10 2. ALGEBRAIC DISCRETE MORSE THEORY Before we proceed to the applications, we give a simple example that shows that the finiteness assumption of Theorem 2.4 indeed is needed. Example 2.5. Let C• be the (non-reduced) chain complex of the simplicial complex triangulating the real line by 1-simplices [i, i+1] for i Z. Then C0 and C1 are free Z-modules of countably infinite dimension. It is easily seen that matching each 0-cell {i} with the 1-cell [i, i+1] yields an acyclic matching. The corresponding Morse complex is constant 0. In particular, its homology in dimension 0 is 0 being different from the 0-th (non-reduced) homology group of the real line, which is Z. In the following chapters we will use the conclusions of Theorem 2.2 and 2.4 in order to construct minimal free resolutions without explicitly referring to the theorems.
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