Usefulness of Stanley–Reisner Betti numbers for computing local Whitney numbers

Investigate to what extent the interpretation of the absolute values of local Whitney numbers as Betti numbers of Stanley–Reisner rings associated with the simplicial complexes arising from lattice intervals can help determine those local Whitney numbers.

Background

The paper explains that the absolute values of local Whitney numbers for flats of a matroid can be interpreted as certain Betti numbers of Stanley–Reisner rings associated with the simplicial complexes formed from the corresponding lattice intervals. This follows from the concentration of homology in the top dimension and an application of Hochster’s formula.

Although this interpretation potentially provides an algebraic method for computing the homology numbers—and consequently the generalized weight polynomials—the paper does not establish how effective or useful the Betti-number perspective is for determining them. The unresolved issue concerns the practical and mathematical extent of that usefulness.