On the number of generators of ideals in local rings
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.
scientific article; zbMATH DE number 3939477