Determine Betti numbers of monomial ideals

Determine the graded Betti numbers β_{i,j}(R/I) for arbitrary monomial ideals I ⊂ K[x_1, …, x_n], where R = K[x_1, …, x_n] is a polynomial ring over a field K; specifically, ascertain methods and formulas to compute these Betti numbers in general beyond the special classes treated in existing work.

Background

The paper studies extremal Betti numbers for a specific family of two-dimensional monomial ideals associated to unions of weighted hyperplanes in projective space. While it provides explicit formulas for extremal Betti numbers in several regimes of the parameters (α, β) and classifies pseudo-Gorenstein cases, the authors emphasize that broader tasks concerning Betti numbers of monomial ideals are not fully resolved.

In the introduction, they explicitly note that many questions in this area remain open and highlight as an important question the determination of the Betti numbers of monomial ideals in general, underscoring the difficulty of obtaining explicit formulas even for extremal Betti numbers.