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.

Gotzmann ideals of the polynomial ring

“…However, an ideal having the smallest graded Betti numbers among all ideals for a fixed Hilbert function need not exist. Recently, existence and non-existence of the smallest graded Betti numbers of ideals for a fixed Hilbert function are studied in several papers (see e.g., [4,6,21] First, we recall fundamental properties on Macaulay representations and the minimal growth of Hilbert functions. Given positive integers a and d, there exists the unique representation of a, called the dth Macaulay representation of a, of the form…”

“…Hence I has a segmentwise critical Hilbert function. However, for any linear transformation ϕ on S, if ϕ(I ) 4 is a canonical critical space, then ϕ(I ) 6 is not a canonical critical space.…”

“…However, an ideal having the smallest graded Betti numbers among all ideals for a fixed Hilbert function need not exist. Recently, existence and non-existence of the smallest graded Betti numbers of ideals for a fixed Hilbert function are studied in several papers (see e.g., [4,6,21]). An extremal example of a Hilbert function having an ideal with the smallest graded Betti numbers is a Hilbert function H for which all ideals with the Hilbert function H have the same graded Betti numbers.…”

Betti Numbers of Monomial Ideals and Shifted Skew Shapes

“…Among these ideals, the lex-segment ideal has the maximal Betti numbers according to Bigatti Hulett,and Pardue ([4] [17], [25]). However, in general there is no common lower bound for these ideals (see, e.g., [12] and the references therein). In comparison, the novelty of our approach is that instead of the Hilbert function we fix the number of minimal generators of the ideals under consideration.…”

Reverse lex ideals

“…In general, there are examples of Hilbert functions for which no ideal has minimal Betti numbers [Ric01,DMMR07]. There are techniques for finding upper bounds on Betti numbers; obtaining lower bounds is much harder.…”