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Spring 2021 On the Hilbert coefficients, depth of associated graded rings and reduction numbers
J. Commut. Algebra 13(1): 103-115 (Spring 2021). DOI: 10.1216/jca.2021.13.103

## 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 $J$.

## Citation

Amir Mafi. Dler Naderi. "On the Hilbert coefficients, depth of associated graded rings and reduction numbers." J. Commut. Algebra 13 (1) 103 - 115, Spring 2021. https://doi.org/10.1216/jca.2021.13.103

## Information

Received: 1 November 2017; Revised: 17 September 2019; Accepted: 16 September 2019; Published: Spring 2021
First available in Project Euclid: 28 May 2021

Digital Object Identifier: 10.1216/jca.2021.13.103

Subjects:
Primary: 13A30, 13D40, 13H10  