Morse inequalities for the Koszul complex of multi-persistence

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of the relative homology of nested filtration pieces and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for phonological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.