.13. In the algebraic setting, the 

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