Dependence of Betti Numbers on Characteristic

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 and Stanley–Reisner ideals of vertex-decomposable complexes. 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.

Key Words:

• Free resolutions
• Simplicial complexes
• Stanley–Reisner rings

2010 Mathematics Subject Classification:

• 13D02
• 13F55

ACKNOWLEDGMENTS

We thank J. Herzog and the referee for helpful comments. Parts of this work were completed at the Pan American Scientific Institute Summer School on “Commutative Algebra and its Connections to Geometry” in Olinda, Brazil, and when the second author visited the University of Missouri; we thank both institutions for their hospitality. The computer algebra system \textttMacaulay2 provided valuable assistance in studying examples.

Notes

Communicated by I. Swanson.