Ideals attaining a given Hilbert function

Abstract

We improve the result of Charalambous and Evans [C-E] to show that the Betti number sequence in their example of incomparable minimals among the resolutions for a fixed Hilbert function is indeed minimal. Their example was dependent upon the graded betti numbers. We give an example of a finite length Hilbert function and two cyclic finite length modules attaining the Hilbert function for which the betti number sequences are incomparable, i.e., independent of the grading.