  To any finite simplicial complex X, we associate a natural filtration
starting from Chari and Joswig's discrete Morse complex and abutting to the
matching complex of X. This construction leads to the definition of several
homology theories, which we compute in a number of examples. We also completely
determine the graded object associated to this filtration in terms of the
homology of simpler complexes. This last result provides some connections to
the number of vertex-disjoint cycles of a graph.
