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

Authors : Amir Mafi, Dler Naderi

Pages : 766-769

View : 12 | Download : 8

Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :Let $(R,\fm)$ 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: $(i)$ $P_I(n)=H_I(n)$ for $n=1,2$; $(ii)$ $P_I(n)=H_I(n)$ for all $n\geq 1$; $(iii)$ $I^3=JI^2$. Moreover, if $\Dim R=3$, $n(I)\leq 1$ and $\grade gr_I(R)_+>0$, then the reduction number $r(I)$ is independent.
Keywords : Cohen-Macaulay rings, Hilbert-Samuel functions