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

we first characterize $tau$-complemented modules with relative (pre)-covers. we also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

We study a variety for graded maximal Cohen–Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only finite number isomorphism classes modules with fixed Hilbert series over algebras countable representation type.