On the Hilbert coefficients, depth of associated graded rings and reduction numbers

Abstract

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and $\sum _{n=1}^{\infty }\lambda (I^{n+1}\cap J_{d-1})/(J{I}^n\cap J_{d-1})=i$ where i=0,1, then depth $G(I)\ge d-i-1$. Moreover, we prove that if $e_2(I)={\sum }_{n=2}^{\infty }(n-1)\lambda (I^n/J{I}^{n-1})-2;$ or if $I$ is integrally closed and $e_2(I)={\sum }_{n=2}^{\infty }(n-1)\lambda (I^n/J{I}^{n-1})-i$ where $i=3,4$, then $e_1(I)={\sum }_{n=1}^{\infty }\lambda (I^n/J{I}^{n-1})-1.$ In addition, we show that $r(I)$ is independent. Furthermore, we study the independence of $r(I)$ with some other conditions.

arXiv commentto appear in JCA