Abstract :
It is known that given a Hilbert function , there need not exist a module which has uniquely the smallest graded Betti numbers among all modules attaining . In this paper we extend the previous example of this behavior to an infinite family and demonstrate with a second infinite family that even when the given Hilbert function is that of a complete intersection, a module with uniquely smallest graded Betti numbers need not exist. Finally we prove a conjecture of Geramita, Harima, and Shin concerning the non-existence of uniquely smallest graded Betti numbers among all Gorenstein rings attaining a given Hilbert function.
