Maximal minimal resolutions   
S. Iyengar and K. Pardue

Abstract.
Given two Hilbert series h_1(s) and h_2(s) and an integer d, 
we give a sharp bound on the graded Betti numbers of a cyclic 
R-module R/J, where R is a graded algebra generated by R_1 
over a field k such that R has Hilbert series h_1(s), and R/J 
has Hilbert series h_2(s) and depth at least d. We also exhibit 
a particular ring R and ideal J such that R/J attains this bound 
for every graded Betti number. We also prove similar statements 
for local rings and raise further questions about maximal Betti 
numbers. The results in this paper are strong generalizations 
of earlier results of Bigatti, Hulett, Pardue, Peeva, Elias, 
Eliahou and Kervaire.