# Graded Version of Some Basic Theorems on Local Cohomology to a Pair of
Ideals

P. H. Lima and V. H. Jorge Pérez Work partially supported by CAPES-Brazil
10056/12-2.Work partially supported by CNPq-Brazil - Grants 309316/2011-1, and
FAPESP Grant 2012/20304-1. 2000 Mathematics Subject Classification 13D45,
13A02, 13E10. Key words: Coefficient ideals, integral closure, reduction,
multiplicity

###### Abstract

In this paper, we prove some well-known results on local cohomology with
respect to a pair of ideals in graded version, such as, Independence Theorem,
Lichtenbaum-Harshorne Vanishing Theorem, Basic Finiteness and Vanishing
Theorem, among others. Besides, we present a generalized version of Melkersson
Theorem about Artinianess of modules and a result concerning Artinianess of
local cohomology modules.

##  1 Introduction

Local cohomology with respect to a pair of ideals was firstly defined in
[TYY], where the authors generalized the usual notion of local cohomology
module and studied its various properties such as the relation between the
usual local cohomology module, $H_{I}^{i}(M)$, and the one defined to a pair
of ideals, $H_{I,J}^{i}(M)$, vanishing and nonvanishing theorems, the
Generalized Version of Lichtenbaum-Hartshorne Theorem, among others.

Having the above results as motivation, the aim of this paper is to present in
graded version some basic theorems on cohomology with respect to a pair of
ideals, such as, Independence Theorem, Lichtenbaum-Harshorne Vanishing
Theorem, Basic Finiteness and Vanishing Theorems, and assertions concerning
Artinianess and depth with respect to a pair of ideals.

The organization of this paper is as follows.

Let $R$ be a graded ring and let $I\subseteq R$ be a graded ideal, $J$ an
arbitrary ideal and $M$ a graded $R$-module. Let $R_{+}$ denote the irrelevant
ideal of $R$, that is, the ideal generalized by elements of positive degree.

In section 2, we view the local cohomology ${}^{*}H^{i}_{I,J}(M)$ as a graded
module and express it in terms of usual local cohomology modules. To be more
precise, we denote by ${}^{*}\widetilde{W}(I,J)$ the set of homogeneous ideals
$\mathfrak{c}$ of $R$ such that $I^{n}\subseteq\mathfrak{c}+J$ for some
integer $n$, and then, show that

${}^{*}H_{I,J}^{i}(M)\cong{\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}}^{*}H^{i}_{\mathfrak{c}}(M).$

We also present the graded version of the Independence Theorem and
Lichtenbaum-Harshorne Vanishing theorem for a pair of ideals.

In section 3, we suppose $R$ is a positively graded Noetherian ring which is
standard, that is, $R=R_{0}[R_{1}]$, where $R_{0}$ is local ring. In [RS,
Theorem 2.1], the authors show that if $n={\rm sup}\\{i\ :H^{i}_{R_{+}}(M)\neq
0\\}$ then the $R$-module $H_{R_{+}}^{n}(M)/\mathfrak{m}_{0}H_{R_{+}}^{n}(M)$
is Artinian. We generalize this result for the case of local cohomology with
respect to a pair of homogeneous ideals, besides showing that
$H_{R_{+},J}^{i}(M)/(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M)$ is actually
Artinian for all $i\geq 0$. Furthermore, we prove that

$\dim M/(\mathfrak{m}_{0}R+J)M={\rm sup}\\{i\ :H^{i}_{R_{+},J}(M)\neq 0\\},$

a generalization for [BH, Lemma 3.4]. We also present a new version, with
respect to a pair of ideals, for Melkersson’s Theorem about Artinianess.

In section 4, the module $M$ and the ring $R$ are assumed to be a Cohen-
Macaulay, and then, it is obtained an expression to the number

${\rm inf}\\{i\in\mathbb{N}_{0}|H_{R_{+},J}^{i}(M)\neq 0\\}.$

In section 5, $R_{0}$ is assumed to be a local ring with infinite residual
field. It is well-known that, for all $i\geq 0$, $H_{R_{+}}^{i}(M)_{n}$ is
finitely generated $R_{0}$-module for $n\in\mathbb{Z}$ and
$H_{R_{+}}^{i}(M)_{n}=0,$ for $n$ sufficiently large. We give a positive
answer for this in the case of cohomology modules with respect to a pair of
ideals. If $J$ is generated by elements of zero degree then, for $i\geq 1$,
$H_{R_{+},J}^{i}(M)_{n}=0,$ for $n$ sufficiently large and
$H_{R_{+},J}^{i}(M)_{n}$ is a finitely generated $R_{0}$-module for all
$n\in\mathbb{Z}$. Finally, we prove a result about asymptotical stable
(Theorem 5.6).

##  2 Graded versions for a pair of ideals

In this section, we introduce a grading to $H^{i}_{I,J}(M)$, making this a
graded module. The result [TYY, Theorem 3.2], Independence theorem for a pair
of ideals and Generalized Version of Lichtenbaum-Hartshorne Theorem are again
presented now from the point of view of graded modules.

(A) Let $R$ be a graded ring and let $I\subseteq R$ be a graded ideal and $J$
an arbitrary ideal. If $M$ is a graded $R$-module then $\Gamma_{I,J}(M)$ is a
graded submodule of $M$. In fact, pick $m=(m_{k})\in\Gamma_{I,J}(M)$, so
$I^{n}\subseteq(0:m)+J$. It is easy to see that $I^{n}\subseteq(0:m_{k})+J$,
for all $i$ and each $k$. Then define $\Gamma_{I,J}(M)_{i}=\\{m\in
M_{i}:mI^{n}\subseteq mJ\text{ for some positive integer }n\geq 1\\}$.

(B) For a homomorphism $f:M\rightarrow N$, we have
$f(\Gamma_{I,J}(M))\subseteq\Gamma_{I,J}(N)$, so that there is a mapping
$\Gamma_{I,J}(f):\Gamma_{I,J}(M)\rightarrow\Gamma_{I,J}(N)$, which is the
restriction of $f$ to $\Gamma_{I,J}(M)$. Thus $\Gamma_{I,J}$ is an additive
functor on the category of all graded $R$-modules.

(C) Since the category of the graded modules has enough injectives, we can
form the $i$-th right derived functor of $\Gamma_{I,J}$ (on the category of
the graded modules), which will be denoted by ${}^{*}H^{i}_{I,J}$ ($i\geq 0$).
For a graded $R$-module $M$, we shall refer to ${}^{*}H^{i}_{I,J}(M)$ to be
the $i$-th graded local cohomology module of $M$ with respect to the pair of
ideals $(I,J)$.

(D) One can derive, by functor properties, that given an exact sequence
$0\rightarrow M\rightarrow N\rightarrow P\rightarrow 0$ of graded $R$-modules,
there is a long exact sequence

$\begin{array}[]{ccccccccc}0&\rightarrow&{}^{*}H^{0}_{I,J}(M)&\rightarrow&{}^{*}H^{0}_{I,J}(N)&\rightarrow&{}^{*}H^{0}_{I,J}(P)&\rightarrow&\\\
&\rightarrow&{}^{*}H^{1}_{I,J}(M)&\rightarrow&{}^{*}H^{1}_{I,J}(N)&\rightarrow&{}^{*}H^{1}_{I,J}(P)&\rightarrow&\cdot\cdot\cdot,\\\
\end{array}$

of graded modules with respect to a pair of ideals.

######  Definition 2.1.

We denote by ${}^{*}\widetilde{W}(I,J)$ the set of homogeneous ideals
$\mathfrak{c}$ of $R$ such that $I^{n}\subseteq\mathfrak{c}+J$ for some
integer $n$. We also define a partial order for this set:

$\mathfrak{c}\leq\mathfrak{d}\text{ if
}\mathfrak{c}\supseteq\mathfrak{d},\text{ for
}\mathfrak{c},\mathfrak{d}\in^{*}\widetilde{W}(I,J).$

If $\mathfrak{a}\leq\mathfrak{b}$ we obtain the inclusion map
$\Gamma_{\mathfrak{a}}(M)\hookrightarrow\Gamma_{\mathfrak{b}}(M)$. The order
relation on ${}^{*}\widetilde{W}(I,J)$ and the inclusion maps turn
$\\{\Gamma_{\mathfrak{a}}(M)\\}_{\mathfrak{a}\in^{*}\widetilde{W}(I,J)}$ into
a direct system of graded $R$-modules.

######  Proposition 2.2.

Let $R$ be a graded ring, $I$ a graded ideal, $J$ an arbitrary ideal of $R$
and $M$ a graded $R$-module. Then there is a natural graded isomorphism

${}^{*}H_{I,J}^{i}(M)\cong{\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}}^{*}H^{i}_{\mathfrak{c}}(M)$

###### Proof.

Firstly we observe that
$\Gamma_{I,J}(M)=\bigcup_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\Gamma_{\mathfrak{c}}(M)$.
In fact, if
$x\in\bigcup_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\Gamma_{\mathfrak{c}}(M)$,
we have $I^{m}\subseteq\mathfrak{c}+J$ and $xI^{n}=0$ for positive integers
$n,m$ and some $\mathfrak{c}\in^{*}\widetilde{W}(I,J)$. Since
$I^{mn}\subseteq(\mathfrak{c}+J)^{n}\subseteq\mathfrak{c}^{n}+J$, we have
$I^{mn}x\subseteq Jx$, that is, $x\in\Gamma_{I,J}(M)$. Now let
$x\in\Gamma_{I,J}(M)$ homogeneous. Then $I^{n}\subseteq\mathfrak{c}+J$, where
$\mathfrak{c}=\text{ann}(x)$, so $x\mathfrak{c}=0$ and
$x\in\Gamma_{\mathfrak{c}}(M)$. If $x$ is not homogeneous, we write
$x=x_{1}+...+x_{r}$, where $x_{i}$ is homogeneous. We have then
$\mathfrak{a}_{i}=\text{ann}(x_{i})$ for each $i$. Hence
$x\mathfrak{a}_{1}\cdot\cdot\cdot\mathfrak{a}_{r}=0$. Therefore
$x\in\Gamma_{\mathfrak{c}}(M)$, where
$\mathfrak{c}=\mathfrak{a}_{1}\cdot\cdot\cdot\mathfrak{a}_{r}$.

Let $0\rightarrow M\rightarrow N\rightarrow P\rightarrow 0$ be an exact
sequence of $R$-modules. For each $\mathfrak{c}\in^{*}\widetilde{W}(I,J)$, we
have a long exact sequence

$\begin{array}[]{ccccccccc}0&\rightarrow&{}^{*}H^{0}_{\mathfrak{c}}(M)&\rightarrow&{}^{*}H^{0}_{\mathfrak{c}}(N)&\rightarrow&{}^{*}H^{0}_{\mathfrak{c}}(P)&\rightarrow&\\\
&\rightarrow&{}^{*}H^{1}_{\mathfrak{c}}(M)&\rightarrow&{}^{*}H^{1}_{\mathfrak{c}}(N)&\rightarrow&{}^{*}H^{1}_{\mathfrak{c}}(P)&\rightarrow&\cdot\cdot\cdot,\\\
\end{array}$

which we can take the limit to obtain a long exact sequence

${\small\begin{array}[]{llllllll}0&\rightarrow&\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\
^{*}H^{0}_{\mathfrak{c}}(M)&\rightarrow&\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\
^{*}H^{0}_{\mathfrak{c}}(N)&\rightarrow&\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\
^{*}H^{0}_{\mathfrak{c}}(P)&\\\
&\rightarrow&\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\
^{*}H^{1}_{\mathfrak{c}}(M)&\rightarrow&\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\
^{*}H^{1}_{\mathfrak{c}}(N)&\rightarrow&\cdots.&\\\ \end{array}}$

Since ${}^{*}H^{i}_{\mathfrak{c}}(E)=0$ for any ∗injective $R$-module $E$ and
any positive integer $i$,
$\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\
^{*}H^{i}_{\mathfrak{c}}(E)=0$. Therefore, one may conclude that

$\\{\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}^{*}H^{i}_{\mathfrak{c}}\
\ \ |\ i=0,1,2,...\\}$

is a system of right derived functors of ${}^{*}\Gamma_{I,J}$ (see [BS,
Theorem 12.3.1]), and that there is the desired graded isomorphism. ∎

######  Remark 2.3.

Consider the above setup. Since
$\Gamma_{I,J}(M)=\bigcup_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}\Gamma_{\mathfrak{c}}(M)$,
one may conclude similarly to the above proof that

$H_{I,J}^{i}(M)\cong\underrightarrow{\lim}_{\mathfrak{c}\in^{*}\widetilde{W}(I,J)}H^{i}_{\mathfrak{c}}(M).$

The following result is obtained from the previous proposition once
${}^{*}H_{\mathfrak{c}}^{i}(E)=0$ for any graded ideal $\mathfrak{c}$ and an
∗injective graded $R$-module $E$.

######  Proposition 2.4.

Let $R$ be a graded ring, I a graded ideal and $J$ an arbitrary ideal. Let $E$
be an ∗injective graded $R$-module. Then $H_{I,J}^{i}(E)=0$.

By Proposition 2.4, we have the following

######  Proposition 2.5.

Let $R$ be a graded ring, $I$ a graded ideal and $J$ an arbitrary ideal. Let
$M$ be an $R$-graded module. There is an isomorphism

${}^{*}H_{I,J}^{i}(M)\cong H_{I,J}^{i}(M),$

for all $i$ as underlying $R$-modules.

######  Theorem 2.6.

_(Graded independence theorem for a pair of ideals)_

Assume $R=\bigoplus_{n\in\mathbb{Z}}R_{n}$ is a graded ring, $I\subset R$ a
graded ideal and $J\subset R$ an arbitrary ideal. Let
$R^{\prime}=\bigoplus_{n\in\mathbb{Z}}R_{n}^{\prime}$ be another Noetherian
graded ring, $f:R\rightarrow R^{\prime}$ a graded homomorphism of rings such
that $f(I)=JR^{\prime}$ and $M^{\prime}$ an $R^{\prime}$-module.

  1. 1.

For each $i\in\mathbb{N}_{0}$, both the cohomology modules

$H_{IR^{\prime},JR^{\prime}}^{i}(M^{\prime})\ \text{ and }\
H_{I,J}^{i}(M^{\prime})$

are graded $R$-modules;

  2. 2.

For each $i\in\mathbb{N}_{0}$, there exists a graded isomorphism

$H_{IR^{\prime},JR^{\prime}}^{i}(M^{\prime})\cong H_{I,J}^{i}(M^{\prime})$

of graded $R$-modules.

###### Proof.

The first item follows because $f$ is homogeneous.

The assumption $f(I)=JR^{\prime}$ gives
$\Gamma_{IR^{\prime},JR^{\prime}}^{i}(M^{\prime})=\Gamma_{I,J}^{i}(M^{\prime})$.
By [TYY, Theorem 2.7], for each graded $R$-module $M^{\prime}$, one has an
isomorphism

$H_{IR^{\prime},JR^{\prime}}^{i}(M^{\prime})\cong H_{I,J}^{i}(M^{\prime}).$

By using the grading on $H_{I,J}^{i}(M^{\prime})$ obtained from first item, we
can turn this isomorphism into a graded isomorphism. On the other hand, given
an $R$-graded homomorphism $f:M^{\prime}\rightarrow N^{\prime}$, we have a
natural commutative diagram

$\begin{array}[]{ccc}H_{I,J}^{i}(M^{\prime})&\longrightarrow&H_{I,J}^{i}(N^{\prime})\\\
\downarrow&&\downarrow\\\
H_{IR^{\prime},JR^{\prime}}^{i}(M^{\prime})&\longrightarrow&H_{IR^{\prime},JR^{\prime}}^{i}(N^{\prime}).\end{array}$

Hence, as $H_{IR^{\prime},JR^{\prime}}^{i}(E)$ equals zero (by Proposition
2.4), these (new) gradings coincide with the ones from item 1. So, one can
conclude the second item. ∎

######  Lemma 2.7.

Let $R$ be a graded local ring of unique graded maximal ideal $\mathfrak{m}$.
Let $d$ be the dimension of $R$, $I$ a graded ideal and $J$ an arbitrary ideal
of $R$. Then $H^{d}_{I,J}(R)\cong
H^{d}_{I\widehat{R},J\widehat{R}}(\widehat{R})$.

###### Proof.

By [CW, Theorem 2.1] (the same proof is true for the non-local case),
$H^{d}_{I,J}(R)$ is Artinian. So, by [TYY, Lemma 4.8], we have

$H^{d}_{I,J}(R)\cong H^{d}_{I,J}(R)\otimes_{R}\widehat{R}\cong
H^{d}_{I,J}(\widehat{R}).$

On the other hand, by [TYY, Theorem 2.7], one has
$H^{d}_{I,J}(\widehat{R})\cong
H^{d}_{I\widehat{R},J\widehat{R}}(\widehat{R})$. The lemma is then concluded.
∎

######  Theorem 2.8.

_(Graded Lichtenbaum-Hartshorne Vanishing Theorem to a pair of ideals)_

Let $R=\oplus_{n\geq 0}R_{n}$ be a positively graded ring and an integral
domain of dimension $d$. Assume $R_{0}$ is a complete local ring. Let $I$ be a
proper graded ideal and $J$ an arbitrary ideal of $R$. Then
$H^{d}_{I,J}(R)=0$.

###### Proof.

If $J=0$, it reduces to the usual case, already proved. Suppose then $J$ is a
nonzero ideal. Let $\mathfrak{m}$ be the unique graded maximal ideal of $R$.
We know the completion $\widehat{R_{\mathfrak{m}}}$ is isomorphic to the
$\mathfrak{m}$-adic completion $\widehat{R}$ of $R$. Further, since $R_{0}$ is
complete, it is known that $\widehat{R}$ is a domain. By the generalized
Lichtenbaum-Harshorne Vanishing theorem ([TYY, Theorem 4.9]) one can deduce
$H^{d}_{IR_{\mathfrak{m}},JR_{\mathfrak{m}}}(R_{\mathfrak{m}})=0$. On the
other hand, by Lemma 2.7, one obtains

$H^{d}_{IR_{\mathfrak{m}},JR_{\mathfrak{m}}}(R_{\mathfrak{m}})\cong
H^{d}_{I,J}(R).$

Therefore $H^{d}_{I,J}(R)=0$. ∎

######  Proposition 2.9.

_([TYY, Corollary 4.2])_ Let $R$ be a graded local ring of unique graded
maximal ideal $\mathfrak{m}$ and $M$ be a finite graded module over $R$. Let
$I,J$ be graded ideals of $R$. If $H_{I,J}^{i}(M)=0$ for all integers $i>0$,
then $M$ is an $(I,J)$-torsion $R$-module.

###### Proof.

Set $N=M/\Gamma_{I,J}(M)$. We need to prove that $N=0$. Suppose $N\neq 0$. By
[TYY, Corollary 1.13], we have $\Gamma_{I,J}(N)=0$ and
$H_{I,J}(N)=H_{I,J}(M)=0$ for all $i>0$. Since by hypothesis $I,J$ are graded
ideals, we have $\mathfrak{m}\in W(I,J)$, so that

$\inf\\{\text{depth}\hskip 1.42271ptN_{\mathfrak{p}}\ |\ \mathfrak{p}\in
W(I,J)\\}\leq\text{depth}\hskip 1.42271ptN_{\mathfrak{m}}=\text{depth}\hskip
1.42271ptN<\infty.$

By using [TYY, Theorem 4.1], we obtain $H_{I,J}(N)\neq 0$ for some integer
$i\leq\text{depth}N$, which is a contradiction. ∎

##  3 Top local cohomology

In this section, we give a version for Melkersson’s Theorem concerning
Artinianess of module with respect to a pair of ideals (Proposition 3.2). Also
it is obtained a result about Artinianess of local cohomology with respect to
a pair of ideal (Theorem 3.3). To conclude the section, we find the top of the
local cohomology with respect to a pair of ideals (Theorem 3.5).

Throughout this section, let $R=\oplus_{d\geq 0}R_{d}$ denote a positively
graded commutative Noetherian ring, which is standard, that is,
$R=R_{0}[R_{1}]$. Assume $R_{0}$ is a local ring of maximal ideal
$\mathfrak{m}_{0}$. Set $R_{+}=\oplus_{i>0}R_{i}$, the irrelevant ideal of
$R$. Let $M=\oplus_{d\in\mathbb{Z}}M_{d}$ be a finite graded $R$-module. Let
$M[a]$ denote the graded module $a$-shift of $M$, defined by
$M[a]_{i}=M_{i+a}$.

######  Remark 3.1.

It can be derived from Proposition 2.2 that $H_{I,J}^{i}(M[a])\cong
H_{I,J}^{i}(M)[a]$ as homogeneous modules.

######  Proposition 3.2.

_([M, Theorem 1.3])_ Let $M$ be an $(I,J)$-torsion $R$-module for which
$(JM:_{M}I)$ is an Artinian module. Then $M$ is an Artinian module.

###### Proof.

If $M$ is an $(I,J)$-torsion $R$-module, then $M/JM$ is an $I$-torsion module
by [TYY, Corollary 1.9]. Since by assumption $(JM:_{M}I)=(0:_{M/JM}I)$ is an
Artinian module we can use a result due to Melkersson (see [M, Theorem 1.3])
to get $M/JM$ is an Artinian module. Moreover, $JM\subseteq(JM:_{M}I)$ is also
an Artinian module. By an exact sequence one can conclude $M$ is an Artinian
module. ∎

Next theorem is a generalization of Theorem 2.1 in [RS].

######  Theorem 3.3.

Let $J$ be a graded ideal of $R$. Then the $R$-module

$H_{R_{+},J}^{i}(M)/(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M)$

is Artinian for all $i\geq 0$.

###### Proof.

Since $M$ is a finite $R$-module one may verify $R_{+}^{k}\hskip
1.42271pt\Gamma_{R_{+},\mathfrak{b}}(M)\subset J\Gamma_{R_{+},J}(M)$ for some
integer $k$. It implies
$(\mathfrak{m}_{0}+R_{+})^{k}\Gamma_{R_{+},J}(M)\subset\mathfrak{m}_{0}\Gamma_{R_{+},J}(M)+JH_{R_{+},J}^{n}(M),$
so that one may conclude

$\Gamma_{R_{+},J}(M)/(\mathfrak{m}_{0}R+J)\Gamma_{R_{+},J}(M)$

is Artinian.

Now assume, by induction, that $i>0$ and we have shown

$H_{R_{+},J}^{i-1}(M^{\prime})/(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i-1}(M^{\prime})$

is an Artinian module for any finite graded $R$-module $M^{\prime}$. In view
of [TYY, Corollary 1.13], we can assume $M$ is an $(R_{+},J)$-torsion free
$R$-module. Since $\Gamma_{R_{+}}(M)\subseteq\Gamma_{R_{+},J}(M)$, there
exists a homogeneous element $x\in R_{+}$ which is $M$-regular. Say,
$\text{deg}(x)=a$. The exact sequence

$0\rightarrow M\stackrel{{\scriptstyle
x}}{{\rightarrow}}M[-a]\rightarrow(M/xM)[-a]\rightarrow 0$

induces an exact sequence

$\begin{array}[]{r}\cdot\cdot\cdot\rightarrow H_{R_{+},J}^{i-1}(M/xM)\\\
\rightarrow H_{R_{+},J}^{i}(M)\stackrel{{\scriptstyle
x}}{{\rightarrow}}H_{R_{+},J}^{i}(M)[-a]\rightarrow
H_{R_{+},J}^{i}(M/xM)[-a]\rightarrow\cdot\cdot\cdot\end{array}$

Hence, we have the exact sequence

$\frac{H_{R_{+},J}^{i-1}(M/xM)}{(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i-1}(M/xM)}\rightarrow\frac{H_{R_{+},J}^{i}(M)}{(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M)}\stackrel{{\scriptstyle
x}}{{\rightarrow}}$
$\rightarrow\frac{xH_{R_{+},J}^{i}(M)[-a]}{x(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M))[-a]}\rightarrow
0.$

By induction hypothesis, we have
$\frac{H_{R_{+},J}^{i-1}(M/xM)}{(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i-1}(M/xM)}$
is Artinian. It shows the kernel of the multiplication by $x$ on
$\frac{H_{R_{+},J}^{i}(M)}{(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M)}$ is an
Artinian $R$-module. By using [TYY, Corollary 1.13], one can deduce
$\frac{H_{R_{+},J}^{i}(M)}{(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M)}$ is an
$((x),J)$-torsion $R$-module. To conclude the proof, we apply Proposition 3.2
in order to obtain
$\frac{H_{R_{+},J}^{i}(M)}{(\mathfrak{m}_{0}R+J)H_{R_{+},J}^{i}(M)}$ is
Artinian. This completes the inductive step.

∎

######  Theorem 3.4.

Let $J$ be a graded ideal of $R$. If $\dim M/(\mathfrak{m}_{0}R+J)M=d,$ then

$H_{R_{+},J}^{i}(M)=0,$

for all $i>d$.

###### Proof.

We argue by induction. Let $n=\dim_{R}M$. Suppose $n=-1$, so there is nothing
to be proved, as $M=0$. Assume then $n\geq 0$ and that the result is
established for $R$-modules of dimension smaller than $n$. By [TYY, Corollary
2.5], $H_{R_{+},J}^{i}(\Gamma_{J}(M))\cong H_{R_{+}}^{i}(\Gamma_{J}(M))$ for
all $i\geq 0$, once $\Gamma_{J}(M)$ is an $J$-torsion module. By using the
fact that
$\sqrt{\text{ann}\left(\frac{\Gamma_{J}(M)}{\mathfrak{m}_{0}\Gamma_{J}(M)}\right)}=\sqrt{\mathfrak{m}_{0}R+\text{ann}(\Gamma_{J}(M))}$
and
$\sqrt{\text{ann}\left(\frac{M/JM}{\mathfrak{m}_{0}(M/JM)}\right)}=\sqrt{\mathfrak{m}_{0}R+J+\text{ann}(M)}$,
one can conclude
$\dim\frac{\Gamma_{J}(M)}{\mathfrak{m}_{0}\Gamma_{J}(M)}\leq\dim\frac{M/JM}{\mathfrak{m}_{0}(M/JM)}=d.$
We use then [BH, Lemma 3.4] to obtain $H_{R_{+},J}^{i}(\Gamma_{J}{(M)})=0$ for
all $i>d$. Moreover, the exact sequence $0\rightarrow\Gamma_{J}(M)\rightarrow
M\rightarrow M/\Gamma_{J}(M)\rightarrow 0$ yields the long exact sequence

$\cdot\cdot\cdot\rightarrow H_{R_{+},J}^{i}(\Gamma_{J}(M))\rightarrow
H_{R_{+},J}^{i}(M)\rightarrow
H_{R_{+},J}^{i}(M/\Gamma_{J}(M))\rightarrow\cdot\cdot\cdot$ (3.1)

We then derive

$H_{R_{+},J}^{i}(M)\cong H_{R_{+},J}^{i}(M/\Gamma_{J}(M)),$

for all $i>d$. We have

$\dim\frac{M/\Gamma_{J}(M)}{(\mathfrak{m}_{0}R+J)M/\Gamma_{J}(M)}=\dim\frac{M}{(\mathfrak{m}_{0}R+J)M+\Gamma_{J}(M)}\leq
d.$

In this way, we can assume $M$ is an $J$-torsion-free module. So there exists
an homogeneous element $a\in J$ which is a non zero-divisor on $M$. The exact
sequence $0\rightarrow M\stackrel{{\scriptstyle a}}{{\rightarrow}}M\rightarrow
M/aM\rightarrow 0$ yields the long exact sequence

$\cdot\cdot\cdot\rightarrow H_{R_{+},J}^{i}(M)\stackrel{{\scriptstyle
a}}{{\rightarrow}}H_{R_{+},J}^{i}(M)\rightarrow
H_{R_{+},J}^{i}(M/aM)\rightarrow\cdot\cdot\cdot$

Once $\dim_{R}M/aM=n-1$ and
$\dim_{R}\frac{M/aM}{(\mathfrak{m}_{0}R+J)M/aM}=d$, we use the inductive
hypothesis to conclude $H_{R_{+},J}^{i}(M/aM)=0$ for all $i>d$. Then the above
long exact sequence says $aH_{R_{+},J}^{i}(M)=H_{R_{+},J}^{i}(M)$. By
Nakayama’s Lemma, we obtain the desired result. ∎

Next result is a similar result to [BH, Lemma 3.4(b)].

######  Theorem 3.5.

Let $J$ be a graded ideal of $R$. Then

$\dim M/(\mathfrak{m}_{0}R+J)M={\rm sup}\\{i\ :H^{i}_{R_{+},J}(M)\neq 0\\}.$

###### Proof.

Consider the exact sequence

$0\rightarrow(\mathfrak{m}_{0}R+J)M\rightarrow M\rightarrow
M/(\mathfrak{m}_{0}R+J)M\rightarrow 0,$

which yields the long exact sequence

$\cdot\cdot\cdot\rightarrow H_{R_{+},J}^{i}(M)\rightarrow
H_{R_{+},J}^{i}(M/(\mathfrak{m}_{0}R+J)M)\rightarrow
H_{R_{+},J}^{i+1}((\mathfrak{m}_{0}R+J)M)\rightarrow\cdot\cdot\cdot.$

Now note that

$\dim\frac{(\mathfrak{m}_{0}R+J)M}{(\mathfrak{m}_{0}R+J)(\mathfrak{m}_{0}R+J)M}\leq\dim\frac{M}{(\mathfrak{m}_{0}R+J)^{2}M}=\dim\frac{M}{(\mathfrak{m}_{0}R+J)M}=d;$

thus, because of Theorem 3.4, we obtain
$H_{R_{+},\mathfrak{b}}^{i}((\mathfrak{m}_{0}R+J)M)=0$ for $i>d$. On the other
hand, by using [TYY, Corollary 2.5] and [HIO, Corollary 35.20], one sees

$H_{R_{+},J}^{d}(M/(\mathfrak{m}_{0}R+J)M)\cong
H_{R_{+}}^{d}(M/(\mathfrak{m}_{0}R+J)M)\cong
H_{(R/\mathfrak{m}_{0}R)_{+}}^{d}(M/(\mathfrak{m}_{0}R+J)M).$

This last cohomology module is nonzero by [HIO, Corollary 36.19]. By observing
the above long exact sequence we get the desired result. ∎

##  4 Depth $(I,J)$ on graded module

In this section, we work with the concept of depth of a pair homogeneous
ideals of $R$, and obtain an expression for it as $M$ and $R$ are both Cohen-
Macaulay. For the ordinary case the [BS, Theorem 6.2.7] says

${\rm depth}_{I}(M)={\rm inf}\\{i\in\mathbb{N}_{0}\ :\ H_{I}^{i}(M)\neq 0\\},$

for an $R$-ideal $I$ and a finite $R$-module $M$ such that $IM\neq M$. For the
case of a pair $(I,J)$ ideals it was defined in [AAS, Definition 3.1]; we then
recall this definition as follows.

Throughout this section we assume $R$ and $M$ are as in section 3.

######  Definition 4.1.

Let $I,$ $J$ be two homogeneous ideals of the graded ring $R$ and $M$ a graded
$R$-module. We define depth of $(I,J)$ on $M$ by

${\rm depth}(I,J,M)={\rm inf}\\{{\rm depth}({\mathfrak{a}},M)\ |\
{\mathfrak{a}}\in\tilde{W}(I,J)\\}$

it this infimum exists, and $\infty$ otherwise.

######  Remark 4.2.

In the general case, by [TYY, Theorem 4.1 and Theorem 3.2] or [AAS,
Proposition 3.3], we have

${\rm depth}(I,J,M)={\rm inf}\\{i\in\mathbb{N}_{0}|H_{I,J}^{i}(M)\neq 0\\}.$

Let $I,J,K$ be three homogeneous ideals of $R$. Let us introduce the number

$\text{ht}(I,J,K):=\text{inf}\\{\text{ht}(\mathfrak{p})\ |\ \mathfrak{p}\in
W(I,J)\cap V(K)\\}.$

Note that when $J=(0)$,

$\text{ht}(I,J,K)=\text{ht}(I+K).$

######  Proposition 4.3.

Assume $M$ is a Cohen-Macaulay $R$-module and $R$ is a Cohen-Macaulay ring.
Set $I=\sqrt{\emph{ann}_{R}M}$. Then

$\emph{depth}\hskip 1.42271pt(R_{+},J,M)=\emph{ht}(R_{+},I,J)-\emph{ht}(I).$

In particular, $\emph{ht}(R_{+},I,J)-\emph{ht}(I)={\rm
inf}\\{i\in\mathbb{N}_{0}\ |\ H_{R_{+},J}^{i}(M)\neq 0\\}.$

###### Proof.

By [TYY, Theorem 4.1] (or [AAS, Proposition 3.3]),

$\text{depth}\hskip 1.42271pt(R_{+},J,M)=\text{inf}\\{\text{depth}\hskip
1.42271ptM_{\mathfrak{p}}\ |\ \mathfrak{p}\in W(R_{+},J)\\}.$

Note that

$\text{depth}\hskip 1.42271pt(R_{+},J,M)=\text{inf}\\{\text{depth}\hskip
1.42271ptM_{\mathfrak{p}}\ |\ \mathfrak{p}\in W(R_{+},J)\cap V(I)\\},$

once $M_{\mathfrak{p}}=0$ as $\mathfrak{p}\not\in V(I)$, that is,
$\text{depth}\hskip 1.42271ptM_{\mathfrak{p}}=\infty$. Since $M$ is Cohen-
Macaulay, in particular,

$\text{depth}\hskip 1.42271ptM_{\mathfrak{p}}=\dim M_{\mathfrak{p}}$

for each $\mathfrak{p}\in W(R_{+},J)$. But $R$ is also Cohen-Macaulay by
hypothesis, so $\dim M_{\mathfrak{p}}=\dim
R_{\mathfrak{p}}/I_{\mathfrak{p}}=\dim
R_{\mathfrak{p}}-\text{ht}I_{\mathfrak{p}}$. By [L, Lemma 1.2.2], all minimal
primes of $I$ have the same height, so that,
$\text{ht}I_{\mathfrak{p}}=\text{ht}I$ for each $\mathfrak{p}$. By combining
the above arguments, we obtain

$\text{depth}\hskip 1.42271pt(R_{+},J,M)=\text{inf}\\{\dim
R_{\mathfrak{p}}-\text{ht}I\ |\ \mathfrak{p}\in W(R_{+},J)\cap V(I)\\}.$

This last one equals $\text{ht}(R_{+},J,K)-\text{ht}I$, by definition. The
proof is completed. ∎

######  Proposition 4.4.

Let $J$ be a graded ideal of $R$. Then any integer $i$ for which
$H_{R_{+},J}^{i}(M)\neq 0$ satisfies

${\rm depth}(R_{+},J,M)\leq i\leq\dim M/({\mathfrak{m}_{0}}R+J)M.$

###### Proof.

The proof follows by Remark 4.2 and Theorem 3.5. ∎

######  Corollary 4.5.

There is exactly one integer $i$ for which $H_{R_{+},J}^{i}(M)\neq 0$ if and
only if

${\rm depth}(R_{+},J,M)=\dim M/({\mathfrak{m}_{0}}R+J)M.$

##  5 Basic finiteness, vanishing theorem and
asymptotical stability

In this section, we generate two classical results about the graded components
of local cohomology for the case of local cohomology with respect to a pair of
ideals (Proposition 5.2 and Theorem 5.4). Finally, we obtain a result on the
asymptotical stability of the sequence $\\{{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M))_{n}\\}_{n\in\mathbb{Z}}$.

Throughout this section, we will make the following assumptions.

Let $R=\oplus_{d\geq 0}R_{d}$ denote a positively graded commutative
Noetherian ring, which is standard, that is, $R=R_{0}[R_{1}]$ and let $M$ be a
finitely generated module over $R$. Assume $R_{0}$ is a local ring of maximal
ideal $\mathfrak{m}_{0}$ and residual field $R_{0}/\mathfrak{m}_{0}$ is
infinite. Set $R_{+}=\oplus_{i>0}R_{i}$, the irrelevant ideal of $R$.

It is well-known that $H_{R_{+}}^{i}(M)_{n}$ is finitely generated
$R_{0}$-module for all $n\in\mathbb{Z}$ and $H_{R_{+}}^{i}(M)_{n}=0,$ for $n$
sufficiently large. Next assertion gives a positive answer for the case of
cohomology modules with respect to a pair of ideals.

######  Remark 5.1.

Let $I,K,J$ be arbitrary ideals of $R$, and let $M$ be a $K$-torsion
$R$-module. It is easy to check that $H_{I+K,J}^{i}(M)\cong H_{I,J}^{i}(M)$
for all $i\in\mathbb{N}_{0}$.

######  Proposition 5.2.

Let $J$ be an ideal generated by elements of zero degree. Set
$\mathfrak{b}=\mathfrak{b}_{0}+R_{+}$. Then for $i\geq 1$,

$H_{\mathfrak{b},J}^{i}(M)_{n}=0,$

for $n$ sufficiently large.

###### Proof.

Firstly we will prove the assertion as $\mathfrak{b}_{0}=0$. We proceed by
induction on $\dim M$. If $\dim M=0$ then by [TYY, Theorem 4.7(1)], the result
is clearly true in this case. Suppose now $\dim M>0$ and the result is
established for finitely generated modules of dimension less than $\dim M$.

By [TYY, Corollary 2.5], $H_{R_{+},J}^{i}(\Gamma_{J}(M))\cong
H_{R_{+}}^{i}(\Gamma_{J}(M)),$ for all $i\geq 0$. The exact sequence
$0\rightarrow\Gamma_{J}(M)\rightarrow M\rightarrow M/\Gamma_{J}(M)\rightarrow
0$ yields then the long exact sequence

$H_{R_{+}}^{i}(\Gamma_{J}(M))_{n}\rightarrow H_{R_{+},J}^{i}(M)_{n}\rightarrow
H_{R_{+},J}^{i}(M/\Gamma_{J}(M))_{n}\rightarrow
H_{R_{+}}^{i+1}(\Gamma_{J}(M))_{n}.$

One can conclude that $H_{R_{+},J}^{i}(M)_{n}$ is isomorphic to
$H_{R_{+},J}^{i}(M/\Gamma_{J}(M))_{n}$ for all $n$ sufficiently large. Hence,
we can assume that $M$ is an $J$-torsion-free $R$-module. As $J$ is generated
by homogeneous elements of zero degree, by a Prime Avoidance Lemma there
exists an element $x$ in $J$ of zero degree which is a non zero divisor on
$M$.

The exact sequence

$0\rightarrow M\stackrel{{\scriptstyle x}}{{\rightarrow}}M\rightarrow
M/xM\rightarrow 0$

induces an exact sequence

$H_{R_{+},J}^{i}(M)_{n}\stackrel{{\scriptstyle
x}}{{\rightarrow}}H_{R_{+},J}^{i}(M)_{n}\rightarrow
H_{R_{+},J}^{i}(M/xM)_{n}.$

By induction, for every $i\geq 1$, $H_{R_{+},J}^{i}(M/xM)_{n}=0$ for all $n$
sufficiently large; the above exact sequence yields then an epimorphism

$H_{R_{+},J}^{i}(M)_{n}\stackrel{{\scriptstyle
x}}{{\rightarrow}}H_{R_{+},J}^{i}(M)_{n},$

for all $n>>0$, so

$H_{R_{+},J}^{i}(M)_{n}=xH_{R_{+},J}^{i}(M)_{n}.$

Nakayama’s Lemma completes the first part.

Now we conclude the proposition. We proceed by induction on $\dim M$. If $\dim
M=0$, the result is clearly true.

Consider the exact sequence
$0\rightarrow\Gamma_{\mathfrak{b}_{0}}(M)\rightarrow M\rightarrow
M/\Gamma_{\mathfrak{b}_{0}}(M)\rightarrow 0$, which yields, by Remark 5.1, the
long exact sequence

$H_{R_{+}}^{i}(\Gamma_{\mathfrak{b}_{0}}(M))_{n}\rightarrow
H_{R_{+},J}^{i}(M)_{n}\rightarrow
H_{R_{+},J}^{i}(M/\Gamma_{\mathfrak{b}_{0}}(M))_{n}\rightarrow
H_{R_{+}}^{i+1}(\Gamma_{\mathfrak{b}_{0}}(M))_{n}.$

So, by using the first part, we obtain the isomorphism

$H_{R_{+},J}^{i}(M)_{n}\rightarrow
H_{R_{+},J}^{i}(M/\Gamma_{\mathfrak{b}_{0}}(M))_{n}$

for $n$ large. We then may assume there exists an element
$x\in\mathfrak{b}_{0}$ which is a nonzero divisor on $M$. The final result is
obtained by following the same arguments as in the last paragraph of the first
part.

∎

######  Lemma 5.3.

If $J$ is generated by elements of zero degree and $\Gamma_{R_{+},J}(M)_{n}=0$
for $n>>0$, then, for each $i\geq 0$, $H_{R_{+},J}^{i}(M)_{n}$ is a finitely
generated $R_{0}$-module for all integer $n$.

###### Proof.

We proceed by induction on $i$. By hypothesis and by the fact that $M_{n}$ for
$n<<0$, we conclude that there exists $u\in\mathbb{N}$ such that
$R_{+}^{u}H_{R_{+},J}^{0}(M)=0$. Note that
$R_{+}^{i}H_{R_{+},J}^{0}(M)/R_{+}^{i+1}H_{R_{+},J}^{0}(M)$ is a Noetherian
$R/R_{+}$-module, that is, a Noetherian $R_{0}$-module for each $i=0,...,u-1$.
One then obtains $H_{R_{+},J}^{0}(M)$ is a finitely generated $R_{0}$-module,
and so is $H_{R_{+},J}^{0}(M)_{n}$ for all $n\in\mathbb{Z}$.

Suppose now $i>0$ and the result is established for smaller values of $i$.
Because of the graded isomorphism $H_{R_{+},J}^{i}(M)\cong
H_{R_{+},J}^{i}(M/\Gamma_{R_{+},J}(M))$ for $i>0$ (see [TYY, Corollary 1.13]),
we can assume $M$ is an $(R_{+},J)$-torsion-free $R$-module (and so is an
$R_{+}$-torsion-free module). Then there exists an element $x\in R_{+}$ which
is a non-zero divisor on $M$, say $\text{deg}(x)=a$. The exact sequence

$0\rightarrow M\stackrel{{\scriptstyle
x}}{{\rightarrow}}M[a]\rightarrow(M/xM)[a]\rightarrow 0$

induces an exact sequence

$\begin{array}[]{r}H_{R_{+},J}^{i-1}(M)_{n+a}\rightarrow
H_{R_{+},J}^{i-1}(M/xM)_{n+a}\rightarrow
H_{R_{+},J}^{i}(M)_{n}\stackrel{{\scriptstyle
x}}{{\rightarrow}}H_{R_{+},J}^{i}(M)_{n+a}.\end{array}$

From the above exact sequence and Proposition 5.2 one can deduce
$\Gamma_{R_{+},J}(M/xM)_{n}=0$ for $n$ sufficiently large. From the inductive
hypothesis it then follows that $H_{R_{+},J}^{i-1}(M/xM)_{q}$ is finitely
generated for all $q\in\mathbb{Z}$. Again by Proposition 5.2, there exists
$s\in\mathbb{Z}$ such that $H_{R_{+},J}^{i-1}(M/xM)_{n}=0$ for all $n\geq s$
and $H_{R_{+},J}^{i}(M)_{n}=0$ for all $n\geq s-a$.

Fix $n\in\mathbb{Z}$ and let $k\geq 0$ be an integer such that $n+ka\geq s-a$.
So $H_{R_{+},J}^{i}(M)_{n+ka}=0$. For each $j=0,..,k-1$, we have the exact
sequence

$H_{R_{+},J}^{i-1}(M/xM)_{n+(j+1)a}\rightarrow
H_{R_{+},J}^{i}(M)_{n+ja}\stackrel{{\scriptstyle
x}}{{\rightarrow}}H_{R_{+},J}^{i}(M)_{n+(j+1)a}.$

In conclusion, we obtain $H_{R_{+},J}^{i}(M)_{n+ja}$ is a finitely generated
for $j=k-1,k-2,...,1,0$, so that $H_{R_{+},J}^{i}(M)_{n}$ is a finitely
generated for all $n\in\mathbb{Z}$.

∎

######  Theorem 5.4.

If $J$ is generated by elements of zero degree, then, for all $i\geq 1$,
$H_{R_{+},J}^{i}(M)_{n}$ is a finitely generated $R_{0}$-module for all
$n\in\mathbb{Z}$.

###### Proof.

It follows from the graded isomorphism $H_{R_{+},J}^{i}(M)\cong
H_{R_{+},J}^{i}(M/\Gamma_{R_{+},J}(M))$ for $i>0$ and the above lemma. ∎

Consider the following definition. We say ${\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})$ is _asymptotically increasing for_
$n\to-\infty$, if there exists an $n_{0}\in\mathbb{Z}$ such that

${\rm Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})\subseteq{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n+1})$

for all $n\leq n_{0}.$

######  Lemma 5.5.

Let $M$ be a finitely generated graded $R$-module and $J$ an arbitrary ideal.
Let $i\in\mathbb{N}_{0}$ be such that $H_{R_{+},J}^{j}(M)_{n}$ is finitely
generated $R_{0}$-module for all $j<i$ and $n<<0$. Then ${\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})$ is asymptotically increasing for
$n\to-\infty$.

###### Proof.

We prove by induction on $i$. The case $i=0$ is trivial as
$H_{R_{+},J}^{0}(M)_{n}=0$ for all $n\ll 0$. So, let $i>0$. In view of the
natural graded isomorphism $H_{R_{+},J}^{k}(M)\cong
H_{R_{+},J}^{k}(M/\Gamma_{R_{+},J}(M))$ for all $k\geq 1$ (see [TYY, Corollary
1.13 (4)]), we may assume $M$ is an $(R_{+},J)$-torsion-free $R$-module. Since
$\Gamma_{R_{+}}(M)\subseteq\Gamma_{R_{+},J}(M)$, we now use [BS, Lemma 2.1.1]
and [BH, Proposition 1.5.12] to deduce that $R_{1}$ contains an element $x$
which is a non zero-divisor on $M$. The exact sequence $0\rightarrow
M(-1)\stackrel{{\scriptstyle x}}{{\rightarrow}}M\rightarrow M/xM\rightarrow 0$
induces a graded long exact sequence

$H_{R_{+},J}^{k-1}(M)\rightarrow H_{R_{+},J}^{k-1}(M/xM)\rightarrow
H_{R_{+},J}^{k}(M)(-1)\rightarrow H_{R_{+},J}^{k}(M).$

Through this sequence, we obtain $H_{R_{+},J}^{j-1}(M/xM)$ is finitely
generated for all $j<i$. So, by induction, there exists some
$n_{1}\in\mathbb{Z}$ such that

${\rm Ass}_{R_{0}}(H_{R_{+},J}^{j-1}(M/xM)_{n})={\rm
Ass}_{R_{0}}(H_{R_{+},J}^{j-1}(M/xM)_{n_{1}})=:{\mathcal{A}}\,\,\,{\rm for\
all}\,\,n\leq n_{1}.$

Moreover, there exists some $n_{2}<n_{1}$ such that
$H_{R_{+},J}^{i-1}(M)_{n+1}=0$ for all $n\leq n_{2}$. So, for each $n\leq
n_{2}$, we have an exact sequence of $R_{0}$-modules

$0\rightarrow H_{R_{+},J}^{i-1}(M/xM)_{n+1}\rightarrow
H_{R_{+},J}^{i}(M)_{n}\rightarrow H_{R_{+},J}^{i}(M)_{n+1},$

induced by the above exact sequence. This shows that

${\mathcal{A}}\subseteq{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})\subseteq{\mathcal{A}}\cup{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n+1}),\ \text{for all}\ n\leq n_{2}.$

One can then conclude that

${\rm Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})\subseteq{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n+1}),\ \text{for all}\ n<n_{2}.$

∎

######  Theorem 5.6.

Let $M$ be a finitely generated graded $R$-module, $J$ an ideal of $R$
generated by elements of zero degree, and let $i\in\mathbb{N}$ be such that
$H_{R_{+},J}^{j}(M)_{n}$ is finitely generated $R_{0}$-module for all $j<i$
and $n<<0$. Then there exists a finite subset $X$ of $\emph{Spec}(R_{0})$ such
that ${\rm Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})=X$ for $n<<0$.

###### Proof.

This is a consequence of Lemma 5.5 and Theorem 5.4, once
${\rm Ass}_{R_{0}}(H_{R_{+},J}^{i}(M)_{n})$ is finite for all integer $n$. ∎

If the sequence $\\{{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{i}(M))_{n}\\}_{n\in\mathbb{Z}}$ satisfies the
assertion in Theorem 5.6, we say it is _asymptotically stable for_
$n\to-\infty$.

As an immediate consequence of Corollary 4.5, Lemma 5.5 and Theorem 5.6 we get
the following result.

######  Corollary 5.7.

Let $M$ be a finitely generated graded $R$-module.

  1. _(i)_

If ${\rm depth}(R_{+},J,M)=\dim M/({\mathfrak{m}_{0}}R+J)M$ then, for all
$i\geq 0$,
$\\{{\rm Ass}_{R_{0}}(H_{R_{+},J}^{i}(M))_{n}\\}_{n\in\mathbb{Z}}$ is
asymptotically stable for $n\to-\infty$;

  2. _(ii)_

If $t={\rm depth}(R_{+},J,M)$, then $\\{{\rm
Ass}_{R_{0}}(H_{R_{+},J}^{t}(M))_{n}\\}_{n\in\mathbb{Z}}$ is asymptotically
stable for $n\to-\infty$.

Last Corollary, item (ii), was showed in [BH, Proposition 5.6] for the case
$J=0.$

## References

  * [AAS] Aghapournahr, M.; Ahmadi-amolie, Kh.; Sadeghi, M.Y.;_The Concepts of Depth of a Pair of Ideals $(I,J)$ on modules and $(I,J)$-Cohen-Macaulay Modules_, Arxiv:1301.1015v1, 6 Jan 2013.
  * [BH] Brodmann, M. P.; Hellus, M.; _Cohomological Patterns of coherent sheaves over projective schemes_ , J. Pure Appl. Algebra 172 (2002) 165-182.
  * [BS] M. Brodmann, R.Y. Sharp; _Local Cohomology: an Algebraic Introduction with Geometric Applications_ , Cambridge Univ. Press, Cambridge, UK (1998).
  * [C] Chu, Lizhong; _Top local cohomology modules with respect to a pair of ideals_ , Proc. Amer. Math. Soc. 139 (2011), no. 3, 777 782.
  * [CW] Chu, Lizhong; Wang, Wing; _Some results on local cohomology modules defined by a pair of ideals_ , J. Math. Kyoto Univ. 49 (2009), no. 1, 193 200.
  * [DAE] Divaani-Aazar, K.; Esmkhani, M. A.; _Artiniannness of local cohomology modules of $ZD$-modules_, Commutative Algebra 33 (2008), no. 1, 59–72.
  * [HIO] Herrmann, H.; Ikeda, S.; Orbanz, U.; _Equimultiplicity and blowing up_ , An algebraic study. With an appendix by B. Moonen. Springer-Verlag, (1988) Berlin.
  * [L] Lim, C.; _Graded local cohomology and its associated primes_ , Ph. D. Thesis, Michigan State University (2002).
  * [M] Melkersson, L; _On asymptotic stability for sets of prime ideals connected with the powers of an ideal_ , Math. Proc. Cambridge Philos. Soc. 107 (1990) 267-271.
  * [Ma] Marley, T.; _Finitely graded local cohomology and the depths of graded algebras_ , Proceeding of the American Math. Soc. Volume 123, Number 12, December (1995).
  * [RS] Rotthaus, Christel; M. Sega, Liana; _Some properties of graded local cohomology modules_ , Journal of Algebra (2005), 283(1):232-247.
  * [TYY] Takahashi, Ryo; Yoshino, Yuji; Yoshizawa, Takeshi; _Local cohomology based on a nonclosed support defined by a pair of ideals_ , J. Pure Appl. Algebra 213 (2009), no. 4, 582 600.

Department of Mathematics, Institute of Mathematics and Computer Science,
ICMC, University of São Paulo, BRAZIL.

_E-mail address_<EMAIL_ADDRESS>
Department of Mathematics, Institute of Mathematics and Computer Science,
ICMC, University of São Paulo, BRAZIL.

_E-mail address_<EMAIL_ADDRESS>


11institutetext: Federal University of São Carlos, Brazil
11email<EMAIL_ADDRESS>22institutetext: São Paulo State University,
Brazil
22email<EMAIL_ADDRESS>33institutetext: University of
Wolverhampton, United Kingdom
33email<EMAIL_ADDRESS>
# MaxDropoutV2: An Improved Method to Drop out Neurons in Convolutional Neural
Networks††thanks: The authors are grateful to FAPESP grants #2013/07375-0,
#2014/12236-1, #2019/07665-4, Petrobras grant #2017/00285-6, CNPq grants
#307066/2017-7, and #427968/2018-6, as well as the Engineering and Physical
Sciences Research Council (EPSRC) grant EP/T021063/1.

Claudio Filipi Goncalves do Santos, Mateus Roder, Leandro A. Passos, João P.
Papa Department of Computing
São Paulo State University
Bauru, Brazil
{claudio.., mateus.roder, leandro.passos<EMAIL_ADDRESS>Claudio Filipi
Goncalves do Santos 11 0000-1111-2222-3333 Mateus Roder 22 1111-2222-3333-4444
Leandro A. Passos 33 2222–3333-4444-5555 João P. Papa 22 2222–3333-4444-5555

###### Abstract

In the last decade, exponential data growth supplied the machine learning-
based algorithms’ capacity and enabled their usage in daily life activities.
Additionally, such an improvement is partially explained due to the advent of
deep learning techniques, i.e., stacks of simple architectures that end up in
more complex models. Although both factors produce outstanding results, they
also pose drawbacks regarding the learning process since training complex
models denotes an expensive task and results are prone to overfit the training
data. A supervised regularization technique called MaxDropout was recently
proposed to tackle the latter, providing several improvements concerning
traditional regularization approaches. In this paper, we present its improved
version called MaxDropoutV2. Results considering two public datasets show that
the model performs faster than the standard version and, in most cases,
provides more accurate results.

##  1 Introduction

The last decades witnessed a true revolution in people’s daily life habits.
Computer-based approaches assume the central role in this process, exerting
fundamental influence in basic human tasks, such as communication and
interaction, entertainment, working, studying, driving, and so on. Among such
approaches, machine learning techniques, especially a subfield usually called
deep learning, occupy one of the top positions of importance in this context
since they empower computers with the ability to act reasonably in an
autonomous fashion.

Deep learning regards a family of machine learning approaches that stacks an
assortment of simpler models. The bottommost model’s output feeds the next
layer, and so on consecutively, with a set of possible intermediate operations
among layers. The paradigm experienced exponential growth and magnificent
popularity in the last years due to remarkable results in virtually any field
of application, ranging from medicine [19, 2, 11] and biology [12] to speech
recognition [10] and computer vision [14].

Despite the success mentioned above, deep learning approaches still suffer
from a drawback very commonly observed in real-world applications, i.e., the
lack of sufficient data for training the model. Such a constraint affects the
learning procedure in two main aspects: (i) poor classification rates or (ii)
overfitting to training data. The former is usually addressed by changing to a
more robust model, which generally leads to the second problem, i.e.,
overfitting. Regarding the latter, many works tackled the problem using
regularization approaches, such as the well-known batch normalization [5],
which normalizes the data traveling from one layer to the other, and dropout
[17], which randomly turns-off some neurons and forces the layer to generate
sparse outputs.

Even though dropout presents itself as an elegant solution to solve
overfitting issues, Santos et al. [15] claim that deactivating neurons at
random may impact negatively in the learning process, slowing down the
convergence. To alleviate this impact, the authors proposed the so-called
MaxDropout, an alternative that considers deactivating only the most active
neurons, forcing less active neurons to prosecute more intensively in the
learning procedure and produce more informative features.

MaxDropout obtained significant results considering image classification’s
task, however, at the cost of considerable computational cost. This paper
addresses such an issue by proposing MaxDropoutV2, an improved and optimized
version of MaxDropout capable of obtaining similar results with higher
performance and substantial reduction of the computational burden.

Therefore, the main contributions of this work are presented as follows:

  * •

to propose a novel regularization approach called MaxDropoutV2, which stands
for an improved and optimized version of MaxDropout;

  * •

to evaluate MaxDropoutV2 overall accuracy and training time performance,
comparing with the original MaxDropout and other regularization approaches;
and

  * •

to foster the literature regarding regularization algorithms and deep learning
in general.

The remainder of this paper is organized as follows. Section 2 introduces the
main works regarding Dropout and its variation, while Section 3 presents the
proposed approach. Further, Sections 4 and 5 describe the methodology adopted
in this work and the experimental results, respectively. Finally, Section 6
states conclusions and future work.

##  2 Related Works

The employment of regularization methods for training deep neural networks
(DNNs) architectures is a well-known practice, and its use is almost always
considered by default. The focus of such approaches is helping DNNs to avoid
or prevent overfitting problems, which reduce their generalization capability.
Besides, regularization methods also allow DNNs to achieve better results
considering the testing phase since the model becomes more robust to unseen
data.

Batch Normalization (BN) is a well-known regularization method that employs
the concept of normalizing the output of a given layer at every iteration in
the training process. In its seminal work, Ioffe and Szegedy [5] demonstrated
that the technique is capable of speeding up the convergence regarding the
task of classification. Further, several other works [23, 16, 21] highlighted
its importance, including the current state-of-the-art on image classification
[20].

Among the most commonly employed techniques for DNN regularization is the
Dropout, which is usually applied to train such networks in most of the
frameworks used for the task. Developed by Srivastava et al. [17], Dropout
shows significant improvements in a wide variety of applications of neural
networks, like image classification, speech recognition, and more. The
standard approach has a simple and efficient work procedure, in which a mask
that directly multiplies the weight connections is created at training time
for each batch. Such a mask follows a Bernoulli distribution, i.e., it assign
values $0$ with a probability $p$ and $1$ with a probability $1-p$. The
authors showed that the best value for $p$ in hidden layers is $0.5$. During
training, the random mask varies, which means that some neurons will be
deactivated while others will work normally.

Following the initial development of the Dropout method, Wang and Manning [22]
focused on exploring different sampling strategies, considering that each
batch corresponds to a new subnetwork taken into account since different units
are dropped out. In this manner, the authors highlighted that the Dropout
represents an approximation of a Markov chain executed several times during
training time. Also, the Bernoulli distribution tends to a Normal distribution
in a high dimensional space, such that Dropout performs best without sampling.

Similarly, Kingma et al. [6] proposed the Variational Dropout, which is a
generalization of the Gaussian Dropout with the particularity of learning the
dropout rate instead of randomly select one value. The authors aimed to reduce
the variance of the stochastic gradients considering the variational Bayesian
inference of a posterior over the model parameters, retaining the
parallelization by investigating the reparametrization approach.

Further, Gal et al. [3] proposed a new Dropout variant to reinforcement
learning models. Such a method aims to improve the performance and calibrate
the uncertainties once it is an intrinsic property of the Dropout. The
proposed approach allows the agent to adapt its uncertainty dynamically as
more data is provided. Molchanov et al. [9] explored the Variational Dropout
proposed by Kingma et al. [6]. The authors generalized the method to
situations where the dropout rates are unbounded, giving very sparse solutions
in fully-connected and convolutional layers. Moreover, they achieved a
reduction in the number of parameters up to $280$ times on LeNet architectures
and up to $68$ times on VGG-like networks with a small decrease in accuracy
rates. Such a fact highlights the importance of sparsity for robustness and
parameter reduction, while the overall performance for “simpler” models can be
improved.

Another class of regularization methods emerged in parallel, i.e., techniques
that change the neural network’s input. Among such methods, one can refer to
the Cutout [1], which works by cutting off/removing a region of the input
image and setting such pixels at zero values. Such a simple approach provided
relevant results in several datasets. In a similar fashion emerged the
RandomErasing [24], which works by changing the pixel values at random for a
given region in the input, instead of setting these values for zero.

Roder et al. [13] proposed the Energy-based Dropout, a method that makes
conscious decisions whether a neuron should be dropped or not based on the
energy analysis. The authors designed such a regularization method by
correlating neurons and the model’s energy as an index of importance level for
further applying it to energy-based models, as Restricted Boltzmann Machines.

##  3 MaxDropoutV2 as an improved version of MaxDropout

This section provides an in-depth introduction to MaxDropout-based learning.

###  3.1 MaxDropout

MaxDropout [15] is a Dropout-inspired [18] regularization task designed to
avoid overfitting on deep learning training methods. The main difference
between both techniques is that, while Dropout randomly selects a set of
neurons to be cut off according to a Bernoulli distribution, MaxDropout
establishes a threshold value, in which only neurons whose activation values
higher than this threshold are considered in the process. Results provided in
[15] show that excluding neurons using their values instead of the likelihood
from a stochastic distribution while training convolutional neuron networks
produces more accurate classification rates.

Algorithm 1 implements the MaxDropout approach. Line $2$ generates a
normalized representation of the input tensor. Line $3$ attributes the
normalized value to a vector to be returned. Further, Lines $4$ and $5$ set
this value to $0$ where the normalized tensor is bigger than the threshold.
This process is only performed during training. Concerning the inference, the
original values of the tensor are used.

Algorithm 1: Original MaxDropout code

⬇

1def MaxDropout(tensor, threshold):

2 norm_tensor = normalize(tensor)

3 return_tensor = norm_tensor

4 if norm_tensor > threshold:

5 return_tensor = 0 where

6 return return_tensor

Even though MaxDropout obtained satisfactory results for the task, it was not
tailored-designed for Convolutional Neural Networks (CNNs), thus presenting
two main drawbacks:

  * •

it does not consider the feature map spacial distribution produced from a CNN
layer output since it relies on individual neurons, independently of their
location on a tensor; and

  * •

it evaluates every single neuron from a tensor, which is computationally
expensive.

Such drawbacks motivated the development of an improved version of the model,
namely MaxDropoutV2, which addresses the issues mentioned above and provides a
faster and more effective approach. The following section describes the
technique.

###  3.2 MaxDropoutV2

The main difference between MaxDropout and MaxDropoutV2 is that the latter
relies on a more representative feature space. While MaxDropout compares the
values from each neuron directly, MaxDropoutV2 sums up these feature maps
considering the depth axis, thus providing a bidimensional representation. In
a nutshell, consider a CNN layer output tensor with dimensions $32\times
32\times 64$. The original MaxDropout performs $32\times 32\times 64$, i.e.,
$65,536$ comparisons. The proposed method sums up the values of the tensor
over axis one (which would be the depth of the tensor) for each $32\times 32$
kernel, thus performing only $1,024$ comparisons.

Algorithm 2 provides the implementation of the proposed approach. Line $2$
performs the sum in the depth axis. Similar to Algorithm 1, Line $3$ generates
a normalized representation of the sum in depth of the input tensor. Line $4$
creates the mask that defines what positions of the original tensor should be
dropped, i.e., set to $0$. Notice that the process is performed faster in
MaxDropoutV2 due to the reduced dimensionality of the tensor. Further, in
Lines $5$ and $6$, the tensor is unsqueezed and repeated so the mask can be
used along all the tensor dimensions. Finally, the mask is applied to the
tensor in Line $8$ and returned in Line $9$. These operations are only
performed during training, similar to the original.

Algorithm 2: MaxDropoutV2 code

⬇

1def MaxDropout_V2(tensor, threshold):

2 sum_axis = sum tensor along axis 1

3 sum_axis = normalized(sum_axis)

4 mask = 0 where sum_axis > threshold

5 mask_tensor = tensor.shape[0]

6 repetitions of mask

7

8 return_tensor = tensor * mask_tensor

9 return return_tensor

Fig. 1 depicts an example of application, presenting an original image in Fig.
1a and a simulation of output colors considering Dropout, MaxDropout, and
MaxDropoutV2, for Figs. 1b, 1c, and 1d, respectively.

(a) (b) (c) (d)

Figure 1: Simulation using colored images. The original colored image is
presented in (a), and its outcomes are bestowed after (b) Dropout, (c)
MaxDropout, and (d) MaxDropoutV2 transformations using a dropout rate of
$50$%.

##  4 Methodology

This section provides a brief description of the datasets employed in this
work, i.e., CIFAR-10 and CIFAR-100, as well all the setup considered during
the experiments.

###  4.1 Dataset

In this work, we consider the public datasets CIFAR-10 and CIFAR-100 [7] to
evaluate the performance of MaxDropoutV2 since both datasets are widely
employed in similar regularization contexts [24, 1, 15, 8]. Both datasets
comprise $60,000$ color images of animals, automobiles, and ships, to cite a
few, with a size of $32\times 32$ pixels. Such images are divided such that
$50,000$ instances are employed for training, and $10,000$ samples are
considered for evaluation purposes. The main difference between CIFAR-10 and
CIFAR-100 regards the number of classes, i.e., CIFAR-10 comprises $10$ classes
while CIFAR-100 is composed of $100$ classes.

###  4.2 Experimental Setup

To provide a fair comparison, we adopted the same protocol employed in several
works in literature [1, 15, 24], which evaluate the proposed techniques over
the ResNet-18 [4] neural network. Regarding the pre-processing steps, each
image sample is resized to $32\times 32$ pixels for further extracting random
crops of size $28\times 28$ pixels, with the addition of horizontal flip. The
network hyperparameter setup employs the Stochastic Gradient Descent (SGD)
with Nesterov momentum of $0.9$ and a weight decay of
$5\text{\times}{10}^{-4}$. The initial learning rate is initially set to $0.1$
and updated on epochs $60$, $120$, and $160$ by multiplying its value by
$0.2$. Finally, the training is performed during a total of $200$ epochs and
repeated during five rounds over each dataset to extract statistical measures.
It is important to highlight that this protocol is used in several other works
related to regularization on Deep Learning models [1, 15, 25]. In this work,
we compare our proposed method against other regularizers that explicitly
target to improve the results of CNNs.

Regarding the hardware setup, experiments were conducted using an Intel 2x
Xeon®E5-2620 @ 2.20GHz with 40 cores, a GTX $1080$ Ti GPU, and $128$ GB of
RAM. 111The code will be available in case of the paper acceptance.

##  5 Experimental Results

This section provides an extensive set of experiments where MaxDropoutV2 is
compared against several baselines considering both classification error rate
and time efficiency. Additionally, it also evaluates combining MaxDropoutV2
with other regularization techniques.

###  5.1 Classification Error

Table 1 shows the average error rate for all models and architectures
regarding the task of image classification. Highlighted values denote the best
results, which were obtained over five independent repetitions.

Table 1: Average classification error rate ($\%$) over CIFAR-10 and CIFAR-100 datasets. Notice ResNet18 results are provided as the baseline.  | CIFAR-10 | CIFAR-100
---|---|---
ResNet-18 [4] | $4.72\pm 0.21$ | $22.46\pm 0.31$
Cutout [1] | $\bm{3.99\pm 0.13}$ | $21.96\pm 0.24$
RandomErasing [25] | $4.31\pm 0.07$ | $24.03\pm 0.19$
LocalDrop [8] | $4.3$ | $22.2$
MaxDropout [15] | $4.66\pm 0.13$ | $21.93\pm 0.07$
MaxDropoutV2 (ours) | $4.63\pm 0.04$ | $\bm{21.92\pm 0.23}$

From the results presented in Table 1, one can observe that Cutout obtained
the lowest error rate over the CIFAR-10 dataset. Meanwhile, MaxDropoutV2
achieved the most accurate results considering the CIFAR-100 dataset, showing
itself capable of outperforming its first version, i.e., MaxDropout, over more
challenging tasks composed of a higher number of classes.

Additionally, Figure 2 depicts the convergence evolution of MaxDropout and
MaxDropoutV2 over the training and validation splits, in which the training
partition comprises $50,000$ samples, and the validation contains $10,000$
samples. In Figure 2, V1 stands for the MaxDropout method, and V2 stands for
the proposed approach. One can notice that MaxDropoutV2 does not overpass the
MaxDropout validation accuracy, mainly on the CIFAR-10 dataset. However, when
dealing with more classes and the same number of training samples, both
performances was almost the same, indicating the robustness of MaxDropoutV2.

|
---|---
Figure 2: Convergence analysis regarding the CIFAR-10 and CIFAR-100 datasets.

###  5.2 Combining Regularization Techiniques

A critical point about regularization concerns avoiding overfitting and
improving the results of a given neural network architecture in any case. For
instance, if some regularization approach is already applied, including
another regularization should still improve the outcomes. In this context,
MaxDropoutV2 performs this task with success, as shown in Table 2.

Table 2: Average classification error rate ($\%$) over CIFAR-10 and CIFAR-100 datasets combining MaxDropout and MaxDropoutV2 with Cutout. | CIFAR-10 | CIFAR-100
---|---|---
ResNet-18 [4] | $4.72\pm 0.21$ | $22.46\pm 0.31$
Cutout [1] | $3.99\pm 0.13$ | $21.96\pm 0.24$
MaxDropout [15] | $4.66\pm 0.13$ | $21.93\pm 0.07$
MaxDropoutV2 (ours) | $4.63\pm 0.04$ | $21.92\pm 0.23$
MaxDropout [15] \+ Cutout [1] | $\bm{3.76\pm 0.08}$ | $\bm{21.82\pm 0.13}$
MaxDropoutV2 (ours) + Cutout [1] | $3.95\pm 0.13$ | $\bm{21.82\pm 0.12}$

###  5.3 Performance Evaluation

The main advantage of MaxDropoutV2 over MaxDropout regards its computational
time. In this context, Tables 3 and 4 provides the average time demanded to
train both models considering each epoch and the total consumed time. Such
results confirm the hypothesis stated in Section 3.2 since MaxDropoutV2
performed around $10\%$ faster than the standard version.

Table 3: Time consumed in seconds for training ResNet-18 in CIFAR-10 dataset. | Seconds per Epoch | Total time
---|---|---
MaxDropout [15] | $32.8$ | $6,563$
MaxDropoutV2 (ours) | $\bm{29.8}$ | $\bm{5,960}$
Table 4: Time consumed in seconds for training ResNet-18 in CIFAR-100 dataset. | Seconds per Epoch | Total time
---|---|---
MaxDropout [15] | $33.1$ | $6,621$
MaxDropoutV2 (ours) | $\bm{30.2}$ | $\bm{6,038}$

###  5.4 Evaluating Distinct Drop Rate Scenarios

This section provides an in-depth analysis of MaxDropoutV2 and MaxDropout [15]
results considering a proper selection of the drop rate parameter. Tables 5
and 6 present the models’ results while varying the drop rate from $5$% to
$50$% considering CIFAR-10 and CIFAR-100 datasets, respectively.

Table 5: Mean error (%) concerning CIFAR-10 dataset. Drop Rate | MaxDropoutV2 | MaxDropout
---|---|---
$5$ | $\bm{4.63\pm 0.03}$ | $4.76\pm 0.09$
$10$ | $4.67\pm 0.13$ | $4.71\pm 0.09$
$15$ | $4.76\pm 0.12$ | $\bm{4.63\pm 0.11}$
$20$ | $4.66\pm 0.13$ | $4.70\pm 0.08$
$25$ | $4.75\pm 0.11$ | $4.70\pm 0.06$
$30$ | $4.63\pm 0.16$ | $4.67\pm 0.12$
$35$ | $4.70\pm 0.18$ | $4.71\pm 0.16$
$40$ | $4.74\pm 0.13$ | $4.79\pm 0.20$
$45$ | $4.65\pm 0.16$ | $4.71\pm 0.11$
$50$ | $4.71\pm 0.04$ | $4.75\pm 0.10$

Even though MaxDropoutV2 did not achieve the best results in Table 1, the
results presented in Table 5 show the technique is capable of yielding
satisfactory outcomes considering small drop rate values, i.e., $5\%$, while
the standard model obtained its best results considering a drop rate of
$15\%$. Additionally, one can notice that MaxDropoutV2 outperformed MaxDropout
in eight-out-of-ten scenarios, demonstrating the advantage of the model over
distinct circumstances.

Table 6: Mean error (%) concerning CIFAR-100 dataset. Drop Rate | MaxDropoutV2 | MaxDropout
---|---|---
$5$ | $22.26\pm 0.31$ | $22.05\pm 0.17$
$10$ | $22.19\pm 0.13$ | $22.06\pm 0.32$
$15$ | $22.25\pm 0.23$ | $22.16\pm 0.20$
$20$ | $22.26\pm 0.30$ | $21.98\pm 0.21$
$25$ | $22.02\pm 0.13$ | $\bm{21.93\pm 0.23}$
$30$ | $\bm{21.92\pm 0.23}$ | $22.07\pm 0.24$
$35$ | $22.00\pm 0.07$ | $22.10\pm 0.29$
$40$ | $22.09\pm 0.16$ | $22.16\pm 0.34$
$45$ | $21.95\pm 0.15$ | $22.31\pm 0.29$
$50$ | $22.13\pm 0.19$ | $22.33\pm 0.23$

In a similar fashion, Table 6 provides the mean classification error
considering distinct drop rate scenarios over CIFAR-100 dataset. In this
context, both techniques required larger drop rates to obtain the best
results, i.e., $25$ and $30$ for MaxDropout and MaxDropoutV2, respectively.
Moreover, MaxDropoutV2 outperformed MaxDropout in all cases when the drop
rates are greater or equal to $30$, showing more complex problems demand
higher drop rates.

###  5.5 Discussion

According to the provided results, the proposed method accomplishes at least
equivalent outcomes to the original MaxDropout, outperforming it in terms of
classification error in most cases. Moreover, MaxDropoutV2 presented itself as
a more efficient alternative, performing around $10\%$ faster than the
previous version for the task of CNN training.

The main drawback regarding MaxDropoutV2 is that the model is cemented to the
network architecture, while MaxDropout applicability is available to any
network’s architecture. In a nutshell, MaxDropoutV2 relies on a matrix or high
dimensional tensors designed to accommodate CNNs’ layers outputs, while the
standard MaxDropout works well for any neural network structure, such as
Multilayer Perceptrons and Transformers, for instance.

##  6 Conclusion and Future Works

This paper presented an improved version of the regularization method
MaxDropout, namely MaxDropoutV2, which stands for a tailored made
regularization technique for convolutional neural networks. In short, the
technique relies on a more representative feature space to accommodate the
convolutional layer outputs.

Experimental results showed the method significantly reduced the time demanded
to train the network, performing around $10\%$ faster than the standard
MaxDropout with similar or more accurate results. Moreover, it demonstrated
that MaxDropoutV2 is more robust to the selection of the drop rate parameter.
Regarding future work, we will evaluate MaxDropoutV2 in distinct contexts and
applications, such as object detection and image denoising.

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