# On Hilbert coefficients and sequentially generalized Cohen-Macaulay modules

Nguyen Tu Cuong  Institute of Mathematics, Vietnam Academy of Science and
Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam<EMAIL_ADDRESS>,  Nguyen Tuan Long  National Economics University, 207 Giai Phong Road,
Hanoi, Vietnam<EMAIL_ADDRESS>and  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>
###### Abstract.

This paper shows that if $R$ is a homomorphic image of a Cohen-Macaulay local
ring, then $R$-module $M$ is sequentially generalized Cohen-Macaulay if and
only if the difference between Hilbert coefficients and arithmetic degrees for
all distinguished parameter ideals of $M$ are bounded.

Key words and phrases: Hilbert coefficients, multiplicity, sequentially Cohen-
Macaulay, sequentially generalized Cohen-Macaulay, Arithmetic degree,
dimension filtration, distinguished parameter ideals. 2020 Mathematics Subject
Classification: 13C14, 13D40, 13H15. The first author was supported by Vietnam
National Foundation for Science and Technology Development (NAFOSTED) under
grant number 101.04-2020.10. The last author was supported by B2022-TNA-25.

##  1\. Introduction

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, where
$\mathfrak{m}$ is the maximal ideal. Let $M$ be a finitely generated
$R$-module of dimension $d$. For an $\mathfrak{m}$-primary ideal $I$ of $R$,
it is well-known that there are integers
$\\{\mathrm{e}_{i}(I;M)\\}_{i=0}^{d}$, called the Hilbert coefficients of $M$
with respect to $I$, such that

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

for all $n\gg 0$. Here $\ell_{R}(N)$ denotes the length of an $R$-module $N$.
In particular, the leading coefficient $e_{0}(I;M)$ is said to be the
multiplicity of $M$ with respect to $I$ and $e_{1}(I;M)$ is called by
Vasconselos ([27]) the Chern coefficient of $M$ with respect to $I$. In 2008,
Vasconcelos posed the Vanishing Conjecture: $M$ is Cohen-Macaulay if and only
if $\mathrm{e}_{1}(\mathfrak{q};M)=0$ for some parameter ideal $\mathfrak{q}$
of $M$. It is shown that the relation between Cohen-Macaulayness and the Chern
number of parameter ideals is quite surprising. Motivated by some profound
results of [5, 20] and also by the fact that this is true for $M$ is unmixed
as shown in [10], it was asked whether the characterization of many classes of
non-unmixed rings such as Buchsbaum rings, generalized Cohen-Macaulay rings,
sequentially Cohen-Macaulay rings in terms of the Hilbert coefficients and
other invariants of $M$ (see [11, 5, 22, 23, 18, 24]). The aim of our paper is
to continue this research direction.

To state the results of this paper, first of all let us fix our notation and
terminology. First, a filtration

$\mathcal{D}:M=D_{0}\supset D_{1}\supset\ldots\supset D_{t}=W$

of $R$-submodules of $M$ is called the dimension filtration of $M$, if for all
$1\leq i\leq\ell$, $D_{i}$ is the largest submodule of $D_{i-1}$ with
$\dim_{R}D_{i}<\dim_{R}D_{i-1}$, where $\dim_{R}(0)=-\infty$ for convention.
We say that $M$ is sequentially (generalized) Cohen-Macaulay, if
$C_{i}=D_{i}/D_{i+1}$ is (generalized) Cohen-Macaulay for all $0\leq
i\leq\ell-1$. A system of parameters $\underline{x}=x_{1},x_{2},\ldots,x_{d}$
of $M$ is said to be distinguished, if $(x_{j}\mid d_{i}<j\leq d)D_{i}=(0)$
for all $0\leq i\leq\ell$, where $d_{i}=\dim_{R}D_{i}$ ([19, Definition 2.5]).
A parameter ideal $\mathfrak{q}$ of $M$ is called distinguished, if there
exists a distinguished system $x_{1},x_{2},\ldots,x_{d}$ of parameters of $M$
such that $\mathfrak{q}=(x_{1},x_{2},\ldots,x_{d})$. For each $i=1,\ldots,s$,
set

$\Lambda_{i}(M)=\\{(-1)^{i}e_{i}(\mathfrak{q},M)-\operatorname{adeg}_{i}(\mathfrak{q};M)\mid\mathfrak{q}\text{
is a distinguished parameter ideal of }M\\},$

where
$\operatorname{adeg}_{i}(I;M)=\sum\limits_{\begin{subarray}{l}\mathfrak{p}\in\operatorname{Ass}(M),\\\
\dim
R/\mathfrak{p}=i\end{subarray}}\ell_{R_{\mathfrak{p}}}(H^{0}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}}))e_{0}(I;R/\mathfrak{p})$
is the $i$-th arithmetic degree of $M$ with respect to $I$ (see
[1],[25],[26]). Then we have the following results as in Table 1.

$\Lambda_{1}(M)\subseteq(-\infty,0]$ | $M$ | [5]
---|---|---
$0\in\Lambda_{1}(M)$, ($\star$) |  $M$ is Cohen-Macaulay | [10]
$\left|\Lambda_{1}(M)\right|<\infty$, ($\star$) |  $M$ is generalized Cohen-macaulay | [10, 11]
$0\in\Lambda_{i}(M)$ for all $i=1,\ldots,d$ |  $M$ is sequentially Cohen-Macaulay | [5]
Table 1.  Properties of a finitely generated module $M$ are carried by the
behavior of the specific set. A symbol ($\star$) requires that the module $M$
be unmixed.

This paper aims to extend these results in the sequentially generalized Cohen-
Macaulay case. The answer is affirmative, which we are eager to report in the
present writing.

######  Theorem 1.1 (Theorem 4.1).

Assume that $R$ is a homomorphic image of a Cohen-Macaulay local ring. Then
the following statements are equivalent.

  1. $i)$

$M$ sequentially generalized Cohen-Macaulay.

  2. $ii)$

The set $\Lambda_{i}(M)$ is finite for all $1\leq i\leq d$.

The paper is divided into four sections. The next section presents some
preliminaries. In Section 3, we prove that if $M$ is a sequentially
generalized Cohen-Macaulay module, then the set $\Lambda_{i}(M)$ is finite for
all $i=1\ldots,d$. A characterization of sequentially generalized Cohen-
Macaulay modules by the finiteness of $\Lambda_{i}(M)$ will be shown in the
last section.

##  2\. Preliminaries

In what follows, throughout this paper, let $(R,\mathfrak{m},k)$ be a
Noetherian local ring, where $\mathfrak{m}$ is the maximal ideal and
$k=R/\mathfrak{m}$ is the infinite 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 $d$.

######  Definition 2.1 ([7],[2], [19]).

  1. $i)$

We say that a finite filtration of submodules of $M$

$\mathcal{F}:M=M_{0}\supset M_{1}\supset\ldots\supset M_{s}$

satisfies the dimension condition if $\dim M_{i}>\dim M_{i+1}$, for all
$i=0,\ldots,s-1$ and we say this case that the filtration $\mathcal{F}$ has
the length $s$.

  2. $ii)$

A filtration

$\mathcal{D}:M=D_{0}\supset D_{1}\supset\ldots\supset D_{t}=W$

of submodules of $M$ is said to be the dimension filtration if $D_{i}$ is the
largest submodule of $D_{i-1}$ with $\dim D_{i}<\dim D_{i-1}$ for all
$i=1,\ldots,t.$ Note that the dimension filtration always exists uniquely (see
[7]).

######  Notation 2.2.

  1. $\bullet$

$t$ : the length of the dimension filtration of $M$,

  2. $\bullet$

$\mathcal{D}=\\{D_{i}\\}_{i=0}^{t}$ : the dimension filtration of $M$,

  3. $\bullet$

$d_{i}=\dim D_{i}$ for all $i=0,\ldots,t$,

  4. $\bullet$

$\mathcal{F}=\\{M_{i}\\}_{i=0}^{s}$ : a filtration of submodules of $M$ of
length $s$ satisfying the dimension condition,

  5. $\bullet$

$\mathcal{F}/x\mathcal{F}=\\{({M}_{i}+xM)/xM\\}_{i=0}^{k}$ : a filtration of
submodules of $M/xM$, where $x$ is a parameter element of $M$ and
$k=\left\\{\begin{gathered}s-1\text{ if }\dim(M_{s-1})=1,\hfill\\\
s\quad\quad\text{ otherwise},\hfill\\\ \end{gathered}\right.$

  6. $\bullet$

$\mathcal{F}(M)=\\{\mathcal{F}=\\{M_{i}\\}_{i=0}^{t}\mid\ell(D_{i}/M_{i})<\infty\text{
for all }i=0,\ldots,t\\}$.

######  Definition 2.3.

$i)$ A system of parameters $x_{1},\ldots,x_{d}$ of $M$ is called a
distinguished system of parameters with respect to $\mathcal{F}$ if $(x_{\dim
M_{i}+1},\ldots,x_{d})M_{i}=0$ for all $i=1,\ldots,s$. A distinguished system
of parameters of $M$ with respect to $\mathcal{D}$ is simply called a
distinguished system of parameters of $M$. An ideal $\mathfrak{q}$ is said to
be a distinguished parameter ideal of $M$ with respect to $\mathcal{F}$ if it
is generated by a distinguished system of parameters of $M$ with respect to
$\mathcal{F}$. A distinguished parameter ideal of $M$ with respect to
$\mathcal{D}$ is simply called a distinguished parameter ideal of $M$ ([19]).

$ii)$ A system of parameters $x_{1},\ldots,x_{d}$ is called good system of
parameters of $M$ if $(x_{d_{i}+1},\ldots,x_{d})M\cap D_{i}=0$ for all
$i=1,\ldots,t$. An ideal $\mathfrak{q}$ is said to be a good parameter ideal
of $M$ if it is generated by a good system of parameters of $M$ (see [2]).

Recall that there always exists distinguished systems of parameters of $M$
with respect to $\mathcal{F}$ (see [19, Lemma 2.6]). Note that, if
$x_{1},x_{2},\ldots,x_{d}$ is a distinguished system of parameters of $M$ with
respect to $\mathcal{F}$, then $x_{2},\ldots,x_{d}$ is also a distinguished
system of parameters of $M/x_{1}M$ with respect to
$\mathcal{F}/{x_{1}}\mathcal{F}$. Clearly, a good system of parameters is a
distinguished system of parameters.

######  Definition 2.4 ([1],[25],[26]).

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$. The $i$-th arithmetic
degree of $M$ with respect to $I$ is defined by

$\operatorname{adeg}_{j}(I;M)=\sum\limits_{\mathfrak{p}\in\operatorname{Ass}(M),\>\dim
R/\mathfrak{p}=j}\ell(H^{0}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}}))e_{0}(I;R/\mathfrak{p}).$

The following result is an immediate consequence of Proposition 3.2 in [5].

######  Lemma 2.5 (c.f. [5, Proposition 3.2]).

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$ and
$\mathcal{F}\in\mathcal{F}(M)$. Then

$\operatorname{adeg}_{j}(I;M)=\left\\{\begin{gathered}\ell(H^{0}_{\mathfrak{m}}(M))\quad\,\,\text{
if }i=0,\hfill\\\ e_{0}(I;M_{i})\quad\quad\text{ if }d_{i}=j\text{ for some
}i=0,\ldots,t-1,\hfill\\\ 0\quad\quad\quad\quad\quad\,\,\text{
otherwise}.\hfill\\\ \end{gathered}\right.$

######  Lemma 2.6.

Let $\mathcal{F}\in\mathcal{F}(M)$ and $\mathfrak{q}$ a parameter ideal of
$M$. Suppose that there exists a filter regular element $x\in\mathfrak{q}$ of
$M$ such that $\mathcal{F}/x\mathcal{F}\in\mathcal{F}(M/xM)$. We have

$\operatorname{adeg}_{j}(\mathfrak{q};M/xM)=\operatorname{adeg}_{j+1}(\mathfrak{q};M),$

for all $j\geq 1$. Moreover, if $\mathfrak{q}:=(x:=x_{1},x_{2},\ldots,x_{d})$
is a distinguished parameter ideal of $M$ with respect to $\mathcal{F}$, then
$\operatorname{adeg}_{0}(\mathfrak{q};M/xM)\geq\operatorname{adeg}_{1}(\mathfrak{q};M).$

###### Proof.

Since $\mathcal{F}/x\mathcal{F}\in\mathcal{F}(M/xM)$, we have
$\mathcal{D}/xM\in\mathcal{F}(M/xM)$. Moreover, the length of $\mathcal{D}/xM$
is $t-1$ if $d_{t-1}=1$ and $t$ otherwise. Since $x$ is a filter regular
element of $M$, $x$ is a regular element of $M/D_{i}$ for all $i=1,\ldots,t$.
Thus we have $xM\cap D_{i}=xD_{i}$. By Lemma 2.5 and
$\mathcal{D}/xM\in\mathcal{F}(M/xM)$, we have

$\displaystyle\operatorname{adeg}_{d_{i}-1}(\mathfrak{q};M/xM)$
$\displaystyle=e_{0}(\mathfrak{q};(D_{i}+xM)/xM)$
$\displaystyle=e_{0}(\mathfrak{q};D_{i}/xD_{i})=e_{0}(\mathfrak{q};D_{i})=\operatorname{adeg}_{d_{i}}(\mathfrak{q};M)$

and
$\operatorname{adeg}_{j}(\mathfrak{q};M)=0=\operatorname{adeg}_{j-1}(\mathfrak{q};M/xM)$
for all $i=0,\ldots,t-1$ and $2\leq d_{i}<j<d_{i-1}$. Hence

$\operatorname{adeg}_{j}(\mathfrak{q};M/xM)=\operatorname{adeg}_{j+1}(\mathfrak{q};M),$

for all $j\geq 1$.

Now, if $d_{t-1}>1$, then
$\operatorname{adeg}_{1}(\mathfrak{q};M)=0\leq\operatorname{adeg}_{0}(\mathfrak{q};M)$.
So we can assume that $d_{t-1}=1$. Since $\mathfrak{q}=(x,x_{2},\ldots,x_{d})$
is a distinguished ideal of $M$ with respect to
$\mathcal{F}\in\mathcal{F}(M)$, by Lemma 2.5 we get

$\displaystyle\operatorname{adeg}_{1}(\mathfrak{q};M)$
$\displaystyle=e_{0}(\mathfrak{q};M_{t-1})=e_{0}(x;M_{t-1})=e_{0}(x;D_{t-1})\leq\ell(D_{t-1}/xD_{t-1}).$

Since $H^{0}_{\mathfrak{m}}(M/xM)\supseteq(D_{t-1}+xM)/xM\cong
D_{t-1}/(D_{t-1}\cap xM)=D_{t-1}/xD_{t-1}$, we have

$\operatorname{adeg}_{1}(\mathfrak{q};M)\leq\ell(D_{t-1}/xD_{t-1})\leq\ell(H^{0}_{\mathfrak{m}}(M/xM))=\operatorname{adeg}_{0}(\mathfrak{q};M/xM),$

as required. ∎

The following lemma was proved by [5, Lemma 3.3, Lemma 3.4].

######  Lemma 2.7.

The following statements are true.
$(i)$ Assume that $d\geq 2$. Let $x$ be a superficial element of $M$ for a
parameter ideal $\mathfrak{q}$ of $M$. Then

$e_{i}(\mathfrak{q};M)=e_{i}(\mathfrak{q};M/xM)$

for all $i=0,\ldots,d-2$ and
$(-1)^{d-1}e_{d-1}(\mathfrak{q};M)=(-1)^{d-1}e_{d-1}(\mathfrak{q};M/xM)+\ell(0:_{M}x)$.
$(ii)$ Let $N$ be a submodule of $M$ with $\dim N=s<d$ and $I$ an
$\mathfrak{m}$-primary ideal of $R$. Then

$e_{j}(I;M)=\left\\{\begin{gathered}e_{j}(I;M/N),\text{ if }0\leq j\leq
d-s-1,\hfill\\\ e_{d-s}(I;M/N)+(-1)^{d-s}e_{0}(I,N)\text{ if }j=d-s.\hfill\\\
\end{gathered}\right.$

##  3\. The finiteness of the set $\mathcal{P}_{\mathcal{D}}(M)$

Recall that the definition of a sequentially Cohen-Macaulay module was
introduced first by LT. Nhan and the first author ([7]).

######  Definition 3.1.

A filtration of submodules $\mathcal{F}=\\{M_{i}\\}_{i=0}^{t}$ of $M$ is
called a generalized Cohen-Macaulay filtration if
$\mathcal{F}\in\mathcal{F}(M)$ and $M_{i-1}/M_{i}$ are generalized Cohen-
Macaulay modules for all $i=1,\ldots,t-1$. A module $M$ is called sequentially
generalized Cohen-Macaulay if it has a generalized Cohen-Macaulay filtration.
In particular, $\mathcal{F}(M)$ is the set of all generalized Cohen-Macaulay
filtrations of $M$. If $M$ is a sequentially Cohen-Macaulay module, then $M$
is a sequentially generalized Cohen-Macaulay module.

Now, the function

$H^{ad}_{I,M}(n)=\ell(M/I^{n+1}M)-\sum\limits_{i=0}^{d}\operatorname{adeg}_{i}(I;M)\binom{n+i}{i}$

is called an adjusted Hilbert-Samuel function of $M$ with respect to $I$. It
is well known that $\ell(M/I^{n+1}M)$ becomes a polynomial for large enough
$n>0$. So the function $H^{ad}_{I,M}(n)$ become a polynomial
$P^{ad}_{\mathfrak{q},M}(n)$. Such polynomial is called adjusted Hilbert-
Samuel polynomial and of the form

$P^{ad}_{I,M}(n)=\sum\limits_{i=1}^{d}\left((-1)^{i}e_{i}(I;M)-\operatorname{adeg}_{d-i}(I;M)\right)\binom{n+d-i}{d-i}.$

These integers $a_{i}(I;M)=(-1)^{i}e_{i}(I;M)-\operatorname{adeg}_{d-i}(I;M)$
are called adjusted Hilbert coefficients of $M$ with respect to $I$ for all
$i=1,\ldots,d$. We denote by $\mathcal{P}_{\mathcal{F}}(M)$ the set of all
adjusted Hilbert-Samuel polynomials $P^{ad}_{\mathfrak{q},M}(n)$, where
$\mathfrak{q}$ runs over the set of all distinguished parameter ideals of $M$
with respect to $\mathcal{F}$.

Recall that a system $x_{1},\ldots,x_{m}$ in $R$ is said to be d-sequence on
$M$ (see [13], [21]) if

$(x_{1},\ldots,x_{i-1})M:x_{i}x_{k}=(x_{1},\ldots,x_{i-1})M:x_{k}$

for all $i=1,\ldots,m$ and $k\geq i$. The sequence $\underline{x}$ is said to
be dd-sequence on $M$ (see [4]) if $x_{1}^{n_{1}},\ldots,x_{s}^{n_{s}}$ is a
d-sequence on $M$ and $x^{n_{1}}_{1},\ldots,x^{n_{i}}_{i}$ is a d-sequence on
$M/(x_{i+1}^{n_{i+1}},\ldots,x_{s}^{n_{s}})M$ for all positive integers
$n_{1},\ldots,n_{s}$ and all $i=1,\ldots,s-1$. According to D. T. Cuong and
the first author, if the parameter ideal $\mathfrak{q}$ is generated by a dd-
sequence on $M$, the adjusted Hilbert coefficients are described as follows.

######  Lemma 3.2 ([3, Theorem 6.2]).

Let $M$ be a sequentially generalized Cohen-Macaulay module and
$\mathfrak{q}=(x_{1},\ldots,x_{d})$ a system of parameters of $M$. Assume that
$x_{1},\ldots,x_{d}$ is a dd-sequence on $M$. Then the adjusted Hilbert
coefficients are of the form

$a_{d-d_{k}}(\mathfrak{q};M)=\sum\limits_{j=1}^{d_{k}}\binom{d_{k}-1}{j-1}\ell(H^{j}_{\mathfrak{m}}(M/D_{k}))$

for $k=0,\ldots,t-1$,

$a_{d-i}(\mathfrak{q};M)=\sum\limits_{j=1}^{i}\binom{i-1}{j-1}\ell(H^{j}_{\mathfrak{m}}(M/D_{k}))$

for $d_{k}<i<d_{k-1}$ and $a_{d}(\mathfrak{q};M)=0$.

Now with the above notations, we have the main result in this section.

######  Theorem 3.3.

Let $M$ be a sequentially generalized Cohen-Macaulay module and
$\mathcal{F}\in\mathcal{F}(M)$. Then the set $\mathcal{P}_{\mathcal{F}}(M)$ of
adjusted Hilbert-Samuel polynomials is finite.

######  Setting 3.4.

In this section, from now on, we assume that $M$ is a sequentially generalized
Cohen-Macaulay module. Set $W=H^{0}_{\mathfrak{m}}(M)$, $\overline{M}=M/W$,
and $\overline{N}=(N+W)/W$ for all submodules $N$ of $M$. Let
$\mathcal{F}\in\mathcal{F}(M)$, $\mathfrak{q}:=(x:=x_{1},x_{2},\ldots,x_{d})$
be a distinguished parameter ideal of $M$ with respect to $\mathcal{F}$.

Set

$I(M)=\sup\\{\ell(M/\mathfrak{q}M)-e_{0}(\mathfrak{q};M)\mid\mathfrak{q}\text{
is a parameter ideal of }M\\}.$

and
$I(\mathcal{F},M)=\displaystyle\sum\limits^{t-1}_{i=0}I(M_{i}/M_{i+1})+\ell(M_{t})$.

######  Fact 3.5.

With this notation, we have

  1. $i)$

The filtration of submodules
$\overline{\mathcal{F}}:=\\{\overline{M_{i}}\\}_{i=0}^{t}$ of $\overline{M}$
is a generalized Cohen-Macaulay filtration of $\overline{M}$ and

$I({\mathcal{F}},M)=I(\overline{\mathcal{F}},\overline{M})+\ell(W)$

([6, Lemma 6]).

  2. $ii)$

The module $M/xM$ is sequentially generalized Cohen-Macaulay and
$\mathcal{F}/x\mathcal{F}\in\mathcal{F}(M/xM)$ ([6, Corollary 2]). Moreover,
we have

$I(\mathcal{F}/x\mathcal{F},M/xM)\leq I(\mathcal{F},M)$

([6, Lemma 4]).

Now, let $\mathcal{S}=\bigoplus\limits_{n\geq 0}S_{n}$ be a standard
Noetherian graded ring and $E=\bigoplus\limits_{n\in\mathbb{Z}}E_{n}$ a
finitely generated graded $\mathcal{S}$-module. The Castelnouvo-Mumford
regularity reg$(E)$ of $E$ is defined by

$\operatorname{reg}(E)=\sup\\{n+i\mid[H^{i}_{S_{+}}(E)]_{n}\neq 0,i\geq 0\\}$

and simply called regularity, where $S_{+}=\bigoplus\limits_{n>0}S_{n}$. Let
$N$ be a finitely generated $R$-module and $Q$ a parameter ideal of $N$. We
always denote the associate graded module of $N$ with respect to $Q$ by
$G_{Q}(N)$, i.e $G_{Q}(N)=\bigoplus\limits_{n\geq 0}{Q^{n}N/Q^{n+1}N}$. With
this notation, we have

######  Fact 3.6.

  1. $i)$

There is a constant $C=C_{\mathcal{F}}$ such that

$\operatorname{reg}(G_{\mathfrak{q}}(M))\leq
C=(3I(\mathcal{F},M))^{d!}-2I(\mathcal{F},M),$

for all distinguished parameter ideals $\mathfrak{q}$ of $M$ with respect to
$\mathcal{F}$ ([6, Theorem 4]).

  2. $ii)$

We have $H^{ad}_{\mathfrak{q},M}(n)\geq 0$ for all

$n\geq\operatorname{reg}(G_{\mathfrak{q}}(M))+\binom{\operatorname{reg}(G_{\mathfrak{q}}(M))+d-1}{d-1}I(\mathcal{F},M)+d,$

([15, Theorem 4.4]).

The following result give an upper bound for the adjusted Hilbert-Samuel
function.

######  Lemma 3.7.

We have

$H^{ad}_{\mathfrak{q},M}(n)\leq\sum\limits^{t-1}_{i=0}\binom{n+d_{i}-1}{d_{i}-1}I(M_{i}/M_{i+1})+\ell(M_{t})-\ell(W)),$

for all $n\geq 0$,

###### Proof.

Note that from the following exact sequence

$0\rightarrow(\mathfrak{q}^{n+1}M\cap
M_{1})/{\mathfrak{q}^{n+1}M_{1}\longrightarrow
M_{1}/\mathfrak{q}^{n+1}M_{1}\longrightarrow
M/\mathfrak{q}^{n+1}M\longrightarrow}M/\mathfrak{q}^{n+1}M+M_{1}\rightarrow
0,$

we get
$\ell(M/\mathfrak{q}^{n+1}M)\leq\ell(M/\mathfrak{q}^{n+1}M+M_{1})+\ell(M_{1}/\mathfrak{q}^{n+1}M_{1})$.

Now we argue by the induction on the length $t$ of the dimension filtration of
$M$. The case $t=1,$ it follows from $\mathfrak{q}M_{1}=0$ and Lemma 1.1 in
[16] that we have

$\displaystyle\ell(M/\mathfrak{q}^{n+1}M)$
$\displaystyle\leq\ell(M/\mathfrak{q}^{n+1}M+M_{1})+\ell(M_{1})$
$\displaystyle\leq\binom{n+d}{d}e_{0}(\mathfrak{q};M)+\binom{n+d-1}{d-1}I(M/M_{1})+\ell(M_{1}).$

for all $n\geq 0$.

Now, assume that $t>1$ and that our assertion holds true for $t-1$. By the
inductive hypothesis, we have

$\displaystyle\ell(M/\mathfrak{q}^{n+1}M)$
$\displaystyle\leq\ell(M/\mathfrak{q}^{n+1}M+M_{1})+\ell(M_{1}/\mathfrak{q}^{n+1}M_{1})$
$\displaystyle\leq\binom{n+d}{d}e_{0}(\mathfrak{q};M_{0})+\binom{n+d-1}{d-1}I(M/M_{1})+\ell(M_{1}/\mathfrak{q}^{n+1}M_{1})$
$\displaystyle\leq\sum\limits^{t-1}_{i=0}\binom{n+d_{i}}{d_{i}}e_{0}(\mathfrak{q};M_{i})+\sum\limits^{t-1}_{i=0}\binom{n+d_{i}-1}{d_{i}-1}I({M}_{i}/M_{i+1})+\ell(M_{t}),$

for all $n\geq 0$. By Lemma 2.5, we have

$H^{ad}_{\mathfrak{q},M}(n)\leq\displaystyle\sum\limits^{t-1}_{i=0}\binom{n+d_{i}-1}{d_{i}-1}I({M}_{i}/M_{i+1})+\ell(M_{t})-\ell(W),$

for all $n\geq 0$, as requested. ∎

######  Theorem 3.8.

Let $C=C_{\mathcal{F}}=(3I(\mathcal{F},M))^{d!}-2I(\mathcal{F},M)$ as in Fact
3.6 i). Then we have

  1. $i)$

$\mid e_{1}(\mathfrak{q};M)+\operatorname{adeg}_{d-1}(\mathfrak{q};M)\mid\leq
I(M/M_{1}).$

  2. $ii)$

$\mid(-1)^{i}e_{i}(\mathfrak{q};M)-\operatorname{adeg}_{d-i}(\mathfrak{q};M)\mid\leq
2^{i-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{i-1}I(\mathcal{F},M)$
for all $i=2,\ldots,d-1.$

  3. $iii)$

$\mid e_{d}(\mathfrak{q};M)\mid\leq
2^{d-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M).$

###### Proof.

$i)$ By Fact 3.6 ii), we have
$P^{ad}_{\mathfrak{q},M}(n)=H^{ad}_{\mathfrak{q},M}(n)\geq 0$ for all $n\gg
0$. Thus by Lemma 3.7, we have

$\displaystyle 0\leq P^{ad}_{\mathfrak{q},M}(n)$
$\displaystyle=\binom{n+d-1}{d-1}((-1)^{d-1}e_{1}(\mathfrak{q};M)-\operatorname{adeg}_{d-1}(\mathfrak{q};M))+\text{
lower terms }$ $\displaystyle\leq\binom{n+d-1}{d-1}I(M/M_{1})+\text{ lower
terms }.$

Therefore we have

$0\leq(-1)e_{1}(\mathfrak{q};M)-\operatorname{adeg}_{d-1}(\mathfrak{q};M)\leq
I(M/M_{1}),$

as requested.

$ii)$ and $iii)$. Now we proceed by induction on $d$ to show that $ii)$ and
$iii)$. Set $r=\operatorname{reg}(G_{\mathfrak{q}}(M))$. The case $d=2$, we
have

$e_{2}(\mathfrak{q};M)=H^{ad}_{\mathfrak{q},M}(n)+\ell(H^{0}_{\mathfrak{m}}(M))+(e_{1}(\mathfrak{q};M)+\operatorname{adeg}_{1}(\mathfrak{q};M))\binom{n+1}{1}$

for all $n\geq r+1$. Thus by Lemma 3.7 and Theorem 3.6 we have

$\displaystyle\mid e_{2}(\mathfrak{q};M)\mid$
$\displaystyle\leq\sum\limits^{t}_{i=0}\binom{n+d_{i}-1}{d_{i}-1}I({M}_{i}/M_{i+1})+I(M/M_{1})(n+1)$
$\displaystyle\leq 2(n+1)^{2-1}I(\mathcal{F},M).$

for all $n\geq r+\binom{r+1}{1}I(\mathcal{F},M)+d$.
Now choose $n=(C+1)^{2-1}I(\mathcal{F},M)+d+C+1$. Then $n\geq
r+\binom{r+1}{1}I(\mathcal{F},M)+d$ by Fact 3.6 i). Therefore we have

$\displaystyle\mid e_{2}(\mathfrak{q};M)\mid$ $\displaystyle\leq
2^{2-1}\left((C+1)^{2-1}I(\mathcal{F},M)+d+C+1+1\right)I(\mathcal{F},M)$
$\displaystyle=2^{d-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M),$

as required.

Now suppose that $d>2$ and that our assertion holds true for $d-1$. We have by
Lemma 2.7$(ii)$ that

$e_{i}(\mathfrak{q};M)=e_{i}(\mathfrak{q};\overline{M})$

for all $i=0,\ldots,d-1$ and
$(-1)^{d}e_{d}(\mathfrak{q};M)=(-1)^{d}e_{d}(\mathfrak{q};\overline{M})+\ell(W)$.
Moreover, by Fact 3.5$(i)$ we have

$I(\overline{\mathcal{F}},\overline{M})+\ell(W)=I(\mathcal{F},M).$

Consequently, $C_{\mathcal{F}}\geq C_{\overline{\mathcal{F}}}$. Therefore we
can assume $W=0$. Recall that $\underline{x}=x_{1},\ldots,x_{d}$ is a
distinguished system of parameters of $M$ with respect to $\mathcal{F}$ such
that $\mathfrak{q}=(\underline{x})$ and $x=x_{1}$ is a superficial element of
$M$ for $\mathfrak{q}$. Hence $x$ is a regular element of $M$ since $W=0$. So
by Lemma 2.7$(i)$,

$e_{i}(\mathfrak{q};M)=e_{i}(\mathfrak{q};M/xM)\hskip 142.26378pt(1)$

for all $i=0,\ldots,d-1$. On the other hand, it follows from Lemma 2.6 and
Fact 3.5 $ii)$ that we have

$\operatorname{adeg}_{d-i}(\mathfrak{q};M)=\operatorname{adeg}_{d-1-i}(\mathfrak{q};M/xM)\hskip
85.35826pt(2)$

for all $i=0,\ldots,d-2$ and

$I(\mathcal{F}/x\mathcal{F},M/xM)\leq I(\mathcal{F},M).\hskip 142.26378pt(3)$

Set $C_{x}=C_{\mathcal{F}/x\mathcal{F}}$. It follows that

$C_{x}=C_{\mathcal{F}/x\mathcal{F}}\leq C_{\mathcal{F}}=C.\hskip
142.26378pt(4)$

Note that $x_{2},\ldots,x_{d}$ is a distinguished system of parameters of
sequentially generalized Cohen-Macaulay $M/xM$ with respect to
$\mathcal{F}/x\mathcal{F}\in\mathcal{F}(M/xM)$. Therefore it follows from the
inductive hypothesis and $(1)-(4)$ that we have

$\displaystyle\mid(-1)^{i}e_{i}(\mathfrak{q};M)$
$\displaystyle-\operatorname{adeg}_{d-i}(\mathfrak{q};M)\mid$
$\displaystyle=\mid(-1)^{i}e_{i}(\mathfrak{q};M/xM)-\operatorname{adeg}_{d-1-i}(\mathfrak{q};M/xM)\mid$
$\displaystyle\leq
2^{i-1}\left((C_{x}+1)^{d-2}I(\mathcal{F}/x\mathcal{F},M/xM)+(d-1)+C_{x}+2\right)^{i-1}I(\mathcal{F}/x\mathcal{F},M/xM)$
$\displaystyle\leq
2^{i-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2+I(\mathcal{F},M)\right)^{i-1}I(\mathcal{F},M)$

for all $i=2,\ldots,d-2$.

Fact, if $d_{t-1}>1$ then $\operatorname{adeg}_{1}(\mathfrak{q};M)=0$ by Lemma
2.5. So

$\displaystyle\mid(-1)^{d-1}$ $\displaystyle
e_{d-1}(\mathfrak{q};M)-\operatorname{adeg}_{1}(\mathfrak{q};M)\mid$
$\displaystyle=\mid(-1)^{d-1}e_{d-1}(\mathfrak{q};M/xM)\mid$
$\displaystyle\leq
2^{(d-1)-1}\left((C_{x}+1)^{d-2}I(\mathcal{F}/x\mathcal{F},M/xM)+(d-1)+C_{x}+2\right)^{(d-1)-1}I(\mathcal{F}/x\mathcal{F},M/xM)$
$\displaystyle\leq
2^{d-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M).$

The last inequality is followed by $(3)$ and $(4)$.

Now we can assume that $d_{t-1}=1$. Set $\widetilde{M}=M/D_{t-1}$ and
$\widetilde{N}=(N+D_{t-1})/D_{t-1}$ for all submodules $N$ of $M$. Then
$\widetilde{M}$ is sequentially generalized Cohen-Macaulay,
$\widetilde{\mathcal{F}}=\\{\widetilde{M_{i}}\\}_{i=0}^{t-1}\in\mathcal{F}(\widetilde{M})$
and $\mathfrak{q}$ is also a distinguished parameter ideal of $\widetilde{M}$
with respect to $\widetilde{\mathcal{F}}$. Note that $x_{1},\ldots,x_{d}$ is a
distinguished system of parameters of sequentially generalized Cohen-Macaulay
$\widetilde{M}$ with respect to $\widetilde{\mathcal{F}}$. Using similar above
arguments, it follows from $H^{0}_{\mathfrak{m}}(\widetilde{M})=0$ that we
show that results $1)-4)$ for module $\widetilde{M}$.

By Fact 3.6 i), we have

$\operatorname{reg}(G_{\mathfrak{q}}(\widetilde{M}))\leq\widetilde{C}:=(3I(\widetilde{\mathcal{F}},\widetilde{M}))^{d!}-2I(\widetilde{\mathcal{F}},\widetilde{M}).$

Since $(M_{i}\cap D_{t-1})/(M_{i+1}\cap D_{t-1})=(M_{i+1}+M_{i}\cap
D_{t-1})/M_{i+1}$ is a submodule of the module $D_{t-1}/M_{t-1}$ of finite
length for all $i=0,...,t-2$, we have

$\displaystyle I(\mathcal{F},M)=$
$\displaystyle\sum\limits^{t-2}_{i=0}I(M_{i}/M_{i+1})+I(M_{t-2}/M_{t})+\ell(M_{t})$
$\displaystyle\geq\sum\limits^{t-2}_{i=0}\left(I(M_{i}/M_{i+1})-l((M_{i}\cap
D_{t-1})/(M_{i+1}\cap D_{t-1}))\right)+I(M_{t-1})$
$\displaystyle=\sum\limits^{t-2}_{i=0}I((M_{i}+D_{t-1})/(M_{i+1}+D_{t-1}))+I(M_{t-1})$
$\displaystyle=I(\widetilde{\mathcal{F}},\widetilde{M})+I(M_{t-1}).$

This implies that $\widetilde{C}\leq C$. We have by Lemma 2.7$(ii)$ and Lemma
2.5 that

$(-1)^{d-1}e_{d-1}(\mathfrak{q},M)-\operatorname{adeg}_{1}(\mathfrak{q},M)=(-1)^{d-1}e_{d-1}(\mathfrak{q},\widetilde{M}).$

Therefore it follows from the inductive hypothesis and similar results $1)-4)$
for module $\widetilde{M}$ that we have

$\displaystyle\mid(-1)^{d-1}e_{d-1}(\mathfrak{q};M)-$
$\displaystyle\operatorname{adeg}_{1}(\mathfrak{q};M)\mid=\mid(-1)^{d-1}e_{d-1}(\mathfrak{q};\widetilde{M})\mid=\mid(-1)^{d-1}e_{d-1}(\mathfrak{q};\widetilde{M}/x\widetilde{M})\mid$
$\displaystyle\leq
2^{d-2}\left((\widetilde{C}_{x}+1)^{d-1}I(\widetilde{\mathcal{F}}/x\widetilde{\mathcal{F}},\widetilde{M}/x\widetilde{M})+d-1+\widetilde{C}_{x}+2\right)^{d-1}I(\widetilde{\mathcal{F}}/x\widetilde{\mathcal{F}},\widetilde{M}/x\widetilde{M})$
$\displaystyle\leq
2^{d-1}\left((\widetilde{C}+1)^{d-1}I(\widetilde{\mathcal{F}},\widetilde{M})+d+\widetilde{C}+2\right)^{d-1}I(\widetilde{\mathcal{F}},\widetilde{M})$
$\displaystyle\leq
2^{d-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M)$

where
$\widetilde{C}_{x}=C_{\widetilde{\mathcal{F}}/x\widetilde{\mathcal{F}}}$.

Now, we have

$(-1)^{d}e_{d}(\mathfrak{q};M)=H^{ad}_{\mathfrak{q},M}(n)-\sum\limits_{i=1}^{d-1}\left((-1)^{i}e_{i}(\mathfrak{q};M)-\operatorname{adeg}_{d-i}(\mathfrak{q};M)\right)\binom{n+d-i}{d-i},$

for all $n\geq r+1$. Furthermore, for all $n\geq
r+\binom{r+1}{1}I(\mathcal{F},M)+d$, by Lemma 3.7 and Fact 3.6, we have

$\displaystyle\mid e_{d}(\mathfrak{q};M)\mid\leq$
$\displaystyle\sum\limits^{t-1}_{i=0}\binom{n+d_{i}-1}{d_{i}-1}I({M}_{i}/M_{i+1})+\mid(-1)e_{1}(\mathfrak{q};M)-\operatorname{adeg}_{d-1}(\mathfrak{q};M)\mid\binom{n+d-1}{d-1}$
$\displaystyle+\sum\limits_{i=2}^{d-1}\mid(-1)^{i}e_{i}(\mathfrak{q};M)-\operatorname{adeg}_{d-i}(\mathfrak{q};M)\mid\binom{n+d-i}{d-i}$
$\displaystyle\leq\sum\limits^{t-1}_{i=0}(n+1)^{d-1}I({M}_{i}/M_{i+1})+(n+1)^{d-1}I(\mathcal{F},M)$
$\displaystyle+\sum\limits_{i=2}^{d-1}2^{i-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{i-1}I(\mathcal{F},M)(n+1)^{d-i},$

where the last inequality is followed by $i)$, $ii)$. Choose
$n=(C+1)^{d-1}I(\mathcal{F},M)+d+C+1$ and note that $n\geq
r+\binom{r+d-1}{d-1}I(\mathcal{F},M)+d$. Then we have

$\displaystyle\mid e_{d}(\mathfrak{q};M)\mid$ $\displaystyle\leq
2\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M)$
$\displaystyle+\sum\limits_{i=2}^{d-1}2^{i-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M)$
$\displaystyle=2^{d-1}\left((C+1)^{d-1}I(\mathcal{F},M)+d+C+2\right)^{d-1}I(\mathcal{F},M).$

as required. ∎

Now, we are in a position to prove the main theorem in this section.

###### Proof of Theorem 3.3.

This is now immediately seen from Theorem 3.8. ∎

We close this section with the following example, which shows that the
condition $\mathcal{F}\in\mathcal{F}(M)$ in Theorem 3.3 and 3.8 can not be
omitted whenever $M$ is sequentially Cohen-Macaulay.

######  Example 3.9.

Let $R=k[[X,Y,Z]]$ be the formal power series ring over a field $k$. Let
$M=k[[X,Y,Z]]\oplus(k[[X,Y,Z]]/(Z^{2}))$ be $R$-module. For an integer $m\geq
1$, set $\mathfrak{q}_{m}=(X^{m},Y^{m},Z)$. Then we have the following.

  1. $i)$

$M$ is sequentially Cohen-Macaulay of dimension $2$.

  2. $ii)$

$\mathfrak{q}$ is a parameter ideal of $M$ but $\mathfrak{q}$ is not a
distinguished parameter ideal with respect to $F\in\mathcal{F}(M)$.

  3. $iii)$

$H^{ad}_{\mathfrak{q}_{m},M}(n)=-m^{2}\binom{n+1}{1}$ for all $m,n>1$. Hence
the set $\Lambda_{1}(M)$ is finite but the set $\mathcal{P}_{\mathcal{F}}(M)$
and $\Lambda_{2}(M)$ is infinite.

##  4\. Characterization of the finiteness of the set
$\mathcal{P}_{\mathcal{D}}(M)$

In this section, we give a characterization of sequentially generalized Cohen-
Macaulay modules. In particular, we have

######  Theorem 4.1.

Let $R$ be a homomorphic image of a Cohen-Macaulay local ring. Then the
following statements are equivalent:

  1. $i)$

$M$ is sequentially generalized Cohen-Macaulay.

  2. $ii)$

The set $\mathcal{P}_{\mathcal{F}}(M)$ is finite for all
$\mathcal{F}\in\mathcal{F}(M)$.

  3. $iii)$

The set $\mathcal{P}_{\mathcal{F}}(M)$ is finite for some
$\mathcal{F}\in\mathcal{F}(M)$.

  4. $iv)$

The set $\mathcal{P}_{\mathcal{D}}(M)$ is finite.

######  Setting 4.2.

In this section, from now on, we assume that $R$ is a homomorphic image of a
Cohen-Macaulay local ring. Let
$\mathcal{F}=\\{M_{i}\\}_{i=0}^{t}\in\mathcal{F}(M)$ and
$\underline{x}=x_{1},\ldots,x_{d}$ be a distinguished system of parameters of
$M$.

######  Notation 4.3.

  1. $\bullet$

Set $\mathfrak{q}:=(x:=x_{1},x_{2},\ldots,x_{d})$ and
$\mathfrak{q}_{i}=(x_{1},\ldots,x_{i})$ for all $i=1,\ldots,d$ Let
$\underline{n}=(n_{1},\ldots,n_{d})\in\mathbb{N}^{d}$
$\underline{x}_{i}^{\underline{n}_{i}}=x_{i+1}^{n_{i+1}},\ldots,x_{d}^{n_{d}}$
and
$\mathfrak{q}_{i}^{\underline{n}_{i}}=(x_{1}^{n_{1}},\ldots,x_{i}^{n_{i}})$.
We set

$\mathcal{N}(\underline{x};M)=\\{\underline{n}\in\mathbb{N}^{d}\mid\underline{x}_{i}^{\underline{n}_{i}}\text{
is a distinguished system of parameters of
}M/\mathfrak{q}_{i}^{\underline{n}_{i}}M\text{ for all }0\leq i\leq d-1\\}.$

  2. $\bullet$

Let $d_{i+1}<k\leq d_{i}$ for some $i=0,\ldots,t-1$. Then
$\mathcal{F}/\mathfrak{q}_{k}\mathcal{F}:=\\{(M_{i}+\mathfrak{q}_{k}M)/\mathfrak{q}_{k}M\\}_{i=0}^{s}$
is a filtration of submodules of $M$ satisfying the dimension condition. The
length $s$ of a filtration of submodules
$\mathcal{F}/\mathfrak{q}_{k}\mathcal{F}$ of $M$ is $i$, if $k=d_{i}$ and
$i+1$, otherwise. Furthermore, $x_{k+1},\ldots.,x_{d}$ is a distinguished
system of parameters of $M/\mathfrak{q}_{k}M$ with respect to
$\mathcal{F}/\mathfrak{q}_{k}\mathcal{F}$. Stipulating $x_{0}=0$ and
$\mathcal{F}_{0}=\mathcal{F}$,

######  Lemma 4.4.

Suppose that $d\geq 2$ and $\underline{x}$ is a distinguished system of
parameters of $M$ such that $\mathcal{N}(\underline{x};M)\neq\emptyset$. Then
we have

  1. $i)$

$\underline{x}^{\underline{n}}$ is a d-sequence on $M$ for all
$\underline{n}\in\mathcal{N}(\underline{x};M)$.

  2. $ii)$

$(n_{1},n_{2}m_{2},\ldots,n_{d}m_{d})\in\mathcal{N}(\underline{x};M)$ for all
$(m_{2},\ldots,m_{d})\in\mathcal{N}(\underline{x}_{1}^{\underline{n}_{1}};M/x_{1}^{n_{1}}M)$.

  3. $iii)$

If $M$ is a sequentially generalized Cohen-Macaulay module, then there is an
$\underline{n}\in\mathcal{N}(\underline{x};M)$ such that
$\underline{x}^{\underline{n}}$ is a dd-sequence on $M$.

###### Proof.

$i)$. Let $\underline{n}=(n_{1},\ldots,n_{d})\in\mathcal{N}(\underline{x};M)$.
Then $\underline{x}_{i-1}^{\underline{n}_{i-1}}$ is a distinguished system of
parameters of $M/\mathfrak{q}_{i-1}^{\underline{n}_{i-1}}M$. Thus
$(\underline{x}_{i-1}^{\underline{n}_{i-1}})H^{0}_{\mathfrak{m}}(M/\mathfrak{q}_{i-1}^{\underline{n}_{i-1}}M)=0.$
Therefore $\underline{x}^{\underline{n}}$ is a filter regular $M$-sequence.
Moreover,

$\mathfrak{q}_{i-1}^{\underline{n}_{i-1}}M:x_{i}^{n_{i}}=H^{0}_{\mathfrak{m}}(M/\mathfrak{q}_{i-1}^{\underline{n}_{i-1}}M)=\bigcup\limits_{n=1}^{\infty}(\mathfrak{q}_{i-1}^{\underline{n}_{i-1}}M:(\underline{x}^{\underline{n}})^{n})$

for all $i=1,\ldots,d$. So $\underline{x}^{\underline{n}}$ is a d-sequence by
[21, Theorem 1.1$(vii)$].

$ii)$. This is now immediately seen from the definition of the distinguished
system of parameters.

$iii)$. By the Artin-Rees Lemma, there are positive integers
$m_{1},\ldots,m_{d}$ such that $x_{1}^{m_{1}},\ldots,x_{d}^{m_{d}}$ is a good
system of parameters of $M$. We have by [3, Theorem 3.8, Corollary 3.9] that
there are positive integers $k_{1},\ldots,k_{d}$ such that
$x_{1}^{n_{1}},\ldots,x_{d}^{n_{d}}$ is a dd-sequence on $M$, where
$n_{i}=m_{i}k_{i}$ for all $i=1,\ldots,d$. We have
$(n_{1},\ldots,n_{d})\in\mathcal{N}(\underline{x};M)$ by [2, Corollary 3.7]
and the definition of dd-sequence. ∎

######  Lemma 4.5.

There is a distinguished system of parameters
$\underline{x}=x_{1},\ldots,x_{d}$ of $M$ such that
$\mathcal{F}/\mathfrak{q}_{i}\mathcal{F}\in\mathcal{F}(M/\mathfrak{q}_{i}M)$
for all $\mathcal{F}\in\mathcal{F}(M)$ and $i=1,\ldots,d$. Moreover,
$\mathcal{N}(\underline{x};M)\neq\emptyset$.

###### Proof.

The first assertion is now immediately seen from Lemma 2.7 in [5].
Now we will prove that $\mathcal{N}(\underline{x};M)\neq\emptyset$ by
induction on $d$. The case $d=1$ is obvious. Assume that $d>1$. Then
$x_{2},\ldots,x_{d}$ is a distinguished system of parameters of $M/x_{1}M$
with respect to $\mathcal{D}/{x_{1}}\mathcal{D}$. Since
$\mathcal{D}/{x_{1}}\mathcal{D}\in\mathcal{F}(M/x_{1}M)$, there are positive
integers $n_{2},\ldots,n_{d}$ such that $x_{2}^{n_{2}},\ldots,x_{d}^{n_{d}}$
is a distinguished system of parameters of $M/x_{1}M$. By the inductive
hypothesis, we have
$(m_{2},\ldots,m_{d})\in\mathcal{N}(x_{2}^{n_{2}},\ldots,x_{d}^{n_{d}};M/x_{1}M)$
for some $m_{i}$. Therefore we have
$(1,n_{2}m_{2},\ldots,n_{d}m_{d})\in\mathcal{N}(\underline{x};M)$, as
requested. ∎

######  Lemma 4.6.

Assume that $d\geq 2$ and
$\mathcal{F}=\\{M_{i}\\}_{i=0}^{t}\in\mathcal{F}(M)$. Then,
$H^{1}_{\mathfrak{m}}(M/M_{i+1})$ are of finite length for $i=0,1,...,t-1$ and
$d_{i}\geq 2$.

###### Proof.

We first see that
$\operatorname{Ass}_{R}(M_{i}/M_{i+1})=\operatorname{Assh}_{R}(M_{i}/M_{i+1})\cup\\{\mathfrak{m}\\}$.
It follows by [12, Lemm 3.1] that $H^{1}_{\mathfrak{m}}(M_{i}/M_{i+1})$ are of
finite length for all $i=0,\ldots,t-1$ and $d_{i}\geq 2$. Now we will prove by
induction on $i$. If $i=0$, clearly that $H^{1}_{\mathfrak{m}}(M/M_{1})$ is of
finite length. Assume that $i>0$ and that assertion holds true for $i-1$. Now
the short exact sequence

$0\longrightarrow M_{i}/M_{i+1}\longrightarrow M/M_{i+1}\longrightarrow
M/M_{i}\longrightarrow 0$

induces the long exact sequence

$\ldots\longrightarrow H^{1}_{\mathfrak{m}}(M_{i}/M_{i+1})\longrightarrow
H^{1}_{\mathfrak{m}}(M/M_{i+1})\longrightarrow
H^{1}_{\mathfrak{m}}(M/M_{i})\longrightarrow\ldots.$

If $d_{i}\geq 2$ then $d_{i-1}>d_{i}\geq 2$. Thus
$H^{1}_{\mathfrak{m}}(M/M_{i})$ is of finite length by induction. So
$H^{1}_{\mathfrak{m}}(M/M_{i+1})$ is of finite length. ∎

######  Theorem 4.7.

Assume that $d\geq 2$. Suppose that there is an integer $C$ and a
distinguished parameter ideal $\mathfrak{q}=(x_{1},\ldots,x_{d})$ of $M$ as in
Lemma 4.5 such that

$\mid a_{i}(\mathfrak{q}^{\underline{n}};M)\mid\leq C$

for all $\underline{n}\in\mathcal{N}(\underline{x};M)$ and $i=0,\ldots,d-1$.
Then we have

$\mathfrak{m}^{C}H^{j}_{\mathfrak{m}}(M/D_{k+1})=0$

for all $j=1,\ldots,d_{k}-1,d_{k}\geq 2$ and $k=0,\ldots,t-1$.

###### Proof.

Let $\underline{n}\in\mathcal{N}(\underline{x};M)$. We have by Lemma 2.7$(ii)$
and Lemma 2.5$(i)$ that

$e_{i}(\underline{x}^{\underline{n}};M)=e_{i}(\underline{x}^{\underline{n}};M/W),$
$\operatorname{adeg}_{d-i}(\underline{x}^{\underline{n}};M)=\operatorname{adeg}_{d-i}(\underline{x}^{\underline{n}};M/W)$

for all $i=0,\ldots,d-1$, where $W=H^{0}_{\mathfrak{m}}(M)$. Thus we can
assume that $W=0$.

Now we argue by the induction on the dimension $d$ of $M$. In the case $d=2$,
$M/D_{1}$ is a generalized Cohen-Macaulay module by Lemma 4.6. Therefore $M$
is sequentially generalized Cohen-Macaulay. By Lemma 4.4, $iii)$, there is a
$(n_{1},n_{2})\in\mathcal{N}(\underline{x};M)$ such that
$x_{1}^{n_{1}},x_{2}^{n_{2}}$ is a dd-sequence on $M$. Therefore, by Lemma
3.2, we have

$\ell(H_{\mathfrak{m}}^{1}(M/D_{1}))=\mid(-1)e_{1}(x_{1}^{n_{1}},x_{2}^{n_{2}};M)-\operatorname{adeg}_{1}(x_{1}^{n_{1}},x_{2}^{n_{2}};M)\mid\leq
C.$

Hence $\mathfrak{m}^{C}H_{\mathfrak{m}}^{1}(M/D_{1})=0$.

Fact, suppose that $d>2$ and our assertion holds true for $d-1$. Since
$\underline{n}^{\prime}=(n_{1},n_{2}m_{2}\ldots,n_{d}m_{d})\in\mathcal{N}(\underline{x};M)$
for all
$\underline{m}=(m_{2},\ldots,m_{d})\in\mathcal{N}(\underline{x}_{1}^{\underline{n}_{1}};M/x_{1}^{n_{1}}M)$
by Lemma 4.4$(ii)$,
$\underline{x}^{\underline{n}^{\prime}}=x_{1}^{n_{1}},x_{2}^{n_{2}m_{2}}\ldots,x_{d}^{n_{d}m_{d}}$
is a d-sequence on $M$ because of Lemma 4.4$(i)$. Set $y=x_{1}^{n_{1}}$ and
$\underline{y}=x_{2}^{n_{2}}\ldots,x_{d}^{n_{d}}$. Therefore $y$ is a
superficial of $M$ for ideals $(y,\underline{y}^{\underline{m}})$ for all
$\underline{m}\in\mathcal{N}(\underline{y};M/yM)$. It follows by Lemma
2.7$(i)$ and Lemma 2.6 that

$\displaystyle\mid(-1)^{i}e_{i}(\underline{y}^{\underline{m}};M/yM)-\operatorname{adeg}_{d-i-1}(\underline{y}^{\underline{m}};M/yM)\mid$
$\displaystyle=\mid(-1)^{i}e_{i}(y,\underline{y}^{\underline{m}};M)-\operatorname{adeg}_{d-i}(y,\underline{y}^{\underline{m}};M)\mid$
$\displaystyle=\mid(-1)^{i}e_{i}(\underline{x}^{\underline{n}^{\prime}};M)-\operatorname{adeg}_{d-i}(\underline{x}^{\underline{n}^{\prime}};M)\mid\leq
C,$

for all $i=0,\ldots,d-2$ and for all
$\underline{m}\in\mathcal{N}(\underline{y};M/yM)$.

Now set $\overline{M}=M/yM$ and and $\overline{N}=(N+yM)/yM$ for all
submodules $N$ of $M$. Let
$\mathcal{D}^{\prime}=\\{D_{i}^{\prime}\\}_{i=0}^{s}$ be the dimension
filtration of $\overline{M}$, where $s$ is $t$ if $d_{t-1}>1$ and $t-1$,
otherwise. By the inductive hypothesis, we have

$\mathfrak{m}^{C}H^{j}_{\mathfrak{m}}(\overline{M}/D^{{}^{\prime}}_{i+1})=0$

for all $\dim D^{{}^{\prime}}_{i}=d_{i}-1\geq 2$ and $j=1,\ldots,d_{i}-2$.
Since $\mathcal{D}/{y}\mathcal{D}\in\mathcal{F}(M/yM)$ by Lemma 4.5, we have
$\ell(D_{i+1}^{{}^{\prime}}/\overline{D}_{i+1})<\infty$ for all
$i=0,\ldots,s-1.$ Therefore we get
$H^{j}_{\mathfrak{m}}(\overline{M}/D^{{}^{\prime}}_{i+1})\cong
H^{j}_{\mathfrak{m}}(\overline{M}/\overline{D}_{i+1}))$ for all $j\geq 1$ and
$i=0,\ldots,s-1$. This implies that

$\mathfrak{m}^{C}H^{j}_{\mathfrak{m}}(M/(yM+D_{i+1}))=0$

for all $d_{i}\geq 3$ and $j=1,\ldots,d_{i}-2$. Since $y$ is
$M/D_{i+1}$-regular for all $i=0,\ldots,t-1$, it follows from short exact
sequences

$0\longrightarrow\frac{M}{D_{i+1}}\xrightarrow{y^{n}}\frac{M}{D_{i+1}}\longrightarrow\frac{M}{D_{i+1}+y^{n}M}\longrightarrow
0$

that long sequences

$\ldots\longrightarrow
H^{j-1}_{\mathfrak{m}}(\frac{M}{D_{i+1}+y^{n}M})\longrightarrow
H^{j}_{\mathfrak{m}}(\frac{M}{D_{i+1}})\xrightarrow{y^{n}}H^{j}_{\mathfrak{m}}(\frac{M}{D_{i+1}})\longrightarrow\ldots$

are exact for all $n\geq 1$. Thus we have

$\mathfrak{m}^{C}(0:_{H^{j}_{\mathfrak{m}}({M}/{D_{i+1}})}y^{n})=0$

for all $j=2,\ldots,d_{i}-1$, $d_{i}\geq 3$ and $n\geq 1$. Note that $n$ and
$C$ are independent of each other. Hence

$\mathfrak{m}^{C}H^{j}_{\mathfrak{m}}({M}/{D_{i+1}})=0$

for all $j=2,\ldots,d_{i}-1$ and $d_{i}\geq 3$. Moreover,
$\ell(H^{1}_{\mathfrak{m}}(M/D_{i+1}))<\infty$ by Lemma 4.6 for all $d_{i}\geq
2$. So $M$ is a sequentially generalized Cohen-Macaulay module by [3,
Proposition 3.5]. Therefore there is a
$\underline{n}=(n_{1},\ldots,n_{d})\in\mathcal{N}(\underline{x};M)$ such that
$\underline{x}^{\underline{n}}$ is a dd-sequence on $M$. By Lemma 3.2, we
havet $\ell(H^{1}_{\mathfrak{m}}(M/D_{i+1}))\leq C$ for all $d_{i}\geq 2$. The
proof is completed. ∎

###### Proof of Theorem 4.1.

$i)\Rightarrow ii)$ is followed by Theorem 3.3.
$ii)\Rightarrow iii)$ is trivial.
$iii)\Rightarrow iv)$ is followed by
$\mathcal{P}_{\mathcal{D}}(M)\subseteq\mathcal{P}_{\mathcal{F}}(M)$ for all
$\mathcal{F}\in\mathcal{F}(M)$.
$iv)\Rightarrow i)$. Let $\underline{x}=x_{1},\ldots,x_{d}$ be a distinguished
system of parameters of $M$ as in Lemma 4.5. Since
$\mathcal{P}_{\mathcal{D}}(M)$ is finite, so is
$\\{P^{ad}_{\underline{x}^{\underline{n}},M}(n)\mid\underline{n}\in\mathcal{N}(\underline{x};M)\\}$.
We have by Theorem 4.7 and [3, Proposition 3.5] that $M$ is sequentially
generalized Cohen-Macaulay, as required. ∎

## References

  * [1] D. Bayer and D. Mumford, What can be computed on algebraic geometry?, Computational Algebraic Geometry and Commutative algebra, Proceedings. Cortona 1991(D. Eisenbud and L. Robbiano. Eds), Cambridge University Press, 1993, pp. 1-48.
  * [2] N.T. Cuong, D.T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J. 30 (2007) 409-428.
  * [3] N. T. Cuong and D. T. Cuong, On the structure of sequentially generalized Cohen-Macaulay modules, J. Algebra 317 (2007), 714-742.
  * [4] N. T. Cuong and D. T. Cuong dd-sequences and partial Euler-Poincaré characteristics of Koszul complex, Vietnam J. Math. 31 (2003), 353-358.
  * [5] N. T. Cuong, S. Goto and H. L. Truong, The Hilbert coefficients and sequentially Cohen-Macaulay module, J. Pure Appl. Algebra, 217 (2013), 470-480.
  * [6] N. T. Cuong, N. T. Long and H. L. Truong, Uniform bounds in Sequentially generalized Cohen - Macaulay Modules, Vietnam J. Math. 43 (2015), 343-356.
  * [7] N. T. Cuong and L. T. Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra, 267 (2003), 156-177.
  * [8] N. T. Cuong, P. Schenzel, N.V. Trung, Verallgemeinerte Cohen-Macaulay moduln, Math. Nachr. 85 (1978), 57-79.
  * [9] N. T. Cuong and H. L. Truong Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules, Proc. Am. Math. Soc. 137 (2009), 19-26.
  * [10] L. Ghezzi, S. Goto, J.-Y. Hong. K. Ozeki, T. T. Phuong, and W. V. Vasconcelos. Cohen-Macaulayness versus vanishing of the first Hilbert coefficient of parameter ideals, J. London Math. Soc. 81 (2010), 679-695.
  * [11] S. Goto and K. Ozeki, Uniform bounds for Hilbert coefficients of parameters, Commutative algebra and its connections to geometry, Contemp. Math., 555, Am. Math. Soc., Providence, RI, (2011), 97-118.
  * [12] S. Goto, Y. Nakamura, Multiplicity and Tight Closures of Parameters, J. Algebra 244 (2001),302-311.
  * [13] C. Huneke, The theory of d-sequences and powers of ideals, Adv. Math. 46 (1982), 249-279.
  * [14] C. Huneke, I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Mathematical Lecture Note Series, Vol. 336, Cambridge University Press, Cambridge (2006).
  * [15] N. T. Long, On adjusted Hilbert-Samuel function, Acta Math. Vietnamica. 40 (2015), 463-477.
  * [16] C. H. Linh and N. V. Trung, Uniform bounds in generalized Cohen-Macaulay rings, J. Algebra 304 (2006), 1147-1159.
  * [17] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.
  * [18] K. Ozeki, H. L. Truong and H. N. Yen, On Hilbert coefficients and sequentially Cohen-Macaulay rings: Proc. Amer. Math. Soc., Vol. 150, no. 6, (2022) 2367–2383.
  * [19] P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Van Oystaeyen, Freddy (ed.), Commutative algebra and algebraic geometry, New York: Marcel Dekker. Lect. Notes Pure Appl. Math., 206(1999), 245-264.
  * [20] N. T. Tam and H.L. Truong, A note on Chern coefficients and Cohen-Macaulay rings, Arkiv för Matematik. 58 vol. 1 (2020), 197-212,
  * [21] N. V. Trung, Absolutely superficial sequence, Math. Proc. Cambrige Phil. Soc, 93 (1983), 35-47.
  * [22] H. L. Truong, Index of reducibility of parameter ideals and Cohen-Macaulay rings, J. Algebra 415 (2014),35–49.
  * [23] H. L. Truong, Chern coefficients and Cohen-Macaulay rings, J. Algebra 490 (2017), 316–329.
  * [24] H. L. Truong and H. N. Yen, On the set of Chern numbers in local rings: 2022, submitted.
  * [25] W. V. Vasconcelos, The degrees of graded modules, Lecture Notes in Summer School on Commutative Algebra, vol. 2, pp 141-196, Centre de Recerca Matematica, Bellaterra (Spain), 1996.
  * [26] W. V. Vasconcelos, Computational Methods in Commutative algebra and Algebraic Geometry, Springer-Verlag Berlin Heidelberg, 1998.
  * [27] W. V. Vasconcelos, The Chern coefficient of local rings, Michigan Math. J., 58, 2008, 725–713.