A note on reduction numbers and Hilbert-Samuel functions of ideals over Cohen-Macaulay rings

Authors : Amir MAFI, Dler NADERI

Pages : 766-769

View : 11 | Download : 7

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $insert ignore into journalissuearticles values(R,\fm);$ be a Cohen--Macaulay local ring of dimension $d\geq 2$ with infinite residue field and $I$ an $\fm$-primary ideal of $R$. Let $I$ be integrally closed and $J$ be a minimal reduction of $I$. In this paper, we show that the following are equivalent: $insert ignore into journalissuearticles values(i);$ $P_Iinsert ignore into journalissuearticles values(n);=H_Iinsert ignore into journalissuearticles values(n);$ for $n=1,2$; $insert ignore into journalissuearticles values(ii);$ $P_Iinsert ignore into journalissuearticles values(n);=H_Iinsert ignore into journalissuearticles values(n);$ for all $n\geq 1$; $insert ignore into journalissuearticles values(iii);$ $I^3=JI^2$. Moreover, if $\Dim R=3$, $ninsert ignore into journalissuearticles values(I);\leq 1$ and $\grade gr_Iinsert ignore into journalissuearticles values(R);_+>0$, then the reduction number $rinsert ignore into journalissuearticles values(I);$ is independent.
Keywords : Cohen Macaulay rings, Hilbert Samuel functions