Cohen-Macaulay local rings with $e_2 = e_1-e+1$

TL;DR

This paper investigates Cohen-Macaulay local rings with a specific relation among their Hilbert coefficients, revealing conditions for the Cohen-Macaulayness of the associated graded ring and classifying possible Hilbert series.

Contribution

It establishes new bounds on the type of the ring based on Hilbert coefficients and characterizes the Hilbert series when the type reaches certain values.

Findings

Type of the ring is at least e - h - 1 when e_2 ≠ 0.

Associated graded ring G(A) is Cohen-Macaulay if type A = e - h - 1.

Hilbert series fully determines the depth of G(A) when type A = e - h.

Abstract

In this paper we study Cohen-Macaulay local rings of dimension $d$, multiplicity $e$ and second Hilbert coefficient $e_2$ in the case $e_2 = e_1 - e + 1$. Let $h = \mu(\mathfrak{m}) - d$. If $e_2 \neq 0$ then in our case we can prove that type $A \geq e - h -1$. If type $A = e - h -1$ then we show that the associated graded ring $G(A)$ is Cohen-Macaulay. In the next case when type $A = e - h$ we determine all possible Hilbert series of $A$. In this case we show that the Hilbert Series of $A$ completely determines depth $G(A)$.