On the Hilbert-Samuel function
type: Item
instance of: scholarly article
title: On the Hilbert-Samuel function
zbMATH Open document ID: 0594.13021
author: Kazuji Kubota
published in: Tokyo Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let A be a Cohen-Macaulay semi-local ring in which each maximal ideal has height d. Let I be an open ideal with Hilbert-Samuel polynomial \(\sum (- 1)^ ie_ i\binom {n+d-i}{d-i}.\) Starting from results, for \(d=1\), of \textit{J. Lipman} [Am. J. Math. 93, 649-685 (1971; Zbl 0228.13008)] on stable ideals, the author proves the equivalence of: (1) the length \(\ell (A/I^{n+1})=e_ 0\binom{n+d}{d}-e_ 1\binom{n+d-1}{d-1}\) for all \(n\geq 0\); (2) \(\ell (A/I)=e_ 0-e_ 1\) and \(\ell (A/I^ 2)=e_ 0(d+1)-e_ 1d,\) (3) \(\ell (A/I^ 2)=e_ 0+d\ell (A/I),\) and also, under suitable conditions, (4) \(I^ 2=XI\) for some ideal X generated by d elements.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 3955077
zbMATH Keywords: Cohen-Macaulay semi-local ring
zbMATH Keywords: Hilbert-Samuel polynomial
zbMATH Keywords: stable ideals
reviewed by: C. P. L. Rhodes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270151224
OpenAlex ID: W2026932239
DOI: 10.3836/TJM/1270151224
description: scientific article; zbMATH DE number 3955077