Betti numbers of monomial ideals and shifted skew shapes

TL;DR

This paper introduces new lower bounds for Betti numbers of monomial ideals, providing explicit resolutions and combinatorial interpretations for specific classes, advancing understanding of algebraic invariants in combinatorial contexts.

Contribution

It establishes two new lower bound problems for Betti numbers, solves them for certain classes, and offers simple cellular resolutions and combinatorial interpretations.

Findings

Lower bounds for total Betti numbers of monomial ideals.

Explicit cellular linear resolutions for strongly stable ideals.

Field-independent combinatorial interpretations of Betti numbers.

Abstract

We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.