Ideals attaining a given Hilbert function (Q1593028)

\textit{H. Charalambous} and \textit{E. G. Evans}, jun. (C-E) [Contemp. Math. 159, 19-26 (1994; Zbl 0799.18009)] have constructed, for a ring \(R=k[x_0, \dots,x_n]\), \(k\) a field, \(n\geq 2\), a Hilbert series for a finite length cyclic \(R\)-module and two incomparable smallest sets of graded Betti numbers, in fine partial ordering, for that Hilbert series. The author, in this paper, improves upon C-E's theorem by proving that the Betti number sequence is in fact smallest. He also proves existence of incomparable minimal resolutions, in the coarse partial ordering, among the resolutions of graded modules with a fixed Hilbert function. He considers only finite length cyclic modules.