On the number of generators of ideals in local rings
type: Item
instance of: scholarly article
title: On the number of generators of ideals in local rings
zbMATH Open document ID: 0586.13014
DOI: 10.1016/0001-8708(86)90039-3
author: Aner Shalev
published in: Advances in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let (R,m) be a Noetherian local ring, and for an ideal I of R let v(I) denote the common size of the minimal generating sets for I. If H denotes the Hilbert function, \(H(n)=v(m^ n)\), and if \(v(R)=\sup \{v(I)| \quad I\quad is\quad an\quad ideal\quad of\quad R\},\) then it is known that \(v(R)<\infty\) iff dim \(R\leq 1\) iff H is bounded. Also for dim R\(=1\), it is obvious that v(R)\(\geq \sup H\geq e(R)\) where e(R) denotes the multiplicity of R. Moreover, if R is Cohen-Macaulay, then these inequalities are equalities. The author considers the non-Cohen-Macaulay case and gives a precise bound for v(R) in terms of sup H. It is also shown for dim R\(=1\) that if \(v_ n(R):=\sup \{v(I)| \quad I\subseteq m^ n\},\) then \(v_ n(R)=e(R)\) for sufficiently large n. For R of arbitrary dimension it is proved that there is a constant C (depending on H) such that \(I\supseteq m^ n\) implies \(v(I)\leq v(m^ n)+C\) for each \(n>0\), and that for R a regular local ring \(C=0\). This latter fact may also be seen from work of Junzo Watanabe on the concept of m-full ideals as defined by D. Rees.
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 3939477
zbMATH Keywords: Noetherian local ring
zbMATH Keywords: minimal generating sets
zbMATH Keywords: Hilbert function
zbMATH Keywords: multiplicity
zbMATH Keywords: non-Cohen-Macaulay
reviewed by: William J. Heinzer
MaRDI profile type: Publication
full work available at URL: 0001-8708(86)90039-3
OpenAlex ID: W2020468149
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description: scientific article; zbMATH DE number 3939477