Dependence of Betti Numbers on Characteristic

TL;DR

This paper investigates how the graded Betti numbers of monomial ideals vary with the characteristic of the base field, providing new insights into their algebraic and topological properties.

Contribution

It offers a detailed analysis of Betti number dependence on field characteristic for various classes of monomial ideals and introduces a method to relate Betti tables over different fields.

Findings

Betti numbers depend on the characteristic for certain monomial ideals.

A description of bipartite graphs related to Betti number behavior.

Betti tables over the rationals can be derived from those over any field via cancellations.

Abstract

We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence of consecutive cancellations.