On the Hilbert function of a graded Cohen-Macaulay domain
type: Item
instance of: scholarly article
title: On the Hilbert function of a graded Cohen-Macaulay domain
zbMATH Open document ID: 0735.13010
DOI: 10.1016/0022-4049(91)90034-Y
author: Richard P. Stanley
published in: Journal of Pure and Applied Algebra
publication date: 1992-06-25T00:00:00Z
review text: 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 \(m+n<s\).
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 12411
zbMATH Keywords: graded algebra
zbMATH Keywords: Hilbert series
zbMATH Keywords: Ehrhardt polynomial
zbMATH Keywords: graded Cohen- Macaulay domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 12411