Cohen-Macaulay local rings with $e_1 = e + 2$

TL;DR

This paper characterizes the Hilbert functions of Cohen-Macaulay local rings with a specific relation between their first Hilbert coefficient and multiplicity, focusing on the case where e_1 equals e plus two.

Contribution

It provides a complete description of possible Hilbert functions for Cohen-Macaulay local rings with e_1 = e + 2, a case not fully understood before.

Findings

Classifies Hilbert functions for the case e_1 = e + 2

Identifies constraints on the structure of such rings

Advances understanding of Cohen-Macaulay ring invariants

Abstract

In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.

Equations

$$P_A(z)=\sum _{i=0}^d(-1{)}^i{e}_i(A)(\genfrac{}{}{0ex}{}{z+d-i}{d-i}).$$

$$H_A(z)=\underset{n\ge 0}{⨁}\ell ({\mathfrak{m}}^n/{\mathfrak{m}}^{n+1})z^n=\frac{h_A(z)}{(1-z{)}^d}\text{where}{h}_A(z)\in \mathbb{Z}[z].$$

Full text

1. introduction

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$. If $M$ is an $A$-module then let $\ell(M)$ denote its length and $\mu(M)$ the number of its minimal generators. The function $H^{(1)}(A,n)=\ell(A/\mathfrak{m}^{n+1})$ is called the Hilbert-Samuel function of $A$ (with respect to $\mathfrak{m}$). It is well known that there exists a polynomial $P_{A}(z)\in\mathbb{Q}[z]$ of degree $d$ such that $H^{(1)}(A,n)=P_{A}(n)$ for all $n\gg 0$. The polynomial $P_{A}(z)$ is called the Hilbert-Samuel polynomial of $A$. We write

[TABLE]

The integers $e_{i}(A)$ are called the $i^{th}$-Hilbert coefficient of $A$. The zeroth Hilbert coefficient $e_{0}(A)$ is called the multiplicity of $A$. We set $e_{i}=e_{i}(A)$ for all $i$ and $e=e_{0}$.

The graded ring $G(A)=\bigoplus_{n\geq 0}\mathfrak{m}^{n}/\mathfrak{m}^{n+1}$ is called the associated graded ring of $A$(with respect to $\mathfrak{m}$). It’s Hilbert series

[TABLE]

If $f$ is a polynomial we use $f^{(i)}$ to denote the $i$-th formal derivative of $f$. It is easy to see that $e_{i}=h_{A}(z)^{(i)}(1)/i!$ for $0\leq i\leq d$. It is also convenient to set $e_{i}=h_{A}(z)^{(i)}(1)/i!$ for all $i\geq 0$. Let $h=\mu(\mathfrak{m})-d$ be the embedding co-dimension of $A$. It is easily seen that $h_{A}(z)=1+hz+\text{higher terms}$. We call $h_{A}(z)$ the h-polynomial of $A$. The function $H(A,n)=\ell(\mathfrak{m}^{n}/\mathfrak{m}^{n+1})$ is called the Hilbert function of $A$ (with respect to $\mathfrak{m}$).

Now assume that $A$ is Cohen-Macaulay. Then Abhyankar proved that $e\geq h+1$, see [A]. Northcott proved that $e_{1}\geq e_{0}-1$, see [N]. Itoh proved that $e_{2}\geq e_{1}-e+1$, see [It, Theorem 12]. It was observed by Sally that when the Hilbert-coefficients satisfy border values then it forces $G(A)$ to have high depth and also forces the $h_{A}(z)$ to have a prescribed form; see [S1], [S2], [S3] and [S4]. See the nice survey article [V] for an introduction to this area of research. In particular see [V] for classification of Hilbert functions when $e=h+1,h+2$ and $e_{1}=e-1,e,e+1$. Let $\operatorname{type}(A)=\ell(\operatorname{Ext}^{d}_{A}(k,A))$ be the Cohen-Macaulay-type of $A$. For classification of Hilbert functions when $e=h+3$ and $\operatorname{type}(A)<h$, see [RV].

In this paper we describe all Hilbert functions that can possibly occur if $e_{1}=e+2$. It turns out that in the process we also have to describe Hilbert functions that can occur when $e_{2}=e_{1}-e+1=3$. It was conjectured by Valla that if $e_{2}=e_{1}-e+1$ then $G(A)$ is Cohen-Macaulay and $\deg h_{A}(z)\leq 2$. This conjecture is true when $\dim A\leq 1$. A counter-example to this conjecture was found by Wang in the case $e_{2}=e_{1}-e+1=3$ and $\dim A=2$ ; see [CPR, 3.10]. We prove

Theorem 1.1**.**

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$. Set $G=G(A)$. If $e_{2}=e_{1}-e+1=3$ then one of the following cases occur

1. (i)

$e=h+4$, $G$ is Cohen-Macaulay  and $h_{A}(z)=1+hz+3z^{2}$. 2. (ii)

$e=h+3$, $d\geq 2$, $0ptG=d-2$, $\operatorname{type}(A)=h$ and

$\displaystyle{h_{A}(z)=1+hz+3z^{3}-z^{4}.}$**

Note that in Theorem 1.1 the $h$-polynomial of $A$ completely determines

$0ptG(A)$. In section 7 we give examples of Cohen-Macaulay local rings having Hilbert functions as described in 1.1.

As a consequence of Theorem 1.1 we can completely classify Hilbert functions that can occur when $e_{1}=e+2$.

Theorem 1.2**.**

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$. Set $G=G(A)$. If $e_{1}=e+2$ then one of the following cases occur

1. (i)

$e=h+2$, $\operatorname{type}(A)=h$, $0ptG=d-1$ and $h_{A}(z)=1+hz+z^{4}$. 2. (ii)

$e=h+4$, $G$ is Cohen-Macaulay  and $h_{A}(z)=1+hz+3z^{2}$. 3. (iii)

$e=h+3$, $e_{2}=4$, $0ptG\geq d-1$ and $h_{A}(z)=1+hz+z^{2}+z^{3}$. 4. (iv)

$e=h+3$, $e_{2}=3$, $d\geq 2$, $0ptG=d-2$, $\operatorname{type}(A)=h$ and

$\displaystyle{h_{A}(z)=1+hz+3z^{3}-z^{4}.}$**

In section 7 we give examples of Cohen-Macaulay local rings having Hilbert functions as described in 1.2.

We now describe in brief the contents of this paper. In section two we describe some preliminary results that we need. In section three we discuss the case of Theorem 1.1 when $d=2$. The most difficult case of Theorem 1.1 is the case when $d=3$. This is done in section four. In section five we prove Theorem 1.1. We prove Theorem 1.2 in section six. Finally in section seven we give examples which illustrate our results.

2. preliminaries

In this section we discuss some preliminaries that we need. In this paper all rings are Noetherian and all modules are assumed to be finitely generated. Let $(A,\mathfrak{m})$ be a local ring of dimension $d$ with residue field $k=A/\mathfrak{m}$.

2.1**.**

If $a$ is a non-zero element of $A$ and if $j$ is the largest integer such that $a\in\mathfrak{m}^{j}$, then we let $a^{*}$ denote the image of $a$ in $\mathfrak{m}^{j}/\mathfrak{m}^{j+1}$.