On the structure of the Sally module and the second normal Hilbert coefficient

TL;DR

This paper characterizes the structure of the Sally module and the Hilbert function of the associated graded ring in Cohen-Macaulay local rings when the second normal Hilbert coefficient is nearly minimal, advancing understanding of Hilbert coefficients and related conjectures.

Contribution

It provides a complete description of the Sally module and Hilbert function in cases of almost minimal second normal Hilbert coefficient, linking to Itoh's conjecture.

Findings

Explicit structure of Sally module in the specified case

Complete description of the Hilbert function of the associated graded ring

Insights into the vanishing of the third Hilbert coefficient

Abstract

The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the Hilbert function of the associated graded ring of the normal filtration. A deep analysis of the vanishing of the third Hilbert coefficient has been necessary. This study is related to a long-standing conjecture stated by S. Itoh.

Equations

$${\ell }_R(R/\overline{I^{n+1}})={\overline{\mathrm{e}}}_0(I)\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)-{\overline{\mathrm{e}}}_1(I)\left(\genfrac{}{}{0ex}{}{n+d-1}{d-1}\right)+\cdots +(-1{)}^d{\overline{\mathrm{e}}}_d(I)$$

$${\overline{\mathrm{e}}}_2(I)\ge {\overline{\mathrm{e}}}_1(I)-{\overline{\mathrm{e}}}_0(I)+{\ell }_R(R/\overline{I})={\overline{g}}_s(I)\ge {\ell }_R(\overline{I^2}/J\overline{I})$$

$$\mathcal{R}(\mathcal{I})={\oplus }_{n\ge 0}{I}_n{t}^n\subseteq R[t],{\mathcal{R}}^{\prime }(\mathcal{I})={\oplus }_{n\in \mathbb{Z}}{I}_n{t}^n\subseteq R[t,t^{-1}],\text{mbox}andG(\mathcal{I})={\mathcal{R}}^{\prime }(\mathcal{I})/t^{-1}{\mathcal{R}}^{\prime }(\mathcal{I})$$

$$S_J(\mathcal{I}):=\frac{\mathcal{R}(\mathcal{I}{)}_{\ge 1}{t}^{-1}}{I_{1}T}\cong {\oplus }_{n\ge 1}{I}_{n+1}/J^n{I}_1$$

$$C^{(\ell )}:=C_J^{(\ell )}(\mathcal{I})=\frac{\mathcal{R}(\mathcal{I}{)}_{\ge \ell }{t}^{-1}}{I_{\ell }T{t}^{\ell -1}}\cong {\oplus }_{n\ge \ell }{I}_{n+1}/J^{n-\ell +1}{I}_{\ell }.$$

$$0\to L^{(\ell )}\to C^{(\ell )}\to C^{(\ell +1)}\to 0.$$

$${\ell }_R(R/I_{n+1})$$

$$0\to {\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}})\to \overline{{\mathcal{R}}^{\prime }}(IS)\to \overline{{\mathcal{R}}^{\prime }}(IS)/{\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}})\to 0$$

$$0\to {\mathrm{H}}_{M^{\prime }}^1({\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}}))\to {\mathrm{H}}_{M^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(I))(-1)\to {\mathrm{H}}_{M^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(I))\to {\mathrm{H}}_{M^{\prime }}^2({\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}}))\to$$

$$0⟶\overline{{\mathcal{R}}^{\prime }}(I)(-1)\stackrel{a_{1}t}{⟶}\overline{{\mathcal{R}}^{\prime }}(I)⟶{\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}})⟶0.$$

$$\sum _{i=0}^d(-1{)}^i{\overline{\mathrm{e}}}_i(I)(\genfrac{}{}{0ex}{}{n+d-i}{d-i})-{\ell }_R(R/\overline{I^{n+1}})=\sum _{k\ge n+2}{\ell }_R(\overline{I^{k}S}/\overline{I^k}S)-\sum _{i=2}^{d-1}(-1{)}^i{\ell }_R({\mathrm{H}}_{N^{\prime }}^i(\overline{{\mathcal{R}}^{\prime }}(IS){)}_{\ge n+2}).$$

$${\overline{\mathrm{e}}}_3(I)={\overline{\mathrm{e}}}_2(I)-{\overline{\mathrm{e}}}_1(I)+{\overline{\mathrm{e}}}_0(I)-{\ell }_R(R/\overline{I})-\sum _{n\ge 2}{\ell }_R(\overline{I^{n}S}/\overline{I^n}S)+{\ell }_R({\mathrm{H}}_{N^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(IS){)}_{\ge 2}).$$

$$0⟶\overline{{\mathcal{R}}^{\prime }}(I)(-1)\stackrel{a_{1}t}{⟶}\overline{{\mathcal{R}}^{\prime }}(I)⟶{\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}})⟶0.$$

$$\cdots \to {\mathrm{H}}_{N^{\prime }}^{i-1}({\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}}))\to {\mathrm{H}}_{N^{\prime }}^i(\overline{{\mathcal{R}}^{\prime }}(I))(-1)\to {\mathrm{H}}_{N^{\prime }}^i(\overline{{\mathcal{R}}^{\prime }}(I))\to {\mathrm{H}}_{N^{\prime }}^i({\mathcal{R}}^{\prime }(\{\overline{I^n}S{\}}_{n\in \mathbb{Z}}))\to$$

$$\sum _{i=2}^d(-1{)}^i\{{\ell }_R({\mathrm{H}}_{N^{\prime }}^i(\overline{{\mathcal{R}}^{\prime }}(I){)}_{\ge n+1})-{\ell }_R({\mathrm{H}}_{N^{\prime }}^i(\overline{{\mathcal{R}}^{\prime }}(I){)}_{\ge n+2})-{\ell }_R({\mathrm{H}}_{N^{\prime }}^{i-1}({\mathcal{R}}^{\prime }(\{\overline{I^n}S\}){)}_{\ge n+2})\}=0$$

$$\sum _{i=0}^d(-1{)}^i{\overline{\mathrm{e}}}_i(I)(\genfrac{}{}{0ex}{}{n+d-i}{d-i})-{\ell }_R(R/\overline{I^{n+1}})=\sum _{i=2}^d(-1{)}^i{\ell }_R({\mathrm{H}}_{N^{\prime }}^i(\overline{{\mathcal{R}}^{\prime }}(I){)}_{n+1})$$

$$1={\overline{\mathrm{e}}}_2(I)-{\overline{\mathrm{e}}}_1(I)+{\overline{\mathrm{e}}}_0(I)-{\ell }_R(R/\overline{I})=\sum _{n\ge 2}(n-1){\ell }_R(\overline{I^{n+1}}/J\overline{I^n}).$$

$$\cdots ⟶{\mathrm{H}}_{N^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(IS))(1)\stackrel{t^{-1}}{⟶}{\mathrm{H}}_{N^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(IS))⟶{\mathrm{H}}_{N^{\prime }}^2(\overline{G}(IS))⟶0$$

$$0⟶\overline{{\mathcal{R}}^{\prime }}(IS)(1)\stackrel{t^{-1}}{⟶}\overline{{\mathcal{R}}^{\prime }}(IS)⟶\overline{G}(IS)⟶0.$$

$${\overline{\mathrm{e}}}_3(I)$$

$${\ell }_R(\overline{I^2}/J\overline{I})+2={\overline{\mathrm{e}}}_2(I)\ge {\overline{\mathrm{e}}}_1(I)-{\overline{\mathrm{e}}}_0(I)+{\ell }_R(R/\overline{I})\ge {\ell }_R(\overline{I^2}/J\overline{I})$$

$$H{S}_{G(I)}(t)=\frac{31+43t+t^2+t^3}{(1-t{)}^3},$$

$${\ell }_R(R/I_{\ell (n+1)})=\sum _{k=0}^n{\ell }_R([\mathrm{G}({\mathcal{I}}^{(\ell )}){]}_k)$$

$${\overline{\mathrm{e}}}_1(I)$$

$$\cdots ⟶{\mathrm{H}}_{N^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(IS){)}_{n+1}⟶{\mathrm{H}}_{N^{\prime }}^2(\overline{{\mathcal{R}}^{\prime }}(IS){)}_n⟶{\mathrm{H}}_{N^{\prime }}^2(\overline{G}(IS){)}_n⟶0$$

$$0⟶\overline{{\mathcal{R}}^{\prime }}(IS)(1)\stackrel{t^{-1}}{⟶}\overline{{\mathcal{R}}^{\prime }}(IS)⟶\overline{\mathrm{G}}(IS)⟶0$$

$$\overline{I^3}S=\overline{I^{3}S}=xS+J\overline{I^{2}S}=(x,a_{2}y,a_{3}y)S+J\overline{I^2}S.$$

$$\overline{I^3}=(x,y_2,y_3)R+J\overline{I^2}+(a_1)\cap \overline{I^3}=(x,y_2,y_3)R+J\overline{I^2}$$

$$[C^{\prime }{]}_n=\{\begin{array}{ll}(0)& \text{mbox}ifn\le 1\\ [(y_2,y_3)J^{n-2}+J^{n-1}\overline{I^2}]/J^{n-1}\overline{I^2}& \text{mbox}ifn\ge 2.\end{array}$$

$$0⟶C^{\prime }⟶\overline{C}⟶\overline{C}/C^{\prime }⟶0(\ast \ast )$$

Full text

SKM is supported by INSPIRE faculty award funded by Department of Science and Technology, Govt. of India. She is also partially supported by a grant from Infosys Foundation. She was supported by INdAM COFUND Fellowships cofunded by Marie Curie actions, Italy, for her research in Genova during which the work has started. KO was partially supported by Grant-in-Aid for Scientific Researches (C) in Japan (18K03241). MER was partially supported by PRIN 2015EYPTSB-008 Geometry of Algebraic Varieties. The fourth author was partially supported by the Vietnam Institute for Advanced Study in Mathematics and the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.14.

1. Introduction

The study of the homological properties of the blow-up algebras of an ideal in a Noetherian local ring is an important problem in commutative algebra and in algebraic geometry. The problem is difficult and it is one of the main obstacles in the resolution of singularities. As a remedy one tries to find available information from the Hilbert polynomial with respect to suitable filtrations. The filtration of the integral closure of the powers of an ideal gives rise to the normal Hilbert polynomial, first investigated by D. Rees in his study of pseudo-rational local rings. This study was carried on by J. Lipman, B. Teissier and more recently by T. Okuma, K.-i. Watanabe, and K.Yoshida with the study of $p_{g}$-ideals which inherit nice properties of integrally closed ideals on rational singularities, see [16, 17]. Let $(R,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring of dimension $d>0$ with infinite residue field $R/\mathfrak{m}$ and $I$ an $\mathfrak{m}$-primary ideal of $R.$ Let $\overline{I}$ denote the integral closure of $I.$ By [21] it is well known that if we consider the normal filtration $\{\overline{I^{n}}\}_{n\in\mathbb{Z}},$ there exist integers $\overline{\mathrm{e}}_{i}(I)$, called the normal Hilbert coefficients of $I$ such that

[TABLE]

for $n\gg 0$. Here $\ell_{R}(N)$ denotes, for an $R$-module $N$, the length of $N$. A large literature was devoted to the study of the integer $\overline{g}_{s}(I):=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I}),$ called by Ooishi the normal sectional genus of $I$ (see [18]).

It is well known that $\overline{g}_{s}(I)\geq 0$ and if the equality holds, then the normal associated graded ring $\overline{G}(I):=\oplus_{n\geq 0}\overline{I^{n}}/\overline{I^{n+1}}$ of $I$ is Cohen-Macaulay and $\overline{I^{n+1}}=J\overline{I^{n}}$ for all $n\geq 1$ where $J$ is any minimal reduction of $I$ $($[9, 11, 15, 18]$)$. Notice that to prove the last equality it is not enough to prove $\overline{I^{2}}=J\overline{I}$ as for the $I$-adic filtration. Okuma produced an interesting geometrical example with $\overline{I^{2}}=J\overline{I},$ but $\overline{I^{3}}\neq J\overline{I^{2}}$ (private communication). This makes it clear how difficult can be in general to get information on the reduction number \overline{\rm{r}}_{J}(I):=\min\{r\geq 0\ |\ \mbox{\overline{I^{n+1}}=J\overline{I^{n}}$for all$n\geq r}\}, an important numerical invariant of $I.$ Recently A. Corso, C. Polini and M. E. Rossi showed that if $\overline{g}_{s}(I)=1$ holds true, then $\mathrm{depth}~{}\overline{G}(I)\geq d-1$ (see CPR16). This case was also explored by T. T. Phuong [20] when $R$ is a generalized Cohen-Macaulay ring.

The normal sectional genus is strictly related to the second normal Hilbert coefficient. By [11, Theorem 2] the following inequalities

[TABLE]

hold true and if either of the inequalities is an equality, then $\overline{\rm{r}}_{J}(I)\leq 2,$ in particular $\overline{G}(I)$ is Cohen-Macaulay (see also [2, Theorem 3.11]). The vanishing of $\overline{\mathrm{e}}_{2}(I)$ is a particular case of this situation. In an excellent normal domain of dimension two, $\overline{\mathrm{e}}_{2}(I)=0$ characterizes the $p_{g}$-ideals, see [16]. It is interesting to notice that if $R$ is an excellent normal local domain of dimension two, then $R$ has a rational singularity (resp. minimally elliptic singularity) if and only if $\overline{\mathrm{e}}_{2}(I)=0$ for every $\mathfrak{m}$-primary ideal $I$ in $R$ (resp. $R$ is Gorenstein and \max\{\overline{\mathrm{e}}_{2}(I):I\mbox{ is \mathfrak{m}-primary}\}=1), see [16].

In this paper we present the structure of the Sally module in the case the second normal coefficient is almost minimal, that is $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1.$ If in addition $\overline{\mathrm{e}}_{3}(I)\neq 0,$ we prove that $\mathrm{depth}~{}\overline{G}(I)\geq d-1$ and $\overline{\rm{r}}_{J}(I)=3,$ see Theorem 3.6. Example 3.8 shows that the result is sharp. If $\overline{\mathrm{e}}_{3}(I)=0$ and $d=3,$ we prove that $\overline{G}(I^{\ell})$ is Cohen-Macaulay for all $\ell\geq 2$ (see Theorem 4.3). Actually, if $R$ is Gorenstein, S. Itoh in [11] conjectured that if $\overline{\mathrm{e}}_{3}(I)=0,$ then $\overline{G}(I)$ is Cohen-Macaulay. Recently the conjecture was studied by several authors, see for instance [3] and [12], but as far as we know it is still open. In this paper a deep analysis of the case $\overline{\mathrm{e}}_{3}(I)=0$ has been presented, see Theorem 4.3. Our hope is that these results will be successfully applied to give new insights to prove (or disprove) the long-standing conjecture by Itoh. An interesting analysis on the vanishing of the higher normal coefficients is presented in [4].

The main tools that we use in this paper are the study of the vanishing of local cohomology modules of the normalized Rees algebra, see Theorem 3.1 and Lemma 3.2 and the Sally module introduced by W. V. Vasconcelos in [25]. In particular we study the structure of a suitable filtration of the Sally module considered by M. Vaz Pinto in [26], see Definition 2.1 and Proposition 2.3, useful for getting information on the reduction number.

Acknowledgements

We thank Manoj Kummini and Claudia Polini for helpful conversations throughout the preparation of the manuscript.

2. Notation and Preliminaries

In [14] the first three authors proved useful properties of the Sally module associated to any $I$-admissible filtration that we will need to prove our main result. In this section we recall part of these results. We refer to [19] and to [14] for details.

Throughout this paper, let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with infinite residue field and $I$ an $\mathfrak{m}$-primary ideal in $R.$ Recall that a a filtration of ideals $\mathcal{I}:=\{I_{n}\}_{n\in\mathbb{Z}}$ is a chain of ideals of $R$ such that $R=I_{0}$ and $I_{n}\supseteq I_{n+1}$ for all $n\in\mathbb{Z}$. We say that a filtration $\mathcal{I}$ is $I$-admissible if for all $m,n\in\mathbb{Z},$ $I_{m}\cdot I_{n}\subseteq I_{m+n},~{}I^{n}\subseteq I_{n}$ and there exists $k\in\mathbb{N}$ such that $I_{n}\subseteq I^{n-k}$ for all $n\in\mathbb{Z}.$ If $R$ is analytically unramified, then $\{\overline{I^{n}}\}_{n\in\mathbb{Z}}$ is an $I$-admissible filtration. In fact, by a classical result of Rees [21], $R$ is analytically unramified if and only if the filtration $\{\overline{I^{n}}\}_{n\in Z}$ is $I$-admissible for some (equiv. for all) $\mathfrak{m}$-primary ideal $I$ in $R.$

For an $I$-admissible filtration $\mathcal{I}=\{I_{n}\}_{n\in\mathbb{Z}},$ let

[TABLE]

denote, respectively, the Rees algebra, the extended Rees algebra, and the associated graded ring of $\mathcal{I}$ where $t$ is an indeterminate over $R$. If $\mathcal{I}=\{I^{n}\}_{n\in\mathbb{Z}}$ (resp. $\{\overline{I^{n}}\}_{n\in\mathbb{Z},}$), we write $\mathcal{R}(I),~{}\mathcal{R^{\prime}}(I)$ and $G(I)$ (resp. $\overline{\mathcal{R}}(I),~{}\overline{\mathcal{R^{\prime}}}(I)$ and $\overline{G}(I)$) for the the Rees algebra, the extended Rees algebra, and the associated graded ring of $\mathcal{I},$ respectively. We set $T=\mathcal{R}(J)=\mathcal{R}(\{J^{n}\}_{n\in\mathbb{Z}})$ where $J$ is a minimal reduction of $I.$ Then $\mathcal{R}(\mathcal{I})$ is a module finite extension of $T.$ Hence $\ell_{R}(R/{I_{n+1}})$ for large $n$ is a polynomial and we denote by ${\mathrm{e}}_{i}(\mathcal{I})$ the Hilbert coefficients of ${\mathcal{I}}.$

Following Vasconcelos [25], we consider

[TABLE]

the Sally module of $\mathcal{I}$ with respect to $J$. Notice that $S_{J}(\mathcal{I})$ is a finite $T$-module. In [26] Pinto introduced a filtration of the Sally module in the case $\mathcal{I}=\{I^{n}\}_{n\in\mathbb{Z}}$. She constructed the following graded modules to decompose the structure of Sally modules.

Definition 2.1**.**

For each $\ell\geq 1$, consider the graded $T$-module

[TABLE]

Let $L^{(\ell)}:=L^{(\ell)}_{J}(\mathcal{I})=[C^{(\ell)}]_{\ell}T$ be the $T$-submodule of $C^{(\ell)}$. Then $L^{(\ell)}\cong\bigoplus_{n\geq\ell}J^{n-\ell}I_{\ell+1}/J^{n-\ell+1}I_{\ell}$. Hence for every $\ell\geq 1$ we have the following natural exact sequence of graded $T$-modules

[TABLE]

Notice that $C^{(1)}=S,$ and since $\mathcal{R}(\mathcal{I})$ is a finite graded $T$-module, $C^{(\ell)}$ and $L^{(\ell)}$ are finitely generated graded $T$-modules for every $\ell\geq 1.$

In this paper, the structure of the graded module $C^{(2)}$ plays a fundamental role. The following lemma was proved for the $I$-adic filtration in [19, Lemma 2.1], but the same proof works for any $I$-admissible filtration.

Lemma 2.2**.**

With the notations as above we have

• $(1)$

$\mathfrak{m}C^{(2)}=(0)$* if and only if $\mathfrak{m}I_{n+1}\subseteq J^{n-1}I_{2}$ for all $n\geq 2$.*

• $(2)$

$C^{(2)}=(0)$* if and only if $I_{n+1}=JI_{n}$ for all $n\geq 2$, and*

• $(3)$

$C^{(2)}=[C^{(2)}]_{2}T$* if and only if $I_{n+1}=JI_{n}$ for all $n\geq 3$.*

In the following proposition we need to assume $J\cap I_{2}=JI_{1}$ where $J$ is a minimal reduction of $\mathcal{I}.$ This condition is automatically satisfied if $\mathcal{I}=\{\mathfrak{m}^{n}\}_{n\in\mathbb{Z}}$ or if $\mathcal{I}=\{\overline{I^{n}}\}_{n\in\mathbb{Z}}$ (see [9, 10]). We set $\mathrm{HS}_{G(\mathcal{I})}(z)$ and $\mathrm{HS}_{C^{(2)}}(z)$ denote the Hilbert series of $G(\mathcal{I})$ and $C^{(2)}$ respectively.

Proposition 2.3**.**

$($[14, Proposition 2.4]$)$* Let $\mathfrak{p}=\mathfrak{m}T$ and suppose that $J\cap I_{2}=JI_{1}$. Then the following assertions hold true.*

1. (1)

$\mathrm{Ass}_{T}C^{(2)}\subseteq\{\mathfrak{p}\}$. Hence $\dim_{T}C^{(2)}=d$, if $C^{(2)}\neq(0)$. 2. (2)

For all $n\geq 0,$

[TABLE] 3. (3)

$\mathrm{e}_{1}(\mathcal{I})=\mathrm{e}_{0}(\mathcal{I})-\ell_{R}(R/I_{1})+\ell_{R}(I_{2}/JI_{1})+\ell_{T_{\mathfrak{p}}}(C^{(2)}_{\mathfrak{p}}).$** 4. (4)

$\mathrm{HS}_{G(\mathcal{I})}(z)=\frac{\ell_{R}(R/I_{1})+(\mathrm{e}_{0}(\mathcal{I})-\ell_{R}(R/I_{1})-\ell_{R}(I_{2}/JI_{1}))z+\ell_{R}(I_{2}/JI_{1})z^{2}}{(1-z)^{d}}-(1-z)\mathrm{HS}_{C^{(2)}}(z)$, 5. (5)

Suppose $C^{(2)}\neq(0)$. Then $\mathrm{depth}~{}G(\mathcal{I})=\mathrm{depth}_{T}C^{(2)}-1$, if $\mathrm{depth}_{T}C^{(2)}<d$. Moreover, $\mathrm{depth}~{}G(\mathcal{I})\geq d-1$ if and only if $C^{(2)}$ is a Cohen-Macaulay $T$-module.

We recall that, by using [14, Proposition 2.11] and [20], we have the following interesting result.

Proposition 2.4**.**

Let $d\geq 2$. Then the graded $T$-module ${C}^{(2)}_{J}(\{\overline{I^{n}}\}_{n\in\mathbb{Z}})$ satisfies the Serre’s property $(S_{2})$ as a $T/\mathrm{Ann}_{T}\,({C}^{(2)}_{J}(\{\overline{I^{n}}\}_{n\in\mathbb{Z}}))$-module.

3. The structure of the Sally module when $\overline{\mathrm{e}}_{3}(I)\neq 0$

In this section we prove the first main result of this paper (Theorem 3.6). We set $\overline{C}=\overline{C}_{J}(I)=C^{(2)}_{J}(\{\overline{I^{n}}\}_{n\in\mathbb{Z}})$. In the following theorem we recall few results on the vanishing of local cohomology modules from [11] (see also [6]). From now onwards we set $M^{\prime}=(t^{-1},\mathfrak{m},It){\mathcal{R}^{\prime}}(I)$ for the graded maximal ideal of ${\mathcal{R}^{\prime}}(I)$ and $N^{\prime}=It{\mathcal{R}^{\prime}}(I)$.

Theorem 3.1**.**

$($[11, Proposition 13]$)$* Suppose that $d\geq 2$. Then we have the following.*

1. (1)

$[\mathrm{H}_{N^{\prime}}^{i}(\overline{\mathcal{R}^{\prime}}(I))]_{n}=(0)$* for all $n\gg 0$ and all $i\geq 0$;* 2. (2)

$\mathrm{H}_{M^{\prime}}^{0}(\overline{\mathcal{R}^{\prime}}(I))=\mathrm{H}_{M^{\prime}}^{1}(\overline{\mathcal{R}^{\prime}}(I))=(0)$; 3. (3)

$[\mathrm{H}_{M^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))]_{n}=(0)$* for $n\leq 0$;* 4. (4)

$\mathrm{H}_{M^{\prime}}^{i}(\overline{\mathcal{R}^{\prime}}(I))\cong\mathrm{H}_{N^{\prime}}^{i}(\overline{\mathcal{R}^{\prime}}(I))$* for $0\leq i\leq d-1$.*

To prove the main result of this section we use induction on the dimension $d.$ One of the main difficulties in applying the induction on $d$ for the normal filtration is that the image of a normal ideal going modulo a superficial element need not be normal. Thanks to [11, Theorem 1] (see also [6]) we may choose $a_{1}\in I$ such that $\overline{I(R/(a_{1}))}=\overline{I}(R/(a_{1}))$, and $\overline{I^{n}(R/(a_{1}))}=\overline{I^{n}}(R/(a_{1}))$ for all $n\gg 0$. In particular $a_{1}t$ is $\overline{G}(I)$-regular. From now onwards we set $S=R/(a_{1})$. We prove the following important lemma which shows that one of the main difficulties in the study of the normal Hilbert coefficients is that in general $\overline{I^{n}S}\neq\overline{I^{n}}S$ for $n\in\mathbb{Z}.$

Lemma 3.2**.**

Suppose that $d\geq 3$. Then

• $(1)$

$\mathrm{H}_{M^{\prime}}^{1}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}}))_{n}\cong\overline{I^{n}S}/\overline{I^{n}}S$* for all $n\in\mathbb{Z}$, and*

• $(2)$

$\mathrm{H}_{M^{\prime}}^{i}(\overline{\mathcal{R}^{\prime}}(IS))\cong\mathrm{H}_{M^{\prime}}^{i}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}}))$* for all $i\geq 2$.*

Proof.

Consider the canonical exact sequence

[TABLE]

of graded $T$-modules. Then, since $\overline{I^{n}S}=\overline{I^{n}}S$ for all $n\leq 0$ and, by Theorem [11, Theorem 1], for all $n\gg 0$, the module $\overline{\mathcal{R^{\prime}}}(IS)/\mathcal{R^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})\cong\bigoplus_{n\in\mathbb{Z}}\overline{I^{n}S}/\overline{I^{n}}S$ is finitely graded. Therefore taking the local cohomology functor $\mathrm{H}_{M^{\prime}}^{i}(*)$ to the above exact sequence and using Theorem 3.1(2), we get $\mathrm{H}^{1}_{M^{\prime}}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}}))\cong\overline{\mathcal{R^{\prime}}}(IS)/\mathcal{R^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})$ and $\mathrm{H}^{i}_{M^{\prime}}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})))\cong\mathrm{H}^{i}_{M^{\prime}}(\overline{\mathcal{R^{\prime}}}(IS))$ for $i\geq 2$ as required. ∎

We remark that the following result works like the Sally’s machine [23, Lemma 1.4], but is not a consequence of it.

Proposition 3.3**.**

Assume $d\geq 3$ and $\mathrm{depth}~{}\overline{\mathrm{G}}(IS)\geq 2$. Then we have $\mathrm{depth}~{}\overline{\mathrm{G}}(I)=\mathrm{depth}~{}\overline{\mathrm{G}}(IS)+1.$

Proof.

Let us look at the exact sequence

[TABLE]

of local cohomology modules induced by the canonical exact sequence

[TABLE]

Because $\mathrm{depth}\ \overline{\mathrm{G}}(IS)\geq 2$, we have $\mathrm{depth}\ \overline{\mathcal{R}^{\prime}}(IS)\geq 3$ so that $\mathrm{H}_{M^{\prime}}^{2}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})\cong\mathrm{H}_{M^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(IS))=(0)$ by Lemma 3.2$(2)$. This gives an epimorphism $\mathrm{H}_{M^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))(-1)\to\mathrm{H}_{M^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))\to 0$ of graded $T$-modules. Then since $\mathrm{H}_{M^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))$ is finitely graded by Theorem 3.1, we get $\mathrm{H}_{M^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))=(0)$ so that $\mathrm{H}_{M^{\prime}}^{1}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}}))=(0)$ by the above exact sequence. Then because $\overline{I^{n}S}=\overline{I^{n}}S$ for all $n\in\mathbb{Z}$ by Lemma 3.2$(1)$, we have $\overline{\mathcal{R}^{\prime}}(IS)={\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})$. Thus, we get $\mathrm{depth}\ \overline{\mathrm{G}}(I)=\mathrm{depth}\ \overline{\mathcal{R}^{\prime}}(I)-1=\mathrm{depth}\ {\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})=\mathrm{depth}\ \overline{\mathcal{R}^{\prime}}(IS)=\mathrm{depth}\ \overline{\mathrm{G}}(IS)+1$ as required. ∎

The following lemma is a consequence of the Grothendieck-Serre formula [1, Theorem 4.1].

Lemma 3.4**.**

1. $(1)$

Suppose $d\geq 3$. Then, for each $n\in\mathbb{Z}$, we have

[TABLE] 2. $(2)$

Suppose $d=3.$ Then

[TABLE]

Proof.

(1) Consider the short exact sequence

[TABLE]

By using Theorem 3.1 we get a long exact sequence of local cohomology modules

[TABLE]

Hence we have

[TABLE]

for all $n\in\mathbb{Z}$. Thanks to [1, Theorem 4.1], for each $n\in\mathbb{Z}$, we have

[TABLE]

because $\mathrm{H}_{N^{\prime}}^{1}(\overline{\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}))_{k}\cong\overline{I^{k}S}/\overline{I^{k}}S$ for all $k\in\mathbb{Z}$ and $\mathrm{H}_{N^{\prime}}^{i}({\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}))\cong\mathrm{H}_{N^{\prime}}^{i}(\overline{\mathcal{R}^{\prime}}(IS))$ as graded $T$-modules for $i\geq 2$ by Lemma 3.2. (2) Follows from (1). ∎

The following result plays a key role for our proof of Theorem 3.6.

Theorem 3.5**.**

Suppose $d\geq 2$. Assume $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$ and $\overline{\mathrm{e}}_{3}(I)\neq 0$ (if $d\geq 3$), then $\ell_{R}(\overline{I^{3}}/J\overline{I^{2}})=1$ and $\overline{I^{n+1}}=J\overline{I^{n}}$ for all $n\geq 3$.

Proof.

We prove the required equalities by induction on $d$. Suppose that $d=2$. Then since $\mathrm{depth}~{}\overline{G}(I)>0,$ using [8, Proposition 4.6] we get $\overline{\mathrm{e}}_{1}(I)=\sum_{n\geq 0}\ell_{R}(\overline{I^{n+1}}/J\overline{I^{n}})$ and $\overline{\mathrm{e}}_{2}(I)=\sum_{n\geq 1}n\ell_{R}(\overline{I^{n+1}}/J\overline{I^{n}})$. Therefore

[TABLE]

This implies that $\ell_{R}(\overline{I^{3}}/J\overline{I^{2}})=1$ and $\overline{I^{n+1}}=J\overline{I^{n}}$ for all $n\geq 3$ as required.

Suppose that $d\geq 3$ and that our assertion holds true for $d-1$. We then have $\overline{\mathrm{e}}_{2}(IS)=\overline{\mathrm{e}}_{1}(IS)-\overline{\mathrm{e}}_{0}(IS)+\ell_{S}(S/\overline{IS})+1$, and $\overline{\mathrm{e}}_{3}(IS)\neq 0$ if $d\geq 4$, by [11, Theorem 1]. The hypothesis of induction on $d$ implies that we have $\ell_{R}(\overline{I^{3}S}/J\overline{I^{2}S})=1$ and $\overline{I^{n+1}S}=J\overline{I^{n}S}$ for all $n\geq 3$. Hence to prove our assertion it is enough to show that $\overline{I^{n}S}=\overline{I^{n}}S$ holds true for all $n\in\mathbb{Z}$.

Suppose $d\geq 4$. By using [14, Theorem 3.1], we have $\mathrm{depth}~{}{\overline{\mathrm{G}}}(IS)\geq(d-1)-1=d-2\geq 2.$ Then $\mathrm{depth}~{}\overline{G}(I)\geq 3$ by Proposition 3.3. Hence $\mathrm{depth}~{}{\mathcal{R}^{\prime}}(\{\overline{I^{n}}S\}_{n\in\mathbb{Z}})=\mathrm{depth}~{}\overline{\mathcal{R^{\prime}}}(I)-1\geq 3$. Therefore $\overline{I^{n}S}=\overline{I^{n}}S$ for all $n\in\mathbb{Z}$ by Lemma 3.2 as required.

Suppose that $d=3$. Consider the exact sequence

[TABLE]

of local cohomology modules which is induced by the canonical exact sequence

[TABLE]

Since $\overline{I^{n+1}S}=J\overline{I^{n}S}$ for all $n\geq 3$, we get ${\rm{a}_{2}}(\overline{G}(IS))+2\leq\overline{\rm{r}}_{JS}(IS)\leq 3$ by [5, Proposition 3.2] $($also [24, Proposition 3.2]$)$ so that $[\mathrm{H}_{N^{\prime}}^{2}(\overline{G}(IS))]_{n}=(0)$ for all $n\geq 2$ where ${\rm{a}_{2}}(\overline{G}(IS)):=\sup\{n\in\mathbb{Z}\ |\ [\mathrm{H}_{N^{\prime}}^{2}(\overline{G}(IS))]_{n}\neq(0)\}$ denotes the ${\rm{a}}$-invariant of $\overline{G}(IS)$. Hence $\mathrm{H}_{N^{\prime}}^{2}(\overline{\mathcal{R^{\prime}}}(IS))_{n}=(0)$ for all $n\geq 2$. Therefore we have

[TABLE]

by Lemma 3.4(2). Therefore, because $\overline{\mathrm{e}}_{3}(I)>0$ by our assumption and [11, Theorem 3], we get $\overline{I^{n}S}=\overline{I^{n}}S$ for all $n\in\mathbb{Z}$ as required. ∎

Let $B:=T/\mathfrak{m}T\cong(R/\mathfrak{m})[X_{1},X_{2},\cdots,X_{d}]$ the polynomial ring with $d$ indeterminates over the field $R/\mathfrak{m}$.

Now we give a complete structure of the Sally module and we describe the Hilbert series of the associated graded ring in the case $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$ and $\overline{\mathrm{e}}_{3}(I)\neq 0.$

Theorem 3.6**.**

Suppose that $d\geq 2$. Then following statements are equivalent:

1. $(1)$

$\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$* and, if $d\geq 3,$ $\overline{\mathrm{e}}_{3}(I)\neq 0$,* 2. $(2)$

$\overline{\mathrm{e}}_{2}(I)=\ell_{R}(\overline{I^{2}}/J\overline{I})+2$, 3. $(3)$

$\overline{C}_{J}(I)\cong B(-2)$* as graded $T$-modules, and* 4. $(4)$

$\ell_{R}(\overline{I^{3}}/J\overline{I^{2}})=1$* and $\overline{I^{n+1}}=J\overline{I^{n}}$ for all $n\geq 3$.*

In this case, the following assertions follow:

• $(i)$

$\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+1$.

• $(ii)$

$\overline{\mathrm{e}}_{3}(I)=1$* if $d\geq 3$, and $\overline{\mathrm{e}}_{i}(I)=0$ for $4\leq i\leq d$.*

• $(iii)$

$HS_{\overline{G}(I)}(z)=\frac{\ell_{R}(R/\overline{I})+(\mathrm{e}_{0}(I)-\ell_{R}(R/\overline{I})-\ell_{R}(\overline{I^{2}}/J\overline{I}))z+(\ell_{R}(\overline{I^{2}}/J\overline{I})-1)z^{2}+z^{3}}{(1-z)^{d}}.$**

• $(iv)$

$\mathrm{depth}~{}\overline{G}(I)\geq d-1$, and $\overline{G}(I)$ is Cohen-Macaulay if and only if $\overline{I^{3}}\nsubseteq J.$

Proof.

$(1)\Rightarrow(4)$ follows from Theorem 3.5. Thanks to [14, Theorem 3.1], $(3)\Leftrightarrow(4)$, $(4)\Rightarrow(2)$, and the last assertions $(i)\sim(iv)$ follow.

$(2)\Rightarrow(1):$ By [11, Theorem 2] and the assumption we have the inequalities

[TABLE]

and the equality $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})$ is true if and only if the equality $\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})$ is true. Hence we have $\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+1$ and $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$. Then, thanks to [14, Theorem 3.1], we get $\overline{\mathrm{e}}_{3}(I)=\binom{m}{2}\neq 0$ for some $m\geq 2$. ∎

By the above result we notice that if $d\geq 3$ and $\overline{\mathrm{e}}_{2}(I)\leq\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1,$ then $\overline{\mathrm{e}}_{3}(I)\leq 1$.

The next result follows immediately from Theorem 3.6 and [14, Corollary 4.1]. Recall that an ideal $I$ is said to be normal if $\overline{I^{n}}=I^{n}$ for all $n\geq 0.$ We set $\mathrm{e}_{i}(I)=\mathrm{e}_{i}(\{I^{n}\}_{n\in\mathbb{Z}})$ denotes the $i$-th Hilbert coefficient of the $I$-adic filtration $\{I^{n}\}_{n\in\mathbb{Z}}$.

Corollary 3.7**.**

Assume $d\geq 2$ and $I$ is normal. Then the following conditions are equivalent:

• $(1)$

$\mathrm{e}_{1}(I)=\mathrm{e}_{0}(I)-\ell_{R}(R/I)+\ell_{R}(I^{2}/JI)+1;$**

• $(2)$

$\mathrm{e}_{2}(I)=\mathrm{e}_{1}(I)-\mathrm{e}_{0}(I)+\ell_{R}(R/I)+1$* and, if $d\geq 3,$ ${\mathrm{e}}_{3}(I)\neq 0;$*

• $(3)$

${C^{(2)}_{J}(\{I^{n}\}_{n\in\mathbb{Z}})}\cong B(-2)$* as graded $T$-modules;*

• $(4)$

$\ell_{R}(I^{3}/JI^{2})=1$* and $I^{4}=JI^{3}$.*

When this is the case, $\mathrm{depth}~{}G(I)\geq d-1$, and $G(I)$ is Cohen-Macaulay if and only if $I^{3}\nsubseteq J$.

The following example, due to Huckaba and Huneke [7, Theorem 3.12], shows that if $I$ is normal, $\mathrm{e}_{2}(I)=\mathrm{e}_{1}(I)-\mathrm{e}_{0}(I)+\ell_{R}(R/I)+1$ and $\mathrm{e}_{3}(I)\neq 0,$ then $G(I)$ need not be Cohen-Macaulay and hence Theorem 3.6 is sharp.

Example 3.8**.**

Let $K$ be a field of characteristic $\not=3$ and set $R=K[\![X,Y,Z]\!]$, where $X,Y,Z$ are indeterminates. Let $N=(X^{4},X(Y^{3}+Z^{3}),Y(Y^{3}+Z^{3}),Z(Y^{3}+Z^{3}))$ and set $I=N+{\mathfrak{m}}^{5}$, where ${\mathfrak{m}}$ is the maximal ideal of $R$. The ideal $I$ is a normal ${\mathfrak{m}}$-primary ideal whose associated graded ring ${G}(I)$ has depth $d-1=2.$ Moreover,

[TABLE]

and hence $\ell_{R}(R/I)=31,~{}\mathrm{e}_{0}(I)=76,~{}\mathrm{e}_{1}(I)=48,~{}\mathrm{e}_{2}(I)=4,~{}\mathrm{e}_{3}(I)=1$. Thus $\mathrm{e}_{2}(I)=\mathrm{e}_{1}(I)-\mathrm{e}_{0}(I)+\ell_{R}(R/I)+1.$ For the computations see [2, Example 3.2].

4. The structure of the Sally module when $\overline{\mathrm{e}}_{3}(I)=0$

In this section we consider the case $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$ and $\overline{\mathrm{e}}_{3}(I)=0$ in three dimensional case. This case faces the difficult problem stated by Itoh in [11] on the vanishing of $\overline{\mathrm{e}}_{3}(I)$ which asserts that if $\overline{\mathrm{e}}_{3}(I)=0$ and $R$ is Gorenstein, then $\overline{G}(I)$ is Cohen-Macaulay or equivalently $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I}).$ Hence for the class of ideals verifying Itoh’s conjecture the assumptions of this section doesn’t occur. This is the case for instance when $\overline{I}=\mathfrak{m}$ and $R$ is Gorenstein, see[11, Theorem 3(2)] (more generally, $R$ satisfying $\ell_{R}(\overline{I^{2}}/J\overline{I})\geq\mathrm{type}(R)-2,$ see [3, 12]). If $R$ is not Gorenstein or $R$ is Gorenstein and $\overline{I}\neq\mathfrak{m},$ our analysis can be useful for proving or disproving Itoh’s conjecture, also because the doubt of the validity of Itoh’s conjecture is growing among the experts.

In Theorem 4.3 we prove that if $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$ and $\overline{\mathrm{e}}_{3}(I)=0$, then $\overline{G}(I^{\ell})$ is Cohen-Macaulay for all $\ell\geq 2.$ For this purpose we need the following proposition which is a consequence of Serre’s formula and it seems to be well known. However, for the sake of completeness we give a proof of this.

We set, for $\ell\in\mathbb{Z}$, ${\mathcal{I}}^{(\ell)}=\{I_{n\ell}\}_{n\in\mathbb{Z}}$, and ${\rm a}_{i}({\rm G}({\mathcal{I}}))=\max\{n\in\mathbb{Z}\ |\ [\mathrm{H}_{M}^{i}({\rm G}({\mathcal{I}})]_{n}\neq(0)\}$ for $i\in\mathbb{Z}$.

Proposition 4.1**.**

Let $\ell>\max\{{\rm a}_{i}({\rm G}({\mathcal{I}}))\,|\,0\leq i\leq d\}$ be an integer. Then we have $\ell_{R}(R/I_{\ell(n+1)})=\sum_{i=0}^{d}(-1)^{i}\mathrm{e}_{i}({\mathcal{I}}^{(\ell)})\binom{n+d-i}{d-i}$ for all $n\geq 0$. In particular, the equality $\ell_{R}(R/I_{\ell})=\sum_{i=0}^{d}(-1)^{i}\mathrm{e}_{i}({\mathcal{I}}^{(\ell)})$ holds true for all $n\geq 0$.

Proof.

We have ${\rm a}_{i}({\rm G}({\mathcal{I}}^{(\ell)}))\leq\lfloor{\rm a}_{i}({\rm G}({\mathcal{I}}))/\ell\rfloor\leq 0$ by [5, Theorem 4.2] where $\lfloor q\rfloor=\max\{n\in\mathbb{Z}\ |\ n\leq q\}$ for $q\in\mathbb{Q}$. Then thanks to Serre’s formula $\ell_{R}([{\rm G}({\mathcal{I}^{(\ell)}})]_{k})=\sum_{i=0}^{d-1}(-1)^{i}\mathrm{e}_{i}({\mathcal{I}^{(\ell)}})\binom{k+d-i-1}{d-i-1}$ for all $k\geq 1$. Hence we have

[TABLE]

for all $n\geq 0$. Therefore, $(-1)^{d}{\mathrm{e}_{d}}({\mathcal{I}}^{(\ell)})=\ell_{R}([{\rm G}({\mathcal{I}^{(\ell)}})]_{0})-\sum_{i=0}^{d-1}(-1)^{i}\mathrm{e}_{i}({\mathcal{I}^{(\ell)}})$ and the equality $\ell_{R}(R/I_{\ell(n+1)})=\sum_{i=0}^{d}(-1)^{i}\mathrm{e}_{i}({\mathcal{I}}^{(\ell)})\binom{n+d-i}{d-i}$ holds true for all $n\geq 0$. ∎

As a consequence of Proposition 4.1 we obtain a result of Rees [22, Theorem 2.6] (see also [9, Theorem 4.5]) in dimension two which states that: $\overline{\mathrm{e}}_{2}(I)=0$ if and only if $\overline{\mathrm{e}}_{1}(I^{\ell})=\mathrm{e}_{0}(I^{\ell})-\ell_{R}(R/\overline{I^{\ell}})$ for all $\ell\geq 1$. In particular, by [23, Theorem 2.9] $\overline{\rm G}(I^{\ell})$ is Cohen-Macaulay for all $\ell\geq 1$. Analogously we obtain the following result on vanishing of $\overline{\mathrm{e}}_{3}(I)$ in dimension three as a consequence of Proposition 4.1. Notice that next result for normal ideals can be also obtained as a consequence of [12, Corollary 5.3.]

Corollary 4.2**.**

Let $d=3$, then the following conditions are equivalent.

• $(1)$

$\overline{\mathrm{e}}_{3}(I)=0$,

• $(2)$

$\overline{\mathrm{e}}_{1}(I^{\ell})=2\overline{\mathrm{e}}_{0}(I^{\ell})+\ell_{R}(R/\overline{I^{\ell}})-\ell_{R}(\overline{I^{\ell}}/\overline{I^{2\ell}})$* for some (equiv. all) $\ell>\max\{{\mathrm{a}}_{i}(\overline{\mathrm{G}}(I))\ |\ 1\leq i\leq 3\}$, and*

• $(3)$

$\overline{\mathrm{e}}_{2}(I^{\ell})=\overline{\mathrm{e}}_{1}(I^{\ell})-\overline{\mathrm{e}}_{0}(I^{\ell})+\ell_{R}(R/\overline{I^{\ell}})$* for some (equiv. all) $\ell>\max\{{\mathrm{a}}_{i}(\overline{\mathrm{G}}(I))\,|\,1\leq i\leq 3\}$.*

In particular, $\overline{\rm G}(I^{\ell})$ is Cohen-Macaulay for all $\ell>\max\{{\mathrm{a}}_{i}(\overline{\mathrm{G}}(I))\ |\ 1\leq i\leq 3\}$ if any of the above conditions are satisfied.

The last assertion of Corollary 4.2 is a consequence of [11, Theorem 2(2)]. Now we are ready to prove the main theorem of this section.

Theorem 4.3**.**

Suppose $d=3$. Then the following conditions are equivalent.

• $(1)$

$\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1$* and $\overline{\mathrm{e}}_{3}(I)=0$;*

• $(2)$

there exists an exact sequence $0\to B(-3)\to B(-2)^{\oplus 3}\to\overline{C}\to 0$ of graded $T$-modules;

• $(3)$

there exists an exact sequence $0\to B(-2)^{\oplus 2}\to\overline{C}\to(B/(a_{1}t))(-2)\to 0$ of graded $T$-modules.

When this is the case, the following assertions hold true:

• $(i)$

$\mathfrak{m}\overline{C}=(0)$* and $\mathrm{rank}_{B}\overline{C}=2$, and $\mathrm{depth}_{T}\overline{C}=2$,*

• $(ii)$

$\mathfrak{m}\overline{I^{3}}\subseteq J\overline{I^{2}}$, $\ell_{R}(\overline{I^{3}}/J\overline{I^{2}})=3$, $\overline{I^{n+1}}=J\overline{I^{n}}$ for all $n\geq 3$,

• $(iii)$

$\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+2$* and $\overline{\mathrm{e}}_{2}(I)=\ell_{R}(\overline{I^{2}}/J\overline{I})+3$,*

• $(iv)$

${\mathrm{HS}}_{\overline{\rm G}(I)}(z)=\frac{\ell_{R}(R/\overline{I})+\{\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})-\ell_{R}(\overline{I^{2}}/J\overline{I})\}z+\{\ell_{R}(\overline{I^{2}}/J\overline{I})-3\}z^{2}+4z^{3}-z^{4}}{(1-z)^{3}}$,

• $(v)$

$\mathrm{depth}~{}\overline{\rm G}(I)=1$, and $\mathrm{H}_{M^{\prime}}^{1}(\overline{\rm G}(I))=[\mathrm{H}_{M^{\prime}}^{1}(\overline{\rm G}(I))]_{0}$, $\ell_{R}([\mathrm{H}_{M^{\prime}}^{1}(\overline{\rm G}(I))]_{0})=1$, ${\rm a}_{2}(\overline{\rm G}(I))=1$, and ${\rm a}_{3}(\overline{\rm G}(I))\leq-1$,

• $(vi)$

$\overline{\rm G}(I^{\ell})$* is Cohen-Macaulay for all $\ell\geq 2$.*

Proof.

: First we prove $(1)\Rightarrow(3)\Rightarrow(2)\Rightarrow(1)$.

$(1)\Rightarrow(3):$ Recall that by [11, Theorem 1] (see also [6]) we can choose $a_{1}\in I$ such that $\overline{I(R/(a_{1}))}=\overline{I}(R/(a_{1}))$, and $\overline{I^{n}(R/(a_{1}))}=\overline{I^{n}}(R/(a_{1}))$ for all $n\gg 0.$ Let $J$ be a minimal reduction of $I$ such that $J=(a_{1},a_{2},a_{3}).$ Let $S=R/(a_{1}).$ We have $\overline{\mathrm{e}}_{2}(IS)=\overline{\mathrm{e}}_{1}(IS)-\overline{\mathrm{e}}_{0}(IS)+\ell_{R}(S/\overline{IS})+1$ by [11, Theorem 1]. Therefore we have

[TABLE]

The last equality is true because $\ell_{R}(R/\overline{I})=\ell_{S}(S/\overline{I}S),$ and $\ell_{R}(\overline{I^{2}}/J\overline{I})=\ell_{R}(\overline{I^{2}}S/J\overline{I}S)$ $($notice that $a_{1}t$ is $\overline{G}(I)$-regular$)$. We also have $\ell_{R}(\overline{I^{3}S}/J\overline{I^{2}S})=1$ and $\overline{I^{n+1}S}=J\overline{I^{n}S}$ for all $n\geq 3$ by Theorem 3.6.

Claim 1**.**

We have $\ell_{R}(\overline{I^{2}S}/\overline{I^{2}}S)=1$ and $\overline{I^{n}S}=\overline{I^{n}}S$ for all $n\geq 3$.

Proof of Claim 1.

Assume that $\overline{I^{2}S}=\overline{I^{2}}S.$ Then we have $\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+1$ by the equality $(*)$. Then, since $\overline{\mathrm{e}}_{2}(I)=\ell_{R}(\overline{I^{2}}/J\overline{I})+2,$ we have $\overline{\mathrm{e}}_{3}(I)=1$ by Theorem 3.6 which contradicts that $\overline{\mathrm{e}}_{3}(I)=0$. Thus, we have $\overline{I^{2}S}\neq\overline{I^{2}}S$.

Since $\overline{\rm{r}}_{JS}(IS)=3$, we have $[\mathrm{H}_{N^{\prime}}^{2}(\overline{G}(IS))]_{n}=(0)$ for all $n\geq 2$ by [5, Proposition 3.2]. Therefore by using the long exact sequence of local cohomology modules

[TABLE]

induced from

[TABLE]

we obtain that $[\mathrm{H}_{N^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(IS))]_{n}=(0)$ for all $n\geq 2.$ Therefore by Lemma 3.4(2), $\overline{\mathrm{e}}_{3}(I)=1-\sum_{n\geq 2}\ell_{R}(\overline{I^{n}S}/\overline{I^{n}}S).$ Then because $\overline{\mathrm{e}}_{3}(I)=0$ by our assumption and $\overline{I^{2}S}\neq\overline{I^{2}}S$, we get $\ell_{R}(\overline{I^{2}S}/\overline{I^{2}}S)=1$ and $\overline{I^{n}S}=\overline{I^{n}}S$ for all $n\geq 3$ as required. ∎

Since $\ell_{R}(\overline{I^{2}S}/\overline{I^{2}}S)=1$ by Claim 1, we choose $y\in R$ such that $\overline{I^{2}S}=yS+\overline{I^{2}}S\neq\overline{I^{2}}S$ and $\mathfrak{m}S\cdot y\subseteq\overline{I^{2}}S$. Recall that we have $\ell_{R}(\overline{I^{3}S}/J\overline{I^{2}S})=1$ and $\overline{I^{n+1}S}=J\overline{I^{n}S}$ for all $n\geq 3.$ Since $\overline{I^{3}S}=\overline{I^{3}}S$, there exists $x\in\overline{I^{3}}$ such that $\overline{I^{3}S}=xS+J\overline{I^{2}S}$. Then, because $\overline{C}_{JS}(IS)\cong B^{\prime}(-2)$ by Theorem 3.6 where $B^{\prime}={\mathcal{R}}(JS)/\mathfrak{m}{\mathcal{R}}(JS)$, we have $\overline{C}_{JS}(IS)=\overline{xt^{2}}^{\prime}B^{\prime}$ where $\overline{xt^{2}}^{\prime}$ denotes the image of $xt^{2}$ in $\overline{C}_{JS}(IS)$. Hence

[TABLE]

Because $\overline{I^{3}}S/J\overline{I^{2}}S$ is a homomorphic image of $\overline{I^{3}}/J\overline{I^{2}}$, there exist elements $y_{2},y_{3}\in\overline{I^{3}}$ which corresponds to $a_{2}y$, and $a_{3}y$ in $S$ respectively. We then have the equality $\overline{I^{3}}S=(x,y_{2},y_{3})S+J\overline{I^{2}}S$ so that the equality

[TABLE]

holds true. We furthermore have, for all $n\geq 3$, $\overline{I^{n+1}}=J\overline{I^{n}}+(a_{1})\cap\overline{I^{n+1}}=J\overline{I^{n}}$, because $\overline{I^{n+1}}S=\overline{I^{n+1}S}=J\overline{I^{n}S}=J\overline{I^{n}}S$ for all $n\geq 3$. Therefore $\overline{C}=(\overline{xt^{2}},\overline{y_{2}t^{2}},\overline{y_{3}t^{2}})T$ where $\overline{xt^{2}}$, $\overline{y_{2}t^{2}}$, and $\overline{y_{3}t^{2}}$ denote the images of $xt^{2}$, $y_{2}t^{2}$, and $y_{3}t^{2}$ in $\overline{C}$, respectively. Let $C^{\prime}=(\overline{y_{2}t^{2}},\overline{y_{3}t^{2}})T$ be the graded submodule of $\overline{C}$. Then for all $n\in\mathbb{Z},$ we have

[TABLE]

Claim 2**.**

We have $C^{\prime}_{2}\cong(R/\mathfrak{m})^{2}$ so that $C^{\prime}$ forms a graded $B$-module.

Proof of Claim 2.

We have $C^{\prime}_{2}\cong[(y_{2},y_{3})+J\overline{I^{2}}]/J\overline{I^{2}}.$ For $u_{2},u_{3}\in R$ define a surjective map $f:(R/\mathfrak{m})^{2}\to C_{2}^{\prime}$ as $f(\overline{u_{2}},\overline{u_{3}})=\overline{u_{2}y_{2}+u_{3}y_{3}}.$ Here $\overline{(.)}$ denotes the image of an element in respective quotient. It is enough to prove that the map $f$ is injective.

Notice that $f$ is well-defined. In fact, since $\mathfrak{m}S\cdot y\subseteq\overline{I^{2}}S,$ we have $\mathfrak{m}\cdot(y_{2},y_{3})S=\mathfrak{m}\cdot(a_{2}y,a_{3}y)S\subseteq J\overline{I^{2}}S.$ Hence $\mathfrak{m}\cdot(y_{2},y_{3})\subseteq J\overline{I^{2}}+(a_{1})\cap\overline{I^{3}}=J\overline{I^{2}}$.

Suppose that $f(\overline{u_{2}},\overline{u_{3}})=0$ for $u_{2},u_{3}\in R.$ Then $u_{2}y_{2}+u_{3}y_{3}\in J\overline{I^{2}}.$ Therefore ${u_{2}}^{\prime}{y_{2}}^{\prime}+{u_{3}}^{\prime}{y_{3}}^{\prime}={u_{2}}^{\prime}(a_{2}{y})^{\prime}+{u_{3}}^{\prime}(a_{3}{y})^{\prime}\in J\overline{I^{2}}S$ where ${(.)}^{\prime}$ denotes the image of an element in $S.$ Write $u_{2}^{\prime}(a_{2}y)^{\prime}+u_{3}^{\prime}(a_{3}y)^{\prime}=a_{2}^{\prime}i_{2}^{\prime}+a_{3}^{\prime}i_{3}^{\prime}$ with $i_{2},i_{3}\in\overline{I^{2}}$. Then $a_{2}^{\prime}((u_{2}y)^{\prime}-i_{2}^{\prime})=a_{3}^{\prime}(i_{3}^{\prime}-(u_{3}y)^{\prime})$ so that $(u_{2}y)^{\prime}-i_{2}^{\prime}\in(a_{3}^{\prime})\cap\overline{I^{2}S}=(a_{3}^{\prime})\overline{I}S\subseteq\overline{I^{2}}S$. Hence $(u_{2}y)^{\prime}\in\overline{I^{2}}S$. Since $y^{\prime}\notin\overline{I^{2}}S$ we get that $u_{2}\in\mathfrak{m}$. By the same argument we get that $u_{3}\in\mathfrak{m}.$ Hence $f$ is injective. Thus $f$ is an isomorphism which proves the claim. ∎

Since $\overline{C}$ is generated by the homogeneous elements of degree two, so is $\overline{C}/C^{\prime}$. We have $\mathfrak{m}\overline{I^{3}}\subseteq(y_{2},y_{3})+J\overline{I^{2}}$ because $\mathfrak{m}\overline{I^{3}S}\subseteq J\overline{I^{2}S}$ $($recall that $\ell_{R}(\overline{I^{3}S}/J\overline{I^{2}S})=1)$. Then because $[\overline{C}/C^{\prime}]_{2}\cong\overline{I^{3}}/[(y_{2},y_{3})+J\overline{I^{2}}]$ and $\overline{I^{3}}=(x)+[(y_{2},y_{3})+J\overline{I^{2}}]$, $\overline{C}/C^{\prime}$ forms a graded $B$-module and $\overline{C}/C^{\prime}=\overline{xt^{2}}^{\prime\prime}B$ where $\overline{xt^{2}}^{\prime\prime}$ denotes the image of $xt^{2}$ in $\overline{C}/C^{\prime}$.

By Claim 1 and the equality $(*)$ we have $\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+2$ and hence $\ell_{T_{\mathfrak{p}}}(\overline{C}_{\mathfrak{p}})=2$ by Proposition 2.3(3). Consider the canonical exact sequence

[TABLE]

of graded $T$-modules. Then $\ell_{T_{\mathfrak{p}}}(\overline{C}_{\mathfrak{p}})=2$ implies that $1\leq\ell_{T_{\mathfrak{p}}}(C^{\prime}_{\mathfrak{p}})\leq 2$.

Suppose that $\ell_{T_{\mathfrak{p}}}(C^{\prime}_{\mathfrak{p}})=1.$ Then, because $\mathrm{Ass}_{T}C^{\prime}\subseteq\mathrm{Ass}_{T}\overline{C}=\{\mathfrak{p}\}$, $C^{\prime}$ is a $B$-torsion free module of rank one. Hence $C^{\prime}\cong\mathfrak{a}(m)$ as graded $B$-modules for some graded ideal $\mathfrak{a}$ in $B$ and for some $m\in\mathbb{Z}$. Since $C^{\prime}$ is not $B$-free $($notice that $C^{\prime}_{n}=(0)$ for all $n\leq 1$ and $\ell_{R}(C^{\prime}_{2})=2)$ and $B$ is a UFD, we may assume that $\mathrm{ht}_{B}~{}\mathfrak{a}\geq 2$. On the other hand we have $\overline{C}/C^{\prime}\cong B(-2)$, because there is an epimorphism $B(-2)\to\overline{C}/C^{\prime}$ of graded $B$-modules, $B$ is a domain, and $\dim_{B}\overline{C}/C^{\prime}=\dim B$. Let $P\in\mathrm{Ass}_{B}(B/\mathfrak{a})$, then thanks to the exact sequence

[TABLE]

we get $\mathrm{depth}_{B_{P}}C^{\prime}_{P}=1$ because $C^{\prime}\cong\mathfrak{a}(m)$ and $\mathrm{depth}~{}B_{P}\geq 2$. Furthermore, the sequence

[TABLE]

is exact. Then, since $\overline{C}$ satisfies the Serre’s property $(S_{2})$ as $T/\mathrm{Ann}_{T}\overline{C}$-module by Corollary 2.4, $\mathrm{depth}_{B_{P}}(\overline{C}/C^{\prime})_{P}=0$. This gives a contradiction, because $\overline{C}/C^{\prime}\cong B(-2)$ and $\mathrm{ht}_{B}~{}P\geq 2$.

Thus we get $\ell_{T_{\mathfrak{p}}}(C^{\prime}_{\mathfrak{p}})=2$. Then, since $C^{\prime}=(\overline{y_{2}t^{2}},\overline{y_{3}t^{2}})B$, the natural surjective map $B(-2)\oplus B(-2)\to C^{\prime}\to 0$ of graded $T$-modules forms an isomorphism. Therefore we have $\mathrm{depth}_{T}~{}\overline{C}/C^{\prime}\geq 2$ by the exact sequence $(**)$ because the ring $B$ is $3$-dimensional Cohen-Macaulay and $\mathrm{depth}_{T}\,\overline{C}\geq 2$ by Proposition 2.3(5). Then for all $n\geq 0$, we have

[TABLE]

Hence by Proposition 2.3(2), we get $\ell_{R}(R/\overline{I^{n+1}})=\overline{\mathrm{e}}_{0}(I)\binom{n+3}{3}-\{\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+2\}\binom{n+2}{2}+\{\ell_{R}(\overline{I^{2}}/J\overline{I})+4\}\binom{n+1}{1}-2-\ell_{R}([\overline{C}/C^{\prime}]_{n}).$ On the other hand, since $\overline{\mathrm{e}}_{1}(I)=\overline{\mathrm{e}}_{0}(I)-\ell_{R}(R/\overline{I})+\ell_{R}(\overline{I^{2}}/J\overline{I})+2$, $\overline{\mathrm{e}}_{2}(I)=\overline{\mathrm{e}}_{1}(I)-\overline{\mathrm{e}}_{0}(I)+\ell_{R}(R/\overline{I})+1=\ell_{R}(\overline{I^{2}}/J\overline{I})+3,$ and $\overline{\mathrm{e}}_{3}(I)=0,$ we have $\ell_{R}([\overline{C}/C^{\prime}]_{n})=\binom{n+1}{1}-2$ for all $n\gg 0$. Hence $\overline{C}/C^{\prime}$ is a $2$-dimensional Cohen-Macaulay $B$-module with multiplicity one.

In the rest of this proof, we show that $\overline{C}/C^{\prime}\cong(B/a_{1}tB)(-2)$ as graded $T$-modules.

Let $\beta:R[t]\to S[t]$ be the natural $R$-algebra map defined by $\beta(t)=t.$ This induces the homomorphism $\psi:\overline{C}\to\overline{C}_{JS}(IS)$ of graded $T$-modules. Since $\beta(y_{2}),\beta(y_{3})\in J\overline{I^{2}S}$ and $C^{\prime}=(\overline{y_{2}t^{2}},\overline{y_{3}t^{2}})B$, $\psi$ in turn induces the graded homomorphism $\overline{\psi}:\overline{C}/C^{\prime}\to\overline{C}_{JS}(IS)$ of graded $T$-modules. Let $\varphi:B(-2)\to\overline{C}/C^{\prime}$ and $\varphi^{\prime}:(B/a_{1}tB)(-2)\to\overline{C}_{JS}(IS)$ denote homomorphisms of graded $B$-modules defined by $\varphi(1)=\overline{xt^{2}}^{\prime\prime}$ and $\varphi^{\prime}(1)=\overline{xt^{2}}^{\prime}$. Consider the following commutative diagram

[TABLE]

of graded $B$-modules where $i:B\to B/a_{1}tB$ denotes the natural map. Then, since $\varphi^{\prime}$ is an isomorphism, we get $[(0):_{B}\overline{xt^{2}}^{\prime\prime}]\subseteq a_{1}tB$.

Since, $\overline{C}/C^{\prime}\cong(B/[(0):_{B}\overline{xt^{2}}^{\prime\prime}])(-2)$ is a 2-dimensional Cohen-Macaulay $B$-module with multiplicity one, the natural surjective map $(B/[(0):_{B}\overline{xt^{2}}^{\prime\prime}])\to(B/a_{1}tB)$ is an isomorphism. This completes the proof of the implication $(1)\Rightarrow(3)$ of Theorem 4.3.

$(3)\Rightarrow(2)$ and $(i)$: Let us consider the exact sequence

[TABLE]

of graded $T$-modules. Since $\overline{C}/\mathrm{Im}~{}\phi\cong(B/a_{1}tB)(-2)$, we have $a_{1}t\overline{C}\subseteq\mathrm{Im}\phi\cong B(-2)^{\oplus 2}$. Hence we have $a_{1}t\mathfrak{m}\overline{C}=(0)$, so that $\mathfrak{m}\overline{C}=(0)$ because $a_{1}t$ is a $\overline{C}$-regular element. Thus, $\overline{C}$ forms a graded $B$-module. We have

[TABLE]

for all $n\geq 1$. Then since $\ell_{R}(\overline{C}_{2})=3$ and $\overline{C}=\overline{C}_{2}\cdot B$ by the above exact sequence, we have $\mathrm{depth}_{B}\overline{C}=2$ and hence the minimal $B$-free resolution

[TABLE]

of $\overline{C}$ as graded $B$-module for some integer $m\geq 3$. Then we have

[TABLE]

for all $n\geq m-2$ so that $m=3$.

$(2)\Rightarrow(1)$, $(iii)$, and $(iv)$: We have $\ell_{R}(\overline{C}_{n})=2\binom{n+2}{2}-3\binom{n+1}{1}$ for $n\geq 1$ and $\mathrm{depth}_{T}\overline{C}=2$ by the exact sequence of our assertion $(2)$. Thus assertions $(1)$, $(iii)$, and $(iv)$ follow by Proposition 2.3(2).

$(ii)$: Since $\mathfrak{m}\overline{C}=(0)$, $\overline{C}=\overline{C}_{2}B$, $\overline{C}_{2}\cong\overline{I^{3}}/J\overline{I^{2}}$, and $\ell_{R}(C_{2})=3$, we have $\mathfrak{m}\overline{I^{3}}\subseteq J\overline{I^{2}}$ and $\overline{I^{n+1}}=J\overline{I^{n}}$ for all $n\geq 3$ by Lemma 2.2, and $\ell_{R}(\overline{I^{3}}/J\overline{I^{2}})=3$.

$(v)$ and $(vi)$: Since $\overline{\mathrm{e}}_{3}(I)=0,$ by [1, Theorem 4.1, Lemma 4.7] we have $\mathrm{H}^{3}_{N^{\prime}}(\overline{\mathcal{R}^{\prime}}(I))_{n}=0$ for all $n\geq 0.$ We have $\ell_{R}(R/\overline{I^{n+1}})=\sum_{i=0}^{3}(-1)^{i}\overline{\mathrm{e}}_{i}(I)\binom{n+3-i}{3-i}$ for all $n\geq 1$ by Proposition 2.3(2) because $\ell_{R}(\overline{C}_{n})=2\binom{n+2}{2}-3\binom{n+1}{1}$ for all $n\geq 1$ as above. Hence, by [1, Theorem 4.1], we have $[\mathrm{H}_{N^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))]_{n}=(0)$ for all $n\geq 2$ so that $\mathrm{H}_{N^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))=[\mathrm{H}_{N^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))]_{1}$ by Theorem 3.1(3). We also get $\ell_{R}([\mathrm{H}_{N^{\prime}}^{2}(\overline{\mathcal{R}^{\prime}}(I))]_{1})=\sum_{i=0}^{3}(-1)^{i}\overline{\mathrm{e}}_{i}(I)-\ell_{R}(R/\overline{I})=1$ by [1, Theorem 4.1] and our assumption.

Now consider the short exact sequence

[TABLE]

By using the induced long exact sequence of local cohomology modules

[TABLE]

we get $(v)$. The assertion $(vi)$ follows by Corollary 4.2. ∎

Remark 4.4**.**

We notice that Theorem 4.3 can be extended to $d\geq 3$ under the assumption $\mathrm{depth}~{}\overline{G}(I)\geq d-2.$ We omit the details because the techniques are standard.