On the Hilbert function of a graded Cohen-Macaulay domain
Let \(R\) be a noetherian graded \(k\)-algebra which is integral over \(k[R_ 1]\). Then the Hilbert series of \(R\), \(\sum_ i\lim_ k R_ it^ i\), has the form \((h_ 0+h_ 1t+\dots+h_ st^ s)/(1-t)^ d\). The following theorem is proved: If \(R\) is a CM-domain, then \(h_ 0+h_ 1+\dots+h_ i\leq h_ s+h_{s-1}+\dots+h_{s-i}\) for all \(i\). An application to the Ehrhardt polynomial of a convex polytope is given. Furthermore, if \(R\) is generated by \(R_ 1\) and a CM-domain of dimension at least 2, then it is shown that \(h_{m+1}+h_{m+2}+\dots+h_{m+n}\geq h_ 1+\dots+h_ n\) if \(m+n<s\).
scientific article; zbMATH DE number 12411