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

Abstract

Let $\left(R,\mathfrak{𝔪}\right)$ be a $d$-dimensional Cohen–Macaulay local ring, $I$ be an $\mathfrak{𝔪}$-primary ideal of $R$ and let $J=\left(x_1,\dots ,x_d\right)$ be a minimal reduction of $I$. We show that if, for $i=0$ or $1$, $J_{d-1}=\left(x_1,\dots ,x_{d-1}\right)$ and ${\sum }_{n=1}^{\infty }\lambda \left(I^{n+1}\cap J_{d-1}\right)∕\left(J“I^n\cap J_{d-1}\right)=i$, then $depthG\left(I\right)\ge d-i-1$. Moreover, we prove that if $e_2\left(I\right)={\sum }_{n=2}^{\infty }\left(n-1\right)\lambda \left(I^n∕J“I^{n-1}\right)-2$, or if $e_2\left(I\right)={\sum }_{n=2}^{\infty }\left(n-1\right)\lambda \left(I^n∕J“I^{n-1}\right)-3$ and $I$ is integrally closed, then $e_1\left(I\right)={\sum }_{n=1}^{\infty }\lambda \left(I^n∕J“I^{n-1}\right)-1$, where the integers $e_i$ are the Hilbert coefficients of $I$. In addition, if $J$ is a minimal reduction of $I$ then we prove that the reduction number $r_J\left(I\right)$ is independent of $J$.