Minimal Betti Numbers

Abstract

We give conditions for determining the extremal behavior for the (graded) Betti numbers of squarefree monomial ideals. For the case of non-unique minima, we give several conditions which we use to produce infinite families, exponentially growing with dimension, of Hilbert functions which have no smallest (graded) Betti numbers among squarefree monomial ideals and all ideals. For the case of unique minima, we give two families of Hilbert functions, one with exponential and one with linear growth as dimension grows, that have unique minimal Betti numbers among squarefree monomial ideals.

Keywords:

• Betti numbers
• Hilbert functions
• Simplicial complexes
• Squarefree monomial ideals

Mathematics Subject Classification:

• Primary 13D02
• Secondary 13F55

ACKNOWLEDGMENT

This material is partially based upon work supported by the National Science Foundation under Grant No. DMS-0353622.

Notes

Communicated by W. Bruns.

∗At the time this paper was completed, the author's affiliations were as follows: Christopher Dodd, Mathematics Department, University of Chicago, Chicago, IL, USA; Andrew Marks, Mathematics Department, Pacific Union College, Angwin, CA, USA.