# On the sectional genera and Cohen-Macaulay rings

Shinya Kumashiro  National Institute of Technology, Oyama College, 771
Nakakuki, Oyama, Tochigi, 323-0806, Japan<EMAIL_ADDRESS>,  Hoang
Le Truong  Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, 10307
Hanoi, Vietnam  Thang Long Institute of Mathematics and Applied Sciences,
Hanoi, Vietnam<EMAIL_ADDRESS>
<EMAIL_ADDRESS>and  Hoang Ngoc Yen  Institute of Mathematics, VAST,
18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam  The Department of Mathematics,
Thai Nguyen University of education. 20 Luong Ngoc Quyen Street, Thai Nguyen
City, Thai Nguyen Province, Vietnam<EMAIL_ADDRESS>
###### Abstract.

We explore the behavior of the sectional genera of certain primary ideals in
Noetherian local rings. In this paper, we provide characterizations of a
Cohen-Macaulay local ring in terms of the sectional genera, the Cohen-Macaulay
type, and the second Hilbert coefficients for certain primary ideals. We also
characterize Gorenstein rings and quasi-Buchsbaum rings in terms of the
sectional genera for certain primary ideals.

Key words and phrases: sectional genera, Hilbert coefficients, Cohen-Macaulay,
Gorenstein, quasi-Buchsbaum. 2020 Mathematics Subject Classification: 13D40,
13H10, 13H15.
The first author was supported by JSPS KAKENHI Grant Number JP21K13766. The
second author was supported by NCXS02.01/22-23. The last author was partially
supported by Grant number ICRTM02-2020.05, awarded in the internal grant
competition of International Center for Research and Postgraduate Training in
Mathematics, Hanoi.

##  1\. Introduction

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring of dimension
$d$, where $\mathfrak{m}$ is the maximal ideal. Let $I$ be an
$\mathfrak{m}$-primary ideal of $R$. It is well-known that there are integers
$\mathrm{e}_{i}(I,R)$, called the Hilbert coefficients of $M$ with respect to
$I$, such that

$\displaystyle\ell_{R}(R/{I^{n+1}})=\mathrm{e}_{0}(I,R)\binom{n+d}{d}-\mathrm{e}_{1}(I,R)\binom{n+d-1}{d-1}+\cdots+(-1)^{d}\mathrm{e}_{d}(I,R)$

for all $n\gg 0$. Here $\ell_{R}(N)$ denotes the length of an $R$-module $N$.
The leading coefficient $\mathrm{e}_{0}(I,R)$ is called the multiplicity of
$R$ with respect to $I$, and $\mathrm{e}_{1}(I,R)$ is named by W. V.
Vasconcelos ([Vas08]) as the Chern number of $R$ with respect to $I$.

In 1987, A. Ooishi ([Ooi87]) introduced the notion of sectional genera in
commutative rings. Let

${\rm sg}(I,R)=\ell_{R}(R/I)-\mathrm{e}_{0}(I,R)+\mathrm{e}_{1}(I,R)$

and call it the sectional genera for $R$ with respect to $I$. For the notion
of sectional genera in Cohen-Macaulay local rings, there is general
recognition that the sectional genera of $R$ with respect to $I$ controls the
depth of the associated graded ring of $I$, and determines the Hilbert-Samuel
function of $I$. Indeed, the results along this line of investigation can be
found in, for example, [Hun87], [Sal92], [Sal93], [Ito95], [CPR05]. On the
other hand, for non-Cohen-Macaulay local rings, not so much is known about the
sectional genera. One progress is that S. Goto and K. Ozeki ([GoO16]) gave a
criterion for a certain equality of the sectional genera of parameter ideals
for modules in 2016. On the other hand, even how the sectional genera
characterizes the ring itself has not yet been investigated.

The purpose of our paper is to study the sectional genera of
$\mathfrak{m}$-primary ideals, provided $R$ is unmixed and a homomorphic image
of a Cohen-Macaulay local ring. We especially focus our attention on
$C$-parameter ideals. We note here only that the notion of $C$-parameter
ideals is a special kind of parameter ideals of $R$, and $C$-parameter ideals
always exist in our assumption. See Section 2 for the precise definition.

Let us recall and fix the terminology and the notations to state our results.
The index of reducibility of $C$-parameter ideals $\mathfrak{q}$ of $R$ is
called the stable value of $R$ and denoted by $\mathcal{N}(R)$ (see [CuQ20]).
We denote by $\mathrm{r}(R)$ the Cohen-Macaulay type
$\dim_{R}((0):_{H_{\mathfrak{m}}^{d}(R)}\mathfrak{m})$ of $R$, where
$H_{\mathfrak{m}}^{d}(R)$ denotes the $d$th local cohomology of $R$, and
$\mathrm{g}(R)$ the g-invariant of $R$ (see Definition 3.4). Using these
notations, our results are stated as follows.

######  Theorem 1.1 (Theorem 3.3, 3.5, Corollary 3.6).

Suppose that $R$ is unmixed with $d=\dim R\geq 2$, that is,
$\dim\widehat{R}/\mathfrak{p}=d$ for all $p\in\operatorname{\mathrm{Ass}}R$
where $\widehat{R}$ denotes the completion of $R$. Then for all parameter
ideals $\mathfrak{q}\subset\mathfrak{m}^{\mathrm{g}(R)}$, we have

$\mathrm{r}(R)-\mathcal{N}(R)\leq{\rm sg}(\mathfrak{q}:\mathfrak{m},R)\leq{\rm
sg}(\mathfrak{q},R).$

Furthermore, for each inequality, it is to be equal if and only if $R$ is a
Cohen-Macaulay local ring.

######  Theorem 1.2 (Theorem 4.1, 4.2, Corollary 4.3).

Suppose that $R$ is unmixed with $d=\dim R\geq 2$. Then for all parameter
ideals $\mathfrak{q}\subset\mathfrak{m}^{\mathrm{g}(R)}$, we have

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\leq\mathrm{e}_{2}(\mathfrak{q},R)\leq{\rm
sg}(\mathfrak{q}:\mathfrak{m},R)+\mathcal{N}(R)-\mathrm{r}(R).$

Furthermore, for each inequality, it is to be equal if and only if $R$ is a
Cohen-Macaulay local ring.

Next we explore the condition under which generalized Northcott’s inequality
is equal when $I=\mathfrak{q}:\mathfrak{m}$. Recall that for a Cohen-Macaulay
local ring $R$ and an $\mathfrak{m}$-primary ideal $I$ of $R$, Northcott’s
inequality holds (see [Nor60]):

${\rm sg}(I,R)\geq 0.$

Then, S. Goto and K. Nishida ([GoN03]) generalizes Northcott’s inequality
without assuming that $R$ is a Cohen–Macaulay ring. In particular, they show

${\rm sg}(I,R)\geq\mathrm{e}_{1}(\mathfrak{q},R),$

where $\mathfrak{q}$ is a minimal reduction of $I$. In the present study, we
characterize rings such that the above inequality is to be an equal. Actually,
the next main result is stated as follows.

######  Theorem 1.3.

Suppose that $R$ is a non-regular unmixed local ring with $d=\dim R\geq 2$.
Then the following statements are equivalent.

  1. $i)$

$R$ is a quasi-Buchsbaum ring.

  2. $ii)$

There exists a $C$-parameter ideal $\mathfrak{q}$ of $R$ such that

${\rm sg}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{e}_{1}(\mathfrak{q},R).$

For the definition of quasi-Buchbaum modules, see after Corollary 4.5.

The remainder of this paper is organized as follows. In Section 2, we prove
some preliminary results on the index of reducibility for parameter ideals,
$C$-parameter ideals, and generalized Cohen-Macaulay rings. Theorem 1.1 is
proven in Section 3. Theorem 1.2 and 1.3 are proven in Section 4. The
assumption throughout this paper is written in the beginning of Section 2.

##  2\. Preliminary

In what follows, throughout this paper, let $(R,\mathfrak{m},k)$ be a
Noetherian local ring of dimension $d$, where $\mathfrak{m}$ is the maximal
ideal and $k=R/\mathfrak{m}$ is the residue field of $R$. Suppose that $R$ is
a homomorphic image of a Cohen-Macaulay local ring. Let $M$ be a finitely
generated $R$-module of dimension $s$. Let $I$ be an $\mathfrak{m}$-primary
ideal of $M$, that is, $M/IM$ is of finite length. It is well-known that there
are integers $\\{\mathrm{e}_{i}(I,M)\\}_{i=0}^{s}$, called the Hilbert
coefficients of $M$ with respect to $I$, such that

$\displaystyle\ell_{R}(M/{I^{n+1}}M)={\mathrm{e}}_{0}(I,M)\binom{n+s}{s}-{\mathrm{e}}_{1}(I,M)\binom{n+s-1}{s-1}+\cdots+(-1)^{s}{\mathrm{e}}_{s}(I,M)$

for $n\gg 0$. Here $\ell_{R}(N)$ denotes the length of an $R$-module $N$. In
particular, the first two coefficients $\mathrm{e}_{0}(I,M)$ and
$\mathrm{e}_{1}(I,M)$ are called the multiplicity of $M$ with respect to $I$
and the Chern number of $M$ with respect to $I$, respectively. We set

${\rm sg}(I,M)=\ell_{R}(M/IM)-\mathrm{e}_{0}(I,M)+\mathrm{e}_{1}(I,M)$

and call it the sectional genera for $M$ with respect to $I$ ([Ooi87,
Definition 1.3]).

######  Lemma 2.1.

Let $I$ be an $\mathfrak{m}$-primary ideal of $M$ and $x\in I$ a superficial
element. Then, we have

  1. $i)$

$\mathrm{e}_{j}(I,M)=\begin{cases}\mathrm{e}_{j}(I,M)&\text{if $0\leq j\leq
s-2$,}\\\ \mathrm{e}_{s-1}(I,M/xM)+(-1)^{s}\ell_{R}(0:_{M}x)&\text{if
}j=s-1\end{cases}$

(see Proposition 22.6 in [Nag62]).

  2. $ii)$

${\rm sg}(I,M)=\begin{cases}{\rm sg}(I,M/xM)&\text{if $s\geq 3$,}\\\ {\rm
sg}(I,M/xM)+\ell_{R}(0:_{M}x)&\text{if }s=2\end{cases}$

(see Lemma 3.1 in [GoO16]).

Let us recall the definitions of $C$-parameter ideals and the stable value of
$M$, which are introduced by N. T. Cuong and P. H. Quy in [CuQ20]. Set
$\mathfrak{a}_{i}=\operatorname{\mathrm{Ann}}_{R}\,H^{j}_{\mathfrak{m}}(M)$
for $j\in\mathbb{Z}$, and
$\mathfrak{a}(M)=\mathfrak{a}_{0}(M)\cdots\mathfrak{a}_{s-1}(M)$. We denote

$\mathfrak{b}(M)=\bigcap\limits_{\underline{x},i=1}^{s}\operatorname{\mathrm{Ann}}_{R}((0):_{M/(x_{1},\ldots,x_{i-1})M}x_{i}),$

where $\underline{x}=x_{1},\ldots,x_{s}$ runs over all systems of parameters
of $M$. With the above notation, a system $x_{1},\ldots,x_{s}$ of parameters
of $M$ is called a $C$-system of parameters of $M$ if
$x_{s}\in\mathfrak{b}(M)^{3}$ and
$x_{i}\in\mathfrak{b}(M/(x_{i+1},\ldots,x_{s})M)^{3}$ for all
$i=s-1,\ldots,1$. A parameter ideal $\mathfrak{q}$ of $M$ is called
$C$-parameter ideal if it is generated by a $C$-system of parameters of $M$
(see [MoQ17]). Note that $C$-systems of parameters of $M$ always exist,
provided $R$ is a homomorphic image of a Cohen-Macaulay local ring (see
[CuC18]) and a $C$-system of parameters of $M$ forms a $d$-sequence of $M$ in
the sense of [Hun82].

On the other hand, a proper submodule $N$ of $M$ is called irreducible if it
can not be written as an intersection of two strictly larger submodules of
$M$. For a submodule $N$ of $M$, the number of irreducible components of an
irredundant irreducible decomposition of $N$, which is independent of the
choice of the decomposition, is called the index of reducibility of $N$ and
denoted by $\operatorname{\mathrm{ir}}_{M}(N)$ ([Noe21]). For a parameter
ideal $\mathfrak{q}$ of $M$, we have

$\operatorname{\mathrm{ir}}_{M}(\mathfrak{q}):=\operatorname{\mathrm{ir}}_{M}(\mathfrak{q}M)=\ell_{R}([\mathfrak{q}M:_{M}\mathfrak{m}]/\mathfrak{q}M).$

In [CuQ20], N. T. Cuong and P. H. Quy show that the index
$\operatorname{\mathrm{ir}}_{M}(\mathfrak{q})$ of reducibility of
$C$-parameter ideals $\mathfrak{q}$ of $M$ are independent of the choice of
$\mathfrak{q}$. This invariant is called the stable value of $M$ and is
denoted by $\mathcal{N}_{R}(M)$.

We further set

$\displaystyle\mathrm{I}(\mathfrak{q},M)=$
$\displaystyle\ell(M/\mathfrak{q}M)-\mathrm{e}_{0}(\mathfrak{q},M)\quad\text{and}$
$\displaystyle\mathrm{I}(M)=$ $\displaystyle{\rm
sup}\,\mathrm{I}(\mathfrak{q},M),$

where $\mathfrak{q}$ runs over all systems of parameters of $M$. An $R$-module
$M$ is said to be a generalized Cohen–Macaulay module if
$\mathrm{I}(M)<\infty$ (see [CST78]). A parameter ideal $\mathfrak{q}$
providing $\mathrm{I}(\mathfrak{q},M)=\mathrm{I}(M)$ is called a standard
parameter ideal of $M$ (see [Tru86]). Once $M$ is to be a generalized Cohen-
Macaulay module, then every $C$-parameter ideal is standard.

We summarize a way to compute the invariants introduced above. Let

$h_{i}(M)=\ell_{R}(H^{i}_{\mathfrak{m}}(M))$ and
$r_{i}(M)=\ell((0):_{H^{i}_{\mathfrak{m}}(M)}\mathfrak{m})$

for $i=1,\ldots,s$.

######  Lemma 2.2.

Suppose that $R$ is generalized Cohen-Macaulay with $d=\dim R>0$ and
$\mathfrak{q}$ is a $C$-parameter ideal of $R$. Then we have the following.

  1. $i)$

$\mathrm{I}(\mathfrak{q},R)=\mathrm{I}(R)=\displaystyle\sum_{i=0}^{d-1}\binom{d-1}{i}h_{i}(R)$
(see [CST78, (3.7)]).

  2. $ii)$

${\rm
ir}_{R}(\mathfrak{q})=\mathcal{N}(R)=\displaystyle\sum_{i=0}^{d}\binom{d}{i}r_{i}(R)$
(see [CuT08, Theorem 1.1]).

  3. $iii)$

$\mathrm{e}_{i}(\mathfrak{q}:\mathfrak{m},R)=\begin{cases}(-1)^{i}\left(\displaystyle\sum_{j=1}^{d-i}\binom{d-i-1}{j-1}h_{i}(R)-\sum_{j=1}^{d-i+1}r_{j}(R)\right)&\text{
if }i=1,\ldots,d-1,\\\ (-1)^{d}(h_{0}(R)-r_{1}(R))&\text{ if }i=d,\end{cases}$

provided $R$ is not regular (see [CQT19, Theorem 5.2]).

  4. $iv)$

$(-1)^{i}\mathrm{e}_{i}(\mathfrak{q},R)=\begin{cases}h_{0}(R)&\text{ if
}i=d,\\\ \displaystyle\sum_{j=1}^{d-i}\binom{d-i-1}{j-1}h_{i}(R)&\text{ if
}0<i<d,\end{cases}$

(see [Sch82, Korollary 3.2]).

The purpose of this paper is to provide characterizations of properties of
rings (e.g., Cohen-Macaulay, Gorenstein, and quasi-Buchsbaum) via the
invariants introduced in this section. Hence, we note the invariants for the
case of Cohen-Macaulay rings, although it is well-known and immediately
follows from Lemma 2.2.

######  Remark 2.3.

Suppose that $R$ is a non-regular Cohen-Macaulay ring of dimension $d\geq 2$
and $\mathfrak{q}$ is a $C$-parameter ideal of $R$. Then we have the
following.

  1. $i)$

$\mathrm{I}(\mathfrak{q},R)=0$.

  2. $ii)$

${\rm ir}_{R}(\mathfrak{q})=\mathcal{N}(R)=r_{d}(R)$.

  3. $iii)$

$\mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)=r_{d}(R)$ and
$\mathrm{e}_{i}(\mathfrak{q}:\mathfrak{m},R)=0$ for all $2\leq i\leq d$.

  4. $iv)$

$\mathrm{e}_{i}(\mathfrak{q},R)=0$ for all $1\leq i\leq d$.

We here note our strategy in Section 3: by passing to a part of a $C$-system
of parameters of $R$, we can reduce our assertions to the $2$-dimensional
case. We then obtain relations of invariants introduced in this section from
Lemma 2.2. Hence, we also summarize the $2$-dimensional case of Lemma 2.2 for
the reader’s convenience.

######  Remark 2.4.

Suppose that $R$ is a non-regular generalized Cohen-Macaulay ring with $d=2$
and $\mathfrak{q}$ is a $C$-parameter ideal of $R$. Then we have the
following.

  1. $i)$

$\mathrm{I}(\mathfrak{q},R)=\mathrm{I}(R)=h_{0}(R)+h_{1}(R)$.

  2. $ii)$

${\rm ir}_{R}(\mathfrak{q})=\mathcal{N}(R)=r_{0}(R)+2r_{1}(R)+r_{2}(R)$.

  3. $iii)$

$\mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)=r_{1}(R)+r_{2}(R)-h_{1}(R)$ and
$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)=h_{0}(R)-r_{1}(R)$.

  4. $iv)$

$\mathrm{e}_{1}(\mathfrak{q},R)=-h_{1}(R)$ and
$\mathrm{e}_{2}(\mathfrak{q},R)=-h_{0}(R)$.

The following example shows a computation of the Hilbert-Samuel functions of
$\mathfrak{q}$ and $\mathfrak{q}:_{R}\mathfrak{m}$.

######  Example 2.5.

Let $R=k[[X,XY^{2},X^{2}Y^{3},X^{3}Y]]$ be a subring of the formal power
series ring $k[[X,Y]]$ over a field $k$. Let $\mathfrak{m}$ be the maximal
ideal of $R$. For an integer $a\geq 6$, set

$\mathfrak{q}_{a}=(X^{a},X^{a}Y^{2a}).$

Then we have the following.

  1. $i)$

$\mathfrak{q}_{a}$ is a $C$-parameter ideal.

  2. $ii)$

$\ell_{R}(R/\mathfrak{q}_{a}^{n+1})=2a^{2}\binom{n+2}{2}+2\binom{n+1}{1}$ for
all $n\geq 0$. Hence,

$\mathrm{e}_{0}(\mathfrak{q}_{a},R)=2a^{2}$,
$\mathrm{e}_{1}(\mathfrak{q}_{a},R)=-2$, and
$\mathrm{e}_{2}(\mathfrak{q}_{a},R)=0$.

  3. $iii)$

$\ell_{R}(R/(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{n+1})=2a^{2}\binom{n+2}{2}-\binom{n+1}{1}-1$
for all $n\geq 0$. Hence,

$\mathrm{e}_{0}(\mathfrak{q}_{a}:_{R}\mathfrak{m},R)=2a^{2}$,
$\mathrm{e}_{1}(\mathfrak{q}_{a}:_{R}\mathfrak{m},R)=1$, and
$\mathrm{e}_{2}(\mathfrak{q}_{a}:_{R}\mathfrak{m},R)=-1$.

###### Proof.

Note that we have $\overline{R}=k[[X,XY,XY^{2}]]$, where $\overline{R}$
denotes the integral closure of $R$. Indeed, it is standard to check that
$k[[X,XY,XY^{2}]]\subseteq\overline{R}$. It follows that
$\overline{R}=k[[X,XY,XY^{2}]]$ since $k[[X,XY,XY^{2}]]$ is a normal domain by
[BrH98, Theorem 6.1.4]. Hence, by applying the local cohomology functor
$H_{\mathfrak{m}}(-)$ to the exact sequence $0\to
R\to\overline{R}\to\overline{R}/R\to 0$, we obtain that

(1) $\displaystyle H_{\mathfrak{m}}^{0}(R)=0,H_{\mathfrak{m}}^{1}(R)\cong
H_{\mathfrak{m}}^{0}(\overline{R}/R)\cong\overline{R}/R,\text{ and
}H_{\mathfrak{m}}^{2}(R)\cong H_{\mathfrak{m}}^{2}(\overline{R}).$

$i)$. By (1), we obtain that
$\mathfrak{a}(R)=\mathfrak{a}_{0}(R)\mathfrak{a}_{1}(R)=\operatorname{\mathrm{Ann}}_{R}(\overline{R}/R)\supseteq\mathfrak{m}^{2}$.
By noting that $\mathfrak{b}(R)\supseteq\mathfrak{a}(R)$ (see [Sch82, Satz
2.4.5]),
$\mathfrak{q}_{a}\subseteq\mathfrak{m}^{6}\subseteq\mathfrak{b}(R)^{3}$ since
$a\geq 6$. It follows that $\mathfrak{q}_{a}$ is a $C$-parameter ideal.

$ii)$. Note that the set of all monomials contained in $R$ is described as
follows.

$1$$2$$3$$\cdots$$1$$2$$Y=2X$$X$$Y$O

With the above description, by noting that
$\mathfrak{q}_{a}^{n+1}=(X^{a(n+1)}Y^{2ai}\mid 0\leq i\leq n+1)$, it is
straightforward to check that
$\ell_{R}(R/\mathfrak{q}_{a}^{n+1})=2a^{2}\binom{n+2}{2}+2\binom{n+1}{1}$ for
all $n\geq 0$.

$iii)$. First, we prove the following claim.

######  Claim 2.6.

$\mathfrak{q}_{a}^{n+1}:_{R}\mathfrak{m}=\mathfrak{q}_{a}^{n}(\mathfrak{q}_{a}:_{R}\mathfrak{m})=(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{n+1}$
for all $n\geq 0$.

###### Proof of Claim 2.6.

It is easy to check that the monomial basis of a $k$-vector space
$(\mathfrak{q}_{a}:_{R}\mathfrak{m})/\mathfrak{q}_{a}$ is $X^{a+2}Y^{2a+1}$,
$X^{2a-1}Y^{2a-1}$, $X^{2a-2}Y^{2a-2}$, and $X^{a+2}Y$. Then, by using the
description in $ii)$, we observe that
$\mathfrak{q}_{a}^{n+1}:_{R}\mathfrak{m}=\mathfrak{q}_{a}^{n}(\mathfrak{q}_{a}:_{R}\mathfrak{m})$
for all $n\geq 0$. Since
$\mathfrak{q}_{a}^{n}(\mathfrak{q}_{a}:_{R}\mathfrak{m})\subseteq(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{n+1}$
is clear, the rest to prove is
$\mathfrak{q}_{a}^{n}(\mathfrak{q}_{a}:_{R}\mathfrak{m})\supseteq(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{n+1}$
for all $n\geq 0$. It is enough to prove that
$\mathfrak{q}_{a}(\mathfrak{q}_{a}:_{R}\mathfrak{m})\supseteq(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{2}$.

Set $I=(X^{a+2}Y^{2a+1},X^{2a-1}Y^{2a-1},X^{2a-2}Y^{2a-2},X^{a+2}Y)$, the
ideal generated by the monomial basis of
$(\mathfrak{q}_{a}:_{R}\mathfrak{m})/\mathfrak{q}_{a}$. Since
$\mathfrak{q}_{a}:_{R}\mathfrak{m}=\mathfrak{q}_{a}+I$, we obtain that

$(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{2}=\mathfrak{q}_{a}(\mathfrak{q}_{a}:_{R}\mathfrak{m})+I^{2}.$

On the other hand, the direct calculation shows that
$I^{2}\subseteq\mathfrak{q}_{a}^{2}$ since $a\geq 6$. It follows that
$\mathfrak{q}_{a}(\mathfrak{q}_{a}:_{R}\mathfrak{m})\supseteq(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{2}$.
∎

By Claim 2.6, we have

$\displaystyle\ell_{R}(R/(\mathfrak{q}_{a}:_{R}\mathfrak{m})^{n+1})=$
$\displaystyle\ell_{R}(R/\mathfrak{q}_{a}^{n+1})-\ell_{R}((\mathfrak{q}^{n+1}:_{R}\mathfrak{m})/\mathfrak{q}_{a}^{n+1})$
$\displaystyle=$ $\displaystyle
2a^{2}\binom{n+2}{2}+2\binom{n+1}{1}-[3(n+1)+1]$ $\displaystyle=$
$\displaystyle 2a^{2}\binom{n+2}{2}-\binom{n+1}{1}-1$

for all $n\geq 0$. ∎

######  Remark 2.7.

In Example 2.5, by (1), we obtain that $h_{0}(R)=r_{0}(R)=0$, $h_{1}(R)=2$,
$r_{1}(R)=1$, and $r_{2}(R)=2$. (Note that $\overline{R}$ is a Gorenstein ring
by [BrH98, Theorem 6.3.5] and is generated by $2$ elements as an $R$-module).
Therefore, by Remark 2.4, we have

$\displaystyle i)\ \mathrm{I}(\mathfrak{q},R)=2,\quad ii)\
\mathcal{N}(R)=4,\quad iii)\ \mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)=1,\
\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)=-1,$ $\displaystyle iv)\
\mathrm{e}_{1}(\mathfrak{q},R)=-2,\ \text{and
}\mathrm{e}_{2}(\mathfrak{q},R)=0$

for all $C$-parameter ideals $\mathfrak{q}$ of $R$.

We further note useful properties of a $C$-system of parameters of $M$.

######  Lemma 2.8.

Let $x_{1},\ldots,x_{s}$ be a $C$-system of parameters of $M$. Then the
following hold true.

  1. $i)$

For each $i=1,\ldots,s$, we have that
$x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{s}$ is a $C$-system of parameters of
$M/x_{i}M$ and

$\mathcal{N}_{R}(M)=\mathcal{N}_{R}(M/x_{i}M)$

(see [CuQ20, Lemma 2.13]).

  2. $ii)$

For all $i=1,\ldots,s$, we have

$\operatorname{\mathrm{Ass}}M/(x_{1},\ldots,x_{i})M\subseteq\operatorname{\mathrm{Assh}}M/(x_{1},\ldots,x_{i})M\cup\\{\mathfrak{m}\\},$

where
$\operatorname{\mathrm{Assh}}M=\\{\mathfrak{p}\in\operatorname{\mathrm{Ass}}M\mid\dim
R/\mathfrak{p}=\dim M\\}$ (see [MoQ17, Remark 3.3]).

Recall that the largest submodule of $M$ of dimension less than $s$ is called
the unmixed component of $M$, and denoted by $U_{M}(0)$. The following result
plays a key role in the arguments used in this paper.

######  Lemma 2.9.

Suppose that $M$ is unmixed with $s=\dim M\geq 2$ and $x_{1},\dots,x_{s}$ is a
$C$-system of parameters of $M$. Then

  1. $i)$

$H_{\mathfrak{m}}^{1}(M/(x_{1},x_{2},\dots,x_{i})M)$ is a finitely generated
$R$-module for all $1\leq i\leq s-2$.

  2. $ii)$

Let $N/(x_{1},\ldots,x_{s-2})M$ denote the unmixed component of
$M/(x_{1},\ldots,x_{s-2})M$. If $M/N$ is a Cohen-Macaulay $R$-module (of
dimension $2$), then $M$ is a Cohen-Macaulay $R$-module (of dimension $s$).

###### Proof.

$i)$ is Lemma 3.1 in [GoN01] or Remark 3.3 in [MoQ17].
$ii).$ For $0\leq i\leq s-2$, let $\mathfrak{q}_{i}=(x_{1},x_{2},\dots,x_{i})$
and $M_{i}=M/\mathfrak{q}_{i}M$. (If $i=0$, then $\mathfrak{q}_{0}=(0)$ and
$M_{0}=M$.) Consider the exact sequence

$0\to N/\mathfrak{q}_{s-2}M\to M_{s-2}\to M/N\to 0.$

Note that $\dim N/\mathfrak{q}_{s-2}M=0$ since $\dim
N/\mathfrak{q}_{s-2}M<\dim M_{s-2}$ and
$\operatorname{\mathrm{Ass}}M_{s-2}\subseteq\operatorname{\mathrm{Assh}}M_{s-2}\cup\\{\mathfrak{m}\\}$
by Lemma 2.8 $ii)$. $M/N$ is a Cohen-Macaulay $R$-module of dimension $2$ by
hypothesis. Hence, by applying the local cohomology functor
$H_{\mathfrak{m}}^{-}(-)$ to the above exact sequence, we obtain that

(2) $\displaystyle H_{\mathfrak{m}}^{1}(M_{s-2})=0.$

######  Claim 2.10.

Suppose that $s\geq 3$. Let $1\leq i\leq s-2$ and $1\leq\ell$ be integers.
Then, $H_{\mathfrak{m}}^{j}(M_{i})=0$ for all $1\leq j\leq\ell$ implies that
$H_{\mathfrak{m}}^{j}(M_{i-1})=0$ for all $1\leq j\leq\ell+1$.

###### Proof of Claim 2.10.

By the multiplication $M_{i-1}\xrightarrow{x_{i}}M_{i-1}$, we have

(3) $\displaystyle 0\to(0):_{M_{i-1}}x_{i}\to M_{i-1}\to
M_{i-1}/[(0):_{M_{i-1}}x_{i}]\to 0\quad\text{and}$ (4) $\displaystyle 0\to
M_{i-1}/[(0):_{M_{i-1}}x_{i}]\to M_{i-1}\to M_{i}\to 0.$

By noting that
$\operatorname{\mathrm{Ass}}M_{i-1}\subseteq\operatorname{\mathrm{Assh}}M_{i-1}\cup\\{\mathfrak{m}\\}$
and $x_{i}$ is a part of a system of parameter of $M_{i-1}$, we have
$\operatorname{\mathrm{Ass}}((0):_{M_{i-1}}x_{i})\subseteq\\{\mathfrak{m}\\}$,
that is, $\dim((0):_{M_{i-1}}x_{i})=0$. Hence, by applying
$H_{\mathfrak{m}}^{-}(-)$ to (3), we obtain that

$H_{\mathfrak{m}}^{j}(M_{i-1}/[(0):_{M_{i-1}}x_{i}])\cong
H_{\mathfrak{m}}^{j}(M_{i-1})$

for all $j>0$. Therefore, by applying $H_{\mathfrak{m}}^{-}(-)$ to (4), it
follows that

$H_{\mathfrak{m}}^{j-1}(M_{i})\to
H_{\mathfrak{m}}^{j}(M_{i-1})\xrightarrow{x_{i}}H_{\mathfrak{m}}^{j}(M_{i-1})\to
H_{\mathfrak{m}}^{j}(M_{i})$

for all $j>0$.

If $2\leq j\leq\ell+1$, then $H_{\mathfrak{m}}^{j-1}(M_{i})=0$, that is, $0\to
H_{\mathfrak{m}}^{j}(M_{i-1})\xrightarrow{x_{i}}H_{\mathfrak{m}}^{j}(M_{i-1})$.
On the other hand, for each element of $H_{\mathfrak{m}}^{j}(M_{i-1})$, large
enough powers of $x_{i}$ annihilate the element. Hence, the injection implies
that $H_{\mathfrak{m}}^{j}(M_{i-1})=0$ for all $2\leq j\leq\ell+1$.

Let $j=1$. Then
$H_{\mathfrak{m}}^{1}(M_{i-1})\xrightarrow{x_{i}}H_{\mathfrak{m}}^{1}(M_{i-1})\to
0$. Since $H_{\mathfrak{m}}^{1}(M_{i-1})$ is finitely generated by i), by
Nakayama’s lemma, we obtain that $H_{\mathfrak{m}}^{1}(M_{i-1})=0$. ∎

Let us complete the proof of Lemma 2.9. If $s=2$, then $M$ is a Cohen-Macaulay
$R$-module by (2) and the assumption that $M$ is reduced. Suppose that $s\geq
3$. By using Claim 2.10 recursively, we derive from (2) that
$H_{\mathfrak{m}}^{j}(M)=0$ for all $1\leq j\leq s-1$. On the other hand, by
noting that $H_{\mathfrak{m}}^{0}(M)=0$ since $M$ is reduced, $M$ is a Cohen-
Macaulay $R$-module. ∎

######  Remark 2.11.

In [Tru19, TaT20] the definition of Goto sequences on $M$ was introduced by
the second author. Moreover, if $x_{1},\ldots,x_{s}$ is a Goto sequence, then
the statement Lemma 2.9 $ii)$ is true. Note that, if $M$ is unmixed, a
$C$-system of parameters of $M$ forms a Goto sequence on $M$. Thus applying
Lemma 2.5 in [Tru19] we also have Lemma 2.9 $ii)$.

##  3\.  The bound of the sectional genera

In this section, we study the bound of ${\rm sg}(\mathfrak{q},R)$ and ${\rm
sg}(\mathfrak{q}:\mathfrak{m},R)$, where $\mathfrak{q}$ runs over all
$C$-parameter ideals of $R$. We maintain the assumption of the beginning of
Section 2. The goal of this section is to provide a characterization of a
Cohen–Macaulay ring in terms of its sectional genera. We begin with the
following.

######  Lemma 3.1.

Assume that $\mathrm{e}_{0}(\mathfrak{m},R)>1$ and $\mathfrak{q}$ is a
$C$-parameter ideal of $R$. Then we have

$\displaystyle{\rm
sg}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{I}(\mathfrak{q},R)+\mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)-\mathcal{N}(R).$

###### Proof.

Put $J=\mathfrak{q}:\mathfrak{m}$. Since $\mathrm{e}_{0}(\mathfrak{m},R)>1$,
it follows from by Proposition 2.3 in [GoN03] and Proposition 11.2.1 in
[SwH06] that $\mathrm{e}_{0}(J,R)=\mathrm{e}_{0}(\mathfrak{q},R)$. Since
$\mathfrak{q}$ is a $C$-parameter ideal, by the definition of the stable
value, we have

$\displaystyle{\rm sg}(J,R)$ $\displaystyle=$
$\displaystyle\ell(R/J)-\mathrm{e}_{0}(J,R)+\mathrm{e}_{1}(J,R)$
$\displaystyle=$
$\displaystyle\ell(R/\mathfrak{q})-\mathrm{e}_{0}(\mathfrak{q},R)+\mathrm{e}_{1}(J,R)-\ell(\mathfrak{q}:\mathfrak{m}/\mathfrak{q})$
$\displaystyle=$
$\displaystyle\mathrm{I}(\mathfrak{q},R)+\mathrm{e}_{1}(J,R)-\mathcal{N}(R),$

as required. ∎

######  Theorem 3.2.

Suppose that $\dim R=1$ and $\mathrm{e}_{0}(\mathfrak{m},R)>1$. Then the
following statements are equivalent.

  1. $i)$

$R$ is Cohen–Macaulay.

  2. $ii)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

${\rm sg}(\mathfrak{q}:\mathfrak{m},R)\geq 0.$

###### Proof.

$i)\Rightarrow ii)$. This immediately follows from the assumption that $R$ is
Cohen-Macaulay by applying Lemma 3.1.
$ii)\Rightarrow i)$. Since $\dim R=1$, $R$ is generalized Cohen-Macaulay. It
follows from Lemma 2.2 and Lemma 3.1 that we have

$-r_{0}(R)={\rm sg}(\mathfrak{q}:\mathfrak{m},R)\geq 0.$

Thus $r_{0}(R)=0$. Hence $R$ is Cohen-Macaulay, as required. ∎

Note that in the case where $\dim R=1$, we have ${\rm sg}(\mathfrak{q},R)=0$
for every $C$-parameter ideal $\mathfrak{q}$ of $R$ by Lemma 3.1 in [GoS03].
Therefore, we have ${\rm sg}(\mathfrak{q}:\mathfrak{m},R)\leq{\rm
sg}(\mathfrak{q},R)$ for every $C$-parameter ideal $\mathfrak{q}$ of $R$. In
the case where $\dim R\geq 2$, the following result shows that the inequality
${\rm sg}(\mathfrak{q}:\mathfrak{m},R)\leq{\rm sg}(\mathfrak{q},R)$ still
holds true for every $C$-parameter ideal of $R$, provided $R$ is unmixed.

######  Theorem 3.3.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then the following statements are equivalent.

  1. $i)$

$R$ is Cohen–Macaulay.

  2. $ii)$

There exists a $C$-parameter ideal $\mathfrak{q}$ such that

${\rm sg}(\mathfrak{q}:\mathfrak{m},R)\geq{\rm sg}(\mathfrak{q},R).$

###### Proof.

Let $\mathfrak{q}$ be a $C$-parameter ideal of $R$ and put
$J=\mathfrak{q}:\mathfrak{m}$. Since $R$ is to be regular if $R$ is unmixed
and $\mathrm{e}_{0}(\mathfrak{m},R)=1$ (see [Nag62, Theorem 40.6]), we have
$\mathrm{e}_{0}(\mathfrak{m},R)>1$.

$i)\Rightarrow ii)$. This immediately follows from the assumption that $R$ is
Cohen-Macaulay by applying Lemma 2.2 and Lemma 3.1.

$ii)\Rightarrow i)$. Let $\underline{x}=x_{1},x_{2},\ldots,x_{d}$ be a
$C$-system of parameters of $R$ such that
$\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$. Put $R_{i}=R/(x_{1},\ldots,x_{i})$
for $i=1,\ldots,d-2$. By Lemma 2.8, for all $i=1,\dots,d-2$, we have

  1. (a)

$JR_{i}=\mathfrak{q}R_{i}:\mathfrak{m}R_{i}$,

  2. (b)

$x_{i}$ is a superficial element of $R_{i-1}$ with respect to
$\mathfrak{q}R_{i-1}$ and $JR_{i-1}$,

  3. (c)

$\mathcal{N}_{R}(R_{i})=\mathcal{N}_{R}(R_{i-1}).$

Now, we put $A=R_{d-2}$. By Lemma 2.1 we have
$\mathrm{e}_{i}(\mathfrak{q},R)=\mathrm{e}_{i}(\mathfrak{q}A,A)$ and
$\mathrm{e}_{i}(J,R)=\mathrm{e}_{i}(JA,A)=\mathrm{e}_{i}(\mathfrak{q}A:\mathfrak{m}A,A)$,
for all $i=0,1$. Therefore we have

(5) $\displaystyle{\rm
sg}(\mathfrak{q},R)=\ell(R/\mathfrak{q})-\mathrm{e}_{0}(\mathfrak{q},R)+\mathrm{e}_{1}(\mathfrak{q},R)=\ell(A/\mathfrak{q}A)-\mathrm{e}_{0}(\mathfrak{q}A,A)+\mathrm{e}_{1}(\mathfrak{q}A,A)={\rm
sg}(\mathfrak{q}A,A)$

and

(6) $\displaystyle\begin{split}{\rm
sg}(J,R)&=\ell(R/J)-\mathrm{e}_{0}(J,R)+\mathrm{e}_{1}(J,R)\\\
&=\ell(R/\mathfrak{q})-\mathrm{e}_{0}(\mathfrak{q},R)+\mathrm{e}_{1}(J,R)-\ell(\mathfrak{q}:\mathfrak{m}/\mathfrak{q})\\\
&=\ell(R/\mathfrak{q})-\mathrm{e}_{0}(\mathfrak{q},R)+\mathrm{e}_{1}(J,R)-\mathcal{N}(R)\\\
&=\ell(A/\mathfrak{q}A)-\mathrm{e}_{0}(\mathfrak{q}A,A)+\mathrm{e}_{1}(JA,A)-\mathcal{N}(A).\end{split}$

On the other hand, it follows from Lemma 2.8 $ii)$ that
$A/H_{\mathfrak{m}}^{0}(A)$ is unmixed with $\dim A=2$. Thus by Lemma 2.9
$i)$, $A/H_{\mathfrak{m}}^{0}(A)$ is generalized Cohen–Macaulay and so $A$ is.
Moreover, it follows from Lemma 2.1 that
$\mathrm{e}_{0}(\mathfrak{m}A,A)=\mathrm{e}_{0}(\mathfrak{m},R)>1$. By Lemma
2.2, (5), and (6), we obtain that

(7) $\displaystyle\begin{split}{\rm
sg}(J,R)&=\mathrm{I}(\mathfrak{q}A,A)+\mathrm{e}_{1}(JA,A)-\mathcal{N}(A)=h_{0}(A)-r_{0}(A)-r_{1}(A)\quad\text{and}\\\
{\rm
sg}(\mathfrak{q},R)&=\mathrm{I}(\mathfrak{q}A,A)+\mathrm{e}_{1}(\mathfrak{q}A,A)=h_{0}(A).\end{split}$

Since ${\rm sg}(J,R)\geq{\rm sg}(\mathfrak{q},R),$ we have
$r_{0}(A)=r_{1}(A)=0$. Therefore, $A$ is Cohen–Macaulay. Hence $R$ is Cohen-
Macaulay by applying Lemma 2.9. ∎

For each integer $n\geq 1$, we denote by ${\underline{x}}^{n}$ the sequence
$x^{n}_{1},x^{n}_{2},\ldots,x^{n}_{d}$. Let $K^{\bullet}(x^{n})$ be the Koszul
complex of $R$ generated by the sequence ${\underline{x}}^{n}$ and let
$H^{\bullet}({\underline{x}}^{n};R)=H^{\bullet}(\operatorname{\mathrm{Hom}}_{R}(K^{\bullet}({\underline{x}}^{n}),R))$
be the Koszul cohomology module of $R$. Then for every $p\in\mathbb{Z}$, the
family $\\{H^{p}({\underline{x}}^{n};R)\\}_{n\geq 1}$ naturally forms an
inductive system of $R$, whose limit

$H^{p}_{\mathfrak{q}}(R)=\lim\limits_{n\to\infty}H^{p}({\underline{x}}^{n};R)$

is isomorphic to the local cohomology module

$H^{p}_{\mathfrak{m}}(R)=\lim\limits_{n\to\infty}{\rm
Ext}_{R}^{p}(R/\mathfrak{m}^{n},R).$

For each $n\geq 1$ and $p\in\mathbb{Z}$, let
$\phi^{p,n}_{{\underline{x}},R}:H^{p}({\underline{x}}^{n};R)\to
H^{p}_{\mathfrak{m}}(R)$ denote the canonical homomorphism into the limit.

######  Definition 3.4 ([GoS03, Lemma 3.12]).

There exists an integer $n_{0}$ such that for all systems of parameters
${\underline{x}}=x_{1},\ldots,x_{d}$ for $R$ contained in
$\mathfrak{m}^{n_{0}}$ and for all $p\in\mathbb{Z}$, the canonical
homomorphisms

$\phi^{p,1}_{{\underline{x}},R}:H^{p}({\underline{x}},R)\to
H^{p}_{\mathfrak{m}}(R)$

into the inductive limit are surjective on the socles. The least integer
$n_{0}$ with this property is called a g-invariant of $R$ and denote by
$\mathrm{g}(R)$.

The following result shows that ${\rm sg}(\mathfrak{q}:\mathfrak{m},R)$ and
${\rm sg}(\mathfrak{q},R)$ are bounded below by the same finite value, where
$\mathfrak{q}$ runs over all $C$-parameters ideals of $R$ contained in
$\mathfrak{m}^{\mathrm{g}(R)}$.

######  Theorem 3.5.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then the following statements are equivalent.

  1. $i)$

$R$ is Cohen–Macaulay.

  2. $ii)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

$\mathrm{r}(R)-\mathcal{N}(R)\geq{\rm sg}(\mathfrak{q},R).$

  3. $iii)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

$\mathrm{r}(R)-\mathcal{N}(R)\geq{\rm sg}(\mathfrak{q}:\mathfrak{m},R).$

###### Proof.

$i)\Rightarrow ii)$. Since $R$ is Cohen-Macaulay, by Lemma 2.2 ii), we have

$\mathrm{r}(R)-\mathcal{N}(R)=\mathrm{r}(R)-\mathrm{r}_{d}(R)=0$

for all $C$-parameter ideals $\mathfrak{q}$ of $R$.
$ii)\Rightarrow iii)$ follows from Theorem 3.3.
$iii)\Rightarrow i)$. Since $R$ is to be regular if $R$ is unmixed and
$\mathrm{e}_{0}(\mathfrak{m},R)=1$ (see [Nag62, Theorem 40.6]), we have
$\mathrm{e}_{0}(\mathfrak{m},R)>1$. Let
$\underline{x}=x_{1},x_{2},\ldots,x_{d}$ be a $C$-system of parameters of $R$
such that $\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$ and put
$J=\mathfrak{q}:\mathfrak{m}$. Now, we put $A=R/(x_{1},\ldots,x_{d-2})$. Using
similar arguments as in the proof of Theorem 3.3 (see (7)), one can show that

${\rm
sg}(J,R)=\mathrm{I}(\mathfrak{q}A,A)+\mathrm{e}_{1}(JA,A)-\mathcal{N}(A)=h_{0}(A)-r_{0}(A)-r_{1}(A).$

On the other hand, it follows from Lemma 3.5 in [OTY21] and Lemma 2.8 that

$\mathrm{r}(R)-\mathcal{N}(R)\leq
r_{2}(A)-\mathcal{N}(A)=-2r_{1}(A)-r_{0}(A).$

Since $\mathrm{r}(R)-\mathcal{N}(R)\geq{\rm sg}(J,R)$ by hypothesis, we have
$h_{0}(A)=r_{1}(A)=0$. Therefore, $A$ is Cohen–Macaulay. Hence $R$ is Cohen-
Macaulay because of Lemma 2.9. The proof is complete. ∎

######  Corollary 3.6.

Assume that $R$ is a non-regular unmixed local ring with $d=\dim R\geq 2$.
Then for all $C$-parameter ideals
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$, we have

$\mathrm{r}(R)-\mathcal{N}(R)\leq{\rm sg}(\mathfrak{q}:\mathfrak{m},R)\leq{\rm
sg}(\mathfrak{q},R).$

Furthermore, each equality occurs if and only if $R$ is Cohen-Macaulay.

###### Proof.

This is now immediately seen from Theorem 3.5 and Theorem 3.3. ∎

The following consequence of Theorem 3.5 provides a characterization of
Gorenstein rings.

######  Corollary 3.7.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then the following statements are equivalent.

  1. $i)$

$R$ is Gorenstein.

  2. $ii)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

${\rm sg}(\mathfrak{q}:\mathfrak{m},R)\leq 1-\mathcal{N}(R).$

###### Proof.

$i)\Rightarrow ii)$. Since $R$ is Gorenstein, we get by Theorem 3.5 that

${\rm
sg}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{r}(R)-\mathcal{N}(R)=1-\mathcal{N}(R)$

for all $C$-parameter ideals $\mathfrak{q}$ of $R$.
$ii)\Rightarrow i)$. We have

${\rm sg}(\mathfrak{q}:\mathfrak{m},R)\leq
1-\mathcal{N}(R)\leq\mathrm{r}(R)-\mathcal{N}(R).$

Thus $R$ is Cohen-Macaulay because of Theorem 3.5. Moreover, we have
$\mathrm{r}(R)=1$. Therefore, $R$ is Gorenstein, and this completes the proof.
∎

##  4\. On the sectional genera and the second Hilbert coefficient

In this section, we will explore the relationship between the sectional genera
and the second Hilbert coefficient.

######  Theorem 4.1.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then the following statements are equivalent.

  1. $i)$

$R$ is Cohen–Macaulay.

  2. $ii)$

There exists a $C$-parameter ideal $\mathfrak{q}$ of $R$ such that

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\geq\mathrm{e}_{2}(\mathfrak{q},R).$

###### Proof.

Note that we have $\mathrm{e}_{0}(\mathfrak{m},R)>1$ since $R$ is to be
regular if $R$ is unmixed and $\mathrm{e}_{0}(\mathfrak{m},R)=1$ (see [Nag62,
Theorem 40.6]).
$i)\Rightarrow ii)$. Since $R$ is Cohen-Macaulay, by Lemma 2.2, we get that

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{e}_{2}(\mathfrak{q},R)=0.$

$ii)\Rightarrow i)$. Let $\underline{x}=x_{1},x_{2},\ldots,x_{d}$ be a
$C$-system of parameters of $R$ such that
$\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$ and put
$J=\mathfrak{q}:\mathfrak{m}$. Note that
$JR_{i}=\mathfrak{q}R_{i}:_{R_{i}}\mathfrak{m}R_{i}$, where
$R_{i}=R/(x_{1},\ldots,x_{i})$ for all $i=1,\ldots,d-2$. Now, we put
$A=R/(x_{1},\ldots,x_{d-2})$. We get
$\mathrm{e}_{2}(\mathfrak{q},R)=\mathrm{e}_{2}(\mathfrak{q}A,A)$ and
$\mathrm{e}_{2}(J,R)=\mathrm{e}_{2}(JA,A)$ by Lemma 2.1. Note that by Lemma
2.8 $ii)$, $A/H_{\mathfrak{m}}^{0}(A)$ is unmixed of dimension $2$. Thus $A$
is generalized Cohen–Macaulay because of Lemma 2.9. Therefore, by Lemma 2.1
and 2.2, we have

$\mathrm{e}_{2}(\mathfrak{q},R)=\mathrm{e}_{2}(\mathfrak{q}A,A)-\ell((0):_{R_{d-3}}x_{d-2})=h_{0}(A)-\ell((0):_{R_{d-3}}x_{d-2})$

and

$\mathrm{e}_{2}(J,R)=\mathrm{e}_{2}(JA,A)-\ell((0):_{R_{d-3}}x_{d-2})=h_{0}(A)-r_{1}(A)-\ell((0):_{R_{d-3}}x_{d-2}).$

Since
$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\geq\mathrm{e}_{2}(\mathfrak{q},R)$,
we have $r_{1}(A)=0$. Therefore, $A/H_{\mathfrak{m}}^{0}(A)$ is
Cohen–Macaulay. Hence $R$ is Cohen-Macaulay by Lemma 2.9. The proof is
complete. ∎

######  Theorem 4.2.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then the following statements are equivalent.

  1. $i)$

$R$ is Cohen–Macaulay.

  2. $ii)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\geq{\rm
sg}(\mathfrak{q}:\mathfrak{m},R)+\mathcal{N}(R)-\mathrm{r}(R).$

  3. $iii)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

$\mathrm{e}_{2}(\mathfrak{q},R)\geq{\rm
sg}(\mathfrak{q}:\mathfrak{m},R)+\mathcal{N}(R)-\mathrm{r}(R).$

  4. $iv)$

There exists a $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$ such that

$\mathrm{e}_{2}(\mathfrak{q},R)\geq{\rm
sg}(\mathfrak{q},R)+\mathcal{N}(R)-\mathrm{r}(R).$

###### Proof.

$i)\Rightarrow ii)$ and $i)\Rightarrow iv)$ are immediately seen from the
assumption that $R$ is Cohen-Macaulay and Lemma 2.2. Moreover $ii)\Rightarrow
iii)$ and $iv)\Rightarrow iii)$ was established in Theorem 4.1. Now we show
$iii)\Rightarrow i)$. Since $R$ is to be regular if $R$ is unmixed and
$\mathrm{e}_{0}(\mathfrak{m},R)=1$ (see [Nag62, Theorem 40.6]), we have
$\mathrm{e}_{0}(\mathfrak{m},R)>1$.

Let $\underline{x}=x_{1},x_{2},\ldots,x_{d}$ be a $C$-system of parameters of
$R$ such that $\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$ and put
$J=\mathfrak{q}:\mathfrak{m}$. Set $A=R/(x_{1},\ldots,x_{d-2})$. Using similar
arguments as in the proof of Theorem 3.3 and 4.1, one can show that $A$ is
generalized Cohen–Macaulay. Furthermore, we also obtain that

${\rm
sg}(J,R)=\mathrm{I}(\mathfrak{q}A,A)+\mathrm{e}_{1}(JA,A)-\mathcal{N}(A),$

and

$\mathrm{e}_{2}(\mathfrak{q},R)=\mathrm{e}_{2}(\mathfrak{q}A,A)-\ell((0):_{R_{d-3}}x_{d-2})=h_{0}(A)-\ell((0):_{R_{d-3}}x_{d-2}).$

On the other hand, by Lemma 3.5 in [OTY21], we have
$r_{2}(R)\geq\mathrm{r}(R)$. Since $A$ is generalized Cohen–Macaulay, it
follows from Lemma 2.2 that

${\rm sg}(J,R)+\mathcal{N}(R)-\mathrm{r}(R)\geq h_{0}(A)+r_{1}(A).$

Since $\mathrm{e}_{2}(\mathfrak{q},R)\geq{\rm
sg}(J,R)+\mathcal{N}(R)-\mathrm{r}(R)$, we have $r_{1}(A)=0$. Therefore
$A/H_{\mathfrak{m}}^{0}(A)$ is Cohen–Macaulay. Hence $R$ is Cohen-Macaulay
because of Lemma 2.8. The proof is complete. ∎

######  Corollary 4.3.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then for all $C$-parameter ideal
$\mathfrak{q}\subseteq\mathfrak{m}^{\mathrm{g}(R)}$ of $R$, we have

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\leq\mathrm{e}_{2}(\mathfrak{q},R)\leq{\rm
sg}(\mathfrak{q}:\mathfrak{m},R)+\mathcal{N}(R)-\mathrm{r}(R)\leq{\rm
sg}(\mathfrak{q},R)+\mathcal{N}(R)-\mathrm{r}(R).$

###### Proof.

This is now immediately seen from Theorem 4.2. ∎

######  Theorem 4.4.

Assume that $R$ is a non-regular unmixed local ring of dimension $d=\dim R\geq
2$. Then the following statements are equivalent.

  1. $i)$

$R$ is Cohen–Macaulay.

  2. $ii)$

There exists a $C$-parameter ideal $\mathfrak{q}$ such that

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\geq{\rm sg}(\mathfrak{q},R).$

###### Proof.

$i)\Rightarrow ii)$. Since $R$ is Cohen-Macaulay, by Lemma 2.2 we have
$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)={\rm sg}(\mathfrak{q},R)=0.$
$ii)\Rightarrow i)$. Using similar arguments as in the proof of Theorem 3.3,
we obtain that $\mathrm{e}_{0}(\mathfrak{m},R)>1$. Let
$\underline{x}=x_{1},x_{2},\ldots,x_{d}$ be a $C$-system of parameters of $R$
such that $\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$. We put
$A=R/(x_{1},\ldots,x_{d-2})$. By Lemma 2.8 $ii)$, $A/H_{\mathfrak{m}}^{0}(A)$
is unmixed with $\dim A=2$. Thus $A$ is generalized Cohen–Macaulay because of
Lemma 2.9. Moreover, it is follows from Lemma 2.1 that
$\mathrm{e}_{0}(\mathfrak{m}A,A)=\mathrm{e}_{0}(\mathfrak{m},R)>1$. Note that
$JR_{i}=\mathfrak{q}R_{i}:_{R_{i}}\mathfrak{m}R_{i}$, where
$R_{i}=R/(x_{1},\ldots,x_{i})$ for all $i=1,\ldots,d-2$. By Lemma 2.1 and 2.2,
we get that

$\mathrm{e}_{2}(J,R)=\mathrm{e}_{2}(JA,A)-\ell((0):_{R_{d-3}}x_{d-2})=h_{0}(R)-r_{1}(A)-\ell((0):_{R_{d-3}}x_{d-2})$

and

${\rm sg}(\mathfrak{q},R)={\rm sg}(\mathfrak{q}A,A)=h_{0}(R).$

Since $\mathrm{e}_{2}(J,R)\geq{\rm sg}(\mathfrak{q},R)$, we have
$h_{0}(R)=r_{1}(A)=0$. Therefore $A$ is Cohen–Macaulay. Hence $R$ is Cohen-
Macaulay because of Lemma 2.8. The proof is complete. ∎

######  Corollary 4.5.

Assume that $R$ is a non-regular unmixed local ring and $d=\dim R\geq 2$. Then
for all $C$-parameter ideal $\mathfrak{q}$ of $R$, we have

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)\leq{\rm sg}(\mathfrak{q},R).$

###### Proof.

This is now immediately seen from Theorem 4.4. ∎

For the reader’s convenience, we recall the notion of quasi-Buchsbaum modules.
We say that an $R$-module $M$ is said to be a quasi-Buchsbaum module if
$\mathfrak{m}H^{i}_{\mathfrak{m}}(M)=0$ for all $i<\dim M$. With this
notation, we are now in a position to prove the second main theorem in this
study.

###### Proof of Theorem 1.3.

Since $R$ is to be regular if $R$ is unmixed and
$\mathrm{e}_{0}(\mathfrak{m},R)=1$, we have
$\mathrm{e}_{0}(\mathfrak{m},R)>1$.
$i)\Rightarrow ii)$. Let $\mathfrak{q}$ be a $C$-parameter ideal of $R$. By
lemma 3.1, we have

${\rm
sg}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{I}(\mathfrak{q},R)+\mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)-\mathcal{N}(R).$

Since $R$ is quasi-Buchsbaum, $r_{i}(R)=h_{i}(R)$ for all $i<d$. By Lemma 2.2,
we get that

${\rm
sg}(\mathfrak{q}:\mathfrak{m},R)=-\displaystyle\sum_{j=1}^{d-1}\binom{d-2}{j-1}h_{j}(R)=\mathrm{e}_{1}(\mathfrak{q},R).$

$ii)\Rightarrow i)$. Let $\underline{x}=x_{1},x_{2},\ldots,x_{d}$ be a
$C$-system of parameters of $R$ such that
$\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$ and put
$J=\mathfrak{q}:\mathfrak{m}$. Now, we put $A=R/(x_{1},\ldots,x_{d-2})$. Using
similar arguments as in the proof of Theorem 3.3, one can show that

${\rm
sg}(J,R)=\mathrm{I}(\mathfrak{q}A,A)+\mathrm{e}_{1}(JA,A)-\mathcal{N}(A)=-r_{1}(A)+h_{0}(R)-r_{0}(A)$

and

$\mathrm{e}_{1}(\mathfrak{q},R)=\mathrm{e}_{1}(\mathfrak{q}A,A)=-h_{1}(R).$

Since ${\rm sg}(J,R)=\mathrm{e}_{1}(\mathfrak{q},R)$, we have
$r_{1}(A)=h_{1}(R)$ and so $h_{0}(R)=r_{0}(A)$. Thus $A$ is quasi-Buchsbaum.
It follows from a $C$-system of parameters forms a $d$-sequence of $R$ and
Theorem 3.6 in [Suz87] that $R$ is quasi-Buchsbaum, as required. ∎

We close this paper with the following example, which shows that Theorem 4.1,
4.2, and 4.4 are not true without the assumption that $R$ is unmixed.

######  Example 4.6.

Let $S=k[[X,Y,Z,W]]$ be the formal power series ring over a field $k$. Put
$R=S/[(X,Y,Z)\cap(W)]$. Then

  1. $i)$

$\dim R=3$ and $R$ is not unmixed. Hence, $R$ is not a Cohen-Macaulay ring.

  2. $ii)$

We have

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{e}_{2}(\mathfrak{q},R)={\rm
sg}(\mathfrak{q}:\mathfrak{m},R)+\mathcal{N}(R)-\mathrm{r}(R)={\rm
sg}(\mathfrak{q},R)$

for all $C$-parameter ideals $\mathfrak{q}$ in $R$.

###### Proof.

It is standard to check that $\dim R=3$ and $R$ is not unmixed. It follows
that $R$ is not a Cohen-Macaulay ring. We put $A=S/(W)$ and $B=S/(X,Y,Z)$.
Note that $A$ and $B$ are Gorenstein rings.

Let $\mathfrak{q}$ be a $C$-parameter ideal of $R$ and put
$J=\mathfrak{q}:\mathfrak{m}$. Then $J^{n}=\mathfrak{q}^{n}:\mathfrak{m}$ for
all $n\geq 0$ (see [OTY21, Lemma 2.6]). From the exact sequence $0\to B\to
R\to A\to 0$ and $A$ is Cohen-Macaulay, we observe that

$0\to B/\mathfrak{q}^{n+1}B\to R/\mathfrak{q}^{n+1}\to
A/\mathfrak{q}^{n+1}A\to 0.$

By applying the functor $\mathrm{Hom}_{R}(R/\mathfrak{m},-)$ to the above
exact sequence, by Lemma 2.6 in [OTY21], we obtain the following:

$0\to[\mathfrak{q}^{n+1}:_{B}\mathfrak{m}]/\mathfrak{q}^{n+1}B\to[\mathfrak{q}^{n+1}:_{R}\mathfrak{m}]/\mathfrak{q}^{n+1}\to[\mathfrak{q}^{n+1}:_{A}\mathfrak{m}]/\mathfrak{q}^{n+1}A\to
0$

Therefore, we have

$\displaystyle\ell_{R}(R/\mathfrak{q}^{n+1})$
$\displaystyle=\ell_{R}(A/\mathfrak{q}^{n+1}A)+\ell_{R}(B/\mathfrak{q}^{n+1}B)$
$\displaystyle=\ell_{R}(A/\mathfrak{q}A)\binom{n+3}{3}+\ell_{R}(B/\mathfrak{q}B)\binom{n+1}{1}$

and

$\displaystyle\ell_{R}([\mathfrak{q}^{n+1}:_{R}\mathfrak{m}]/\mathfrak{q}^{n+1})$
$\displaystyle=\ell_{R}([\mathfrak{q}^{n+1}:_{A}\mathfrak{m}]/\mathfrak{q}^{n+1})+\ell_{R}([\mathfrak{q}^{n+1}:_{B}\mathfrak{m}]/\mathfrak{q}^{n+1})$
$\displaystyle=\binom{n+1}{2}+1$

for all integers $n\geq 0$. Since
$\ell_{R}(R/I^{n+1})=\ell_{R}(R/\mathfrak{q}^{n+1})-\ell_{R}([\mathfrak{q}^{n+1}:_{R}\mathfrak{m}]/\mathfrak{q}^{n+1})$,
we have
$\mathrm{e}_{0}(\mathfrak{q},R)=\mathrm{e}_{0}(\mathfrak{q}:\mathfrak{m},R)=\ell_{R}(A/\mathfrak{q}A),\mathrm{e}_{1}(\mathfrak{q},R)=0,\mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)=1$
and
$\mathrm{e}_{2}(\mathfrak{q},R)=\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)=\ell_{R}(B/\mathfrak{q}B)$.
Thus we have

${\rm
sg}(\mathfrak{q},R)=\ell(R/\mathfrak{q})-\mathrm{e}_{0}(\mathfrak{q},R)+\mathrm{e}_{1}(\mathfrak{q},R)=\ell_{R}(B/\mathfrak{q}B).$

On the other hand, it is easily observed from $0\to B\to R\to A\to 0$ that
$r_{0}(R)=r_{2}(R)=0$ and $r_{1}(R)=r_{3}(R)=1$. Since $R$ is sequentially
Cohen-Macaulay, it follows from Theorem 1.1 in [Tru13] that
$\mathcal{N}(R)=\sum\limits_{i=0}^{3}r_{i}(R)=2$. Hence, by Lemma 3.1, we have

${\rm
sg}(\mathfrak{q}:\mathfrak{m},R)=\ell(R/\mathfrak{q})-\mathrm{e}_{0}(\mathfrak{q},R)+\mathrm{e}_{1}(\mathfrak{q}:\mathfrak{m},R)-\mathcal{N}(R)=\ell_{R}(B/\mathfrak{q}B)-1.$

Consequently, we obtain that

$\mathrm{e}_{2}(\mathfrak{q}:\mathfrak{m},R)=\mathrm{e}_{2}(\mathfrak{q},R)={\rm
sg}(\mathfrak{q}:\mathfrak{m},R)+\mathcal{N}(R)-\mathrm{r}(R)={\rm
sg}(\mathfrak{q},R)=\ell_{R}(B/\mathfrak{q}B).$

The proof is complete. ∎

## References