On the associated prime ideals of local cohomology modules defined by a pair of ideals
Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module. For a non-negative integer $n$ it is shown that, if the sets $\Ass_R(\Ext^{n _{R(R/I,M))$ and $\Supp_R(\Ext^{i_{R(R/I,H^{j_{I,J (M)))$ are finite for all $i \leq n+1$ and all $j< n$, then so is \linebreak$\Ass_R(\Hom_{R(R/I,H^{n_{I,J(M)))$. We also study the finiteness of $\Ass_R(\Ext^{i_{R(R/I,H^{n_{I,J (M)))$ for $i=1,2$.\\ Keywords: local cohomology modules defined by a pair of ideals, spectral sequences, associated prime ideals. \\ MSC(2010): Primary 13D45; Secondary 13E05, 13E10.
1502.04978.tex

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\begin{document}


\title[On the associated prime ideals of local cohomology modules ...]
{On the associated prime ideals of local cohomology modules defined
by a pair of ideals}



\author[Kh. Ahmadi Amoli]{Khadijeh Ahmadi Amoli}
\address[Khadijeh Ahmadi Amoli]{Payame Noor University,  Po Box 19395-3697,  Tehran, Iran. }
 \email{khahmadi@pnu.ac.ir}

\author[Z. Habibi]{Zohreh Habibi}
\address[Zohreh Habibi]{  Payame Noor University,  Po Box 19395-3697 Tehran,
Iran.} \email{z\_habibi@pnu.ac.ir}


\author[M. Jahangiri]{Maryam Jahangiri }
\address[Maryam Jahangiri]{Faculty of Mathematical Sciences and Computer,  Kharazmi
University,  Tehran, Iran  and School of Mathematics,  Institute for
Research in Fundamental Sciences (IPM),  P.O. Box: 19395-5746,
Tehran, Iran. } \email{mjahangiri@ipm.ir and
jahangiri.maryam@gmail.com}

\thanks{The third author was in part supported by a grant from IPM (No.
92130111)}
%------------------------------------------------------------------------------------%

\maketitle


\begin{abstract}
Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$
and $M$ be an $R$-module. For a non-negative integer $n$ it is shown
that, if the sets $\Ass_R(\Ext^{n} _{R}(R/I,M))$ and
 $\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are finite for all $i
\leq n+1$ and all $j< n$, then
 so is \linebreak$\Ass_R(\Hom_{R}(R/I,H^{n}_{I,J}(M)))$. We also study
the finiteness of $\Ass_R(\Ext^{i}_{R}(R/I,H^{n}_{I,J} (M)))$ for
$i=1,2$.\\
\textbf{Keywords:}  local cohomology modules defined by a pair of
ideals,
spectral sequences, associated prime ideals.  \\
\textbf{MSC(2010):}  Primary 13D45; Secondary 13E05, 13E10.
\end{abstract}

\section{\bf Introduction}


Let  $R$ be a commutative Noetherian ring, $I$ and $J$ be two ideals
of $R$ and $M$ be an $R$-module. For all $i\in \N_0$ the $i$-th
local cohomology functor with respect to $(I,J)$, denoted by
$H^{i}_{I,J}(-)$, defined by Takahashi et. all in \cite{TAK} as the
$i$-th right derived functor of the $(I,J)$- torsion functor $\Gamma
_{I,J}(-)$, where $$\Gamma _{I,J}(M):=\{x \in M : I^{n}x\subseteq Jx
\  \text {for} \   n\gg 1\}.$$ This notion coincides  with  the
ordinary local cohomology functor $H^{i}_{I }(-)$ when $J=0$, see
\cite{B-SH}.

The main motivation for this generalization comes from the study of
a dual of ordinary local cohomology modules $H^{i}_{I }(M)$
(\cite{sch}). Basic facts and more information about local
cohomology defined by a pair of ideals can be obtained from
\cite{TAK}, \cite{chu1} and \cite{chu2}.

Hartshorne in \cite{HART} proposed the following conjecture:

`` Let $M$ be a finitely generated $R$-module and $\fa$ be an ideal
of $R$. Then $\Ext^ {i}_{R}(R/\fa, H^{j}_{\fa}(M))$ is finitely
generated for all $i\geq 0$ and $j\geq 0$."


Also, Huneke in \cite{HU} raised some crucial problems on local
cohomology modules. One of them was about the finiteness of the set
of associated prime ideals of the local cohomology modules $H^{i}_{I
}(M)$.

Although there are some counterexamples to theses conjectures, see
\cite{si}, but there are some partial positive answers in some
special cases too, see for example \cite{b} or \cite{bl}.

In this paper, we consider these two problems for local cohomology
modules defined by a pair of ideals over not necessary finitely
generated modules. In particular,  we investigate certain conditions
on these modules such that the set of
 associated prime ideals of \linebreak$\Ext^{i}_{R}(R/I, H^{j}_{I,J} (M))$ is finite.

More precisely, let $n\in \N_0$ and assume that the sets
$\Ass_R(\Ext^{n} _{R}(R/I,M))$ and
\linebreak$\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are finite
for all $i \leq n+ 1$ and all $j< n$ then, we use a spectral
sequence argument to show that
$\Ass_R(\Hom_{R}(R/I,H^{n}_{I,J}(M)))$ is finite, too (Theorem
\ref{ass1}).  Moreover, it is shown that if the sets
$\Ass_R(\Ext^{n+ 1} _{R}(R/I,M))$ and
 $\Supp(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are finite for all $i
\leq n+ 2$ and all $j< n$ then, so is
 $\Ass_R(\Ext^{1}_{R}(R/I,H^{n}_{I,J} (M)))$
 (Theorem \ref{ass2}).

We also present a necessary and sufficient condition for the
finiteness of the set \linebreak$\Ass_R(\Ext^{2}_{R}(R/I,H^{n}_{I,J}
(M)))$ (Theorem \ref{ass3}). These generalize some known results
concerning ordinary local cohomology modules.


 In \cite[3.6]{T-T} the authors study the $\underset{M}{\grade\fp}$ for all $\fp\in \Ass_R (H^{t}_{I,J}(M))$, where
 $$t= inf\{i\in \mathbb{N}_{0}:
H^{i}_{I,J}(M)\neq 0 \}$$
 and $M$ is a finitely generated $R$-module. But their proof is not correct. Actually, they use the equality
 $\Supp_{R}(M_{x})= \{\fp \in \Supp_{R}(M): x \notin  \fp\}$ which is not true. Here, we also made a correction to this
 result for not necessary finite modules
 (Theorem \ref{tal}).






\section{ Associated prime ideals}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, first, we are going to study the set of associated
prime ideals of some $\Ext$-modules of local cohomology modules
defined by a pair of ideals.

The following relation between associated prime ideals of modules in
an exact sequence is frequently used in our results.



\begin{lem}\label{*}
Let $M\rightarrow N\rightarrow K\rightarrow 0$ be an exact sequence
of $R$-modules. Then \emph{$\Ass(K)\subseteq \Supp(M)\cup \Ass(N)$.}
\end{lem}
\begin{proof}
Let $\fp\in \Ass(K)$. Assume that $ \fp \notin \Supp(M)$. Then
$M_{\fp}=0$ and so $N_{\fp}\cong K_{\fp}$. Since $\fp R_{\fp} \in
\Ass_{R_{\fp}}(N_{\fp})$, we get $\fp\in \Ass(N)$.
\end{proof}

Next lemma describes a convergence of Grothendieck spectral
sequences.
\begin{lem}\label{**}

Let $M$ be an $R$-module. Then the following convergence of spectral
sequences exists
$$\Ext^{i}_{R}(R/I,H^{j}_{I,J}(M))\overset{i}{\Rightarrow}\Ext^{i+j}
_{R}(R/I,M).$$

\begin{proof}
It is easy to see that $\Hom_{R}(R/I,\Gamma_{I,J}
(M))=\Hom_{R}(R/I,M)$. Also, for any injective $R$-module $E$,
$\Gamma_{I,J}(E)$ is an injective $R$-module, by \cite[3.2]{TAK} and
\cite[2.1.4]{B-SH}. Now, in view of \cite[10.47]{ROT}, the assertion
follows.
\end{proof}

\end{lem}

The following theorem, which concerns with Hartshorne's problem
mentioned in the introduction, is one of the main results in this
paper.
\begin{thm}\label{ass1}
Let $n$ be a non-negative integer and $M$ be an $R$-module such that
\linebreak \emph{$\Ass_R(\Ext^{n} _{R}(R/I,M))$} and
\emph{$\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J}(M)))$} are finite for
all $i \leq n+1$ and all $j< n$. Then
 so is $\Ass_R(\Hom_{R}(R/I,H^{n}_{I,J}(M))).$
\end{thm}
\begin{proof}
Consider the convergence of spectral sequences in Lemma \ref{**} and
note that $E_{2}^{i,j}=0$ for all $i < 0$. Therefore, for all $2\leq
r \leq n+1$ there exists an exact sequence
\begin{equation}\label{r}
0\rightarrow E_{r+1}^{0,n}\rightarrow
E_{r}^{0,n}\xrightarrow{d_{r}^{0,n}}E_{r}^{r,n+1-r}.
\end{equation}

Since, $E_{r}^{r,n+1-r}$ is a subquotient of $E_{2}^{r,n+1-r}=
\Ext^{r}_{R}(R/I,H^{n+1-r}_{I,J} (M))$, $\Supp_R
 (E_{r}^{r,n+1-r})$ is a finite set. So, the above exact sequence  implies that $\sharp \Ass_R (E_{r}^{0,n})< \infty$ if $\sharp
\Ass_R(E_{r+1}^{0,n})< \infty$. Also, from the fact that
$E_{2}^{i,j}=0$ for all $j < 0$, we have $E_{\infty}^{0,n}\cong
E_{n+2}^{0,n}$. Therefore, to prove the assertion it is enough to
show that $\Ass_R(E_{\infty}^{0,n})$ is a finite set.

Using the concept of the convergence of spectral sequences, there
exists a bounded filtration
$$0=\varphi^{n+1}H^{n}\subseteq \varphi^{n}H^{n}\subseteq...
\subseteq \varphi^{1}H^{n}\subseteq\varphi^{0}H^{n}=
\Ext^{n}_{R}(R/I,M)$$ of submodules of $\Ext^{n}_{R}(R/I,M)$ such
that
$$E_{\infty}^{i,n-i}\cong \varphi^{i}H^{n}/\varphi^{i+1}H^{n} \
\text{for  all} \   i=0,...,n.$$
 Therefore, $E_{n+1}^{n,0}\cong
E_{\infty}^{n,0}\cong \varphi^{n}H^{n}$ is a subquotient of
$E_{2}^{n,0}=\Ext^{n}_{R}(R/I,\Gamma_{I,J}(M))$. So, by assumption,
$\Supp_R(\varphi^{n}H^{n})$ is a finite set. Now, assume inductively
that $\sharp \Supp_R(\varphi^{i}H^{n})< \infty$ for all $1<i\leq n$.
Then, since
$$E_{n+1}^{1,n-1}\cong
E_{\infty}^{1,n-1}\cong \varphi^{1}H^{n}/\varphi^{2}H^{n}$$
 is a
subquotient of $E_{2}^{1,n-1}=\Ext^{1}_{R}(R/I,H^{n-1}_{I,J}(M))$,
we deduce that $\Supp_R(\varphi^{1}H^{n})$ is finite. But,
$$E_{\infty}^{0,n}\cong \Ext^{n}_{R}(R/I,M)/\varphi^{1}H^{n}$$
 and
Lemma \ref{*} implies that $\sharp \Ass_R(E_{\infty}^{0,n})<\infty$,
as desired.

\end{proof}

As an immediate consequence of Theorem \ref{ass1}, we obtain the
following result that is a generalization of \cite[2.3]{B-S-SH}.
 \begin{cor}\label{max}
Let M be a finite R-module. Suppose that there is an integer n such
that for all $i < n$ the set \emph{$\Supp_R(H^{i}_{I,J}(M))$} is
finite. Then \emph{$\Ass_R(H^{n}_{I,J}(M))$} is finite.
\end{cor}

\begin{cor}\label{final}
Let $M$ be a finite $R$-module and
\emph{$t=inf\{i|H^{i}_{I,J}(M)\neq 0 \}$} be an   integer. Then
\emph{$\Ass_R(\Hom_{R}(R/I,H^{t}_{I,J}(M)))$} is finite. If in
addition, \emph{${\underset{M}{\grade I}}=t$}, then for a maximal
$M$-sequence $x_{1},...,x_{t}$ in $I$, we have
\emph{$$\Ass_R(\Hom_{R}(R/I,H^{t}_{I,J}(M)))= \{\fp\in \Ass_R(M/
(x_{1},..,x_{t})M)\cap V(I); \underset{M}{\grade\fp=}t\}.$$}
\end{cor}
\begin{proof}
It is straightforward from Theorem \ref{ass1}, \cite[3.10]{T-T} and
\cite[2.6]{AGH-AM}.
\end{proof}
\begin{cor}\label{max}
Let $M$ be a finite $R$-module. Suppose that   $q=inf \{i:
H^{i}_{I,J} (M)$ is not Artinian $\}$ is an  integer, then
\emph{$\Ass_R (\Hom_{R}(R/I,H^{q}_{I,J} (M)))$} is finite.
\end{cor}

In the rest of this paper we consider the set of associated prime
ideals of some $\Ext$ modules of local cohomology modules defined by
a pair of ideals.
\begin{thm}\label{ass2}
Let $n$ be a non-negative integer and $M$ be an $R$-module such that
\linebreak\emph{$\Ass_R(\Ext^{n+1} _{R}(R/I,M))$} and \emph{$\Supp_R
(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$} are finite for all $i \leq
n+2$ and all $j<n$. Then so is  $\Ass_R(\Ext^{1}
_{R}(R/I,H^{n}_{I,J}(M)))$.
\end{thm}
\begin{proof}

Considering the convergence of the spectral sequences of Lemma
\ref{**}, we have to show that $\Ass_R(E_{2}^{1,n})$ is a finite
set. Using similar arguments as used in Theorem \ref{ass1}, one can
see that it is enough to show that $\Ass_R(E_{\infty}^{1,n})=
\Ass_R(E_{n+2}^{1,n})$ is a finite set.

By the concept of convergence of spectral sequences, there exists a
filtration
$$0=\varphi^{n+2}H^{n+1}\subseteq \varphi^{n+1}H^{n+1}\subseteq...
\subseteq \varphi^{1}H^{n+1}\subseteq\varphi^{0}H^{n+1}=
\Ext^{n+1}_{R}(R/I,M)$$ of submodules of $\Ext^{n+1}_{R}(R/I,M)$
such that $E_{\infty}^{i,n+1-i}\cong
\varphi^{i}H^{n+1}/\varphi^{i+1}H^{n+1}$ for all $i=0,...,n+1$.
Using the fact that $\sharp \Supp_R(E_{2}^{i,j})< \infty$ for all
$i\leq n+2$ and all $j<n$ one can see that
$\Supp_R(\varphi^{i}H^{n+1})$ is a finite set for all $i=2,...,n+2$.
Also, $\sharp \Ass_R(\varphi^{1}H^{n+1})<\infty$. Now, since
$$E_{n+2}^{1,n}\cong E_{\infty}^{1,n}\cong
\varphi^{1}H^{n+1}/\varphi^{2}H^{n+1},$$ using Lemma \ref{*}, we
have $\sharp \Ass_R(E_{\infty}^{1,n})<\infty$, and the result
follows.
\end{proof}

The following theorem presents a necessary and sufficient condition
for the finiteness of the set
$\Ass_R(\Ext^{i}_{R}(R/I,H^{n}_{I,J}(M)))$ when $ i= 1, 2.$

\begin{thm}\label{ass3}
Let $n$ be a non-negative integer and $M$ be an $R$-module such that
the sets \linebreak$\Supp_R(\Ext^{n+1} _{R}(R/I,M))$    and
 $\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$  are finite for
all $i \leq n+2$ and all $j< n$. Then   $\Ass_R
(\Hom_{R}(R/I,H^{n+1}_{I,J}(M)))$ is finite if and only if
 $\Ass_R (\Ext^{2}_{R}(R/I,H^{n}_{I,J}(M)))$  is finite.

\end{thm}
\begin{proof}
$(\Leftarrow)$ Again, consider the convergence of spectral sequences
of Lemma \ref{**} and assume that $Ass_R (E_{2}^{2,n})$ is finite.
Since $E_{2}^{i,j}=0$ for all $i < 0$ or $j < 0$, using similar
arguments as used in Theorem \ref{ass1}, one can see that
$E_{\infty}^{0,n+1}\cong E_{n+3}^{0,n+1}$ and in order to prove that
$\sharp \Ass_R (E_{2}^{0,n+1})<\infty$ we have to show that $\sharp
\Ass_R(E_{\infty}^{0,n+1})<\infty$.

 There exists a filtration
$$0=\varphi^{n+2}H^{n+1}\subseteq \varphi^{n+1}H^{n+1}\subseteq...
\subseteq \varphi^{1}H^{n+1}\subseteq\varphi^{0}H^{n+1}=
\Ext^{n+1}_{R}(R/I,M)$$ of submodules of $\Ext^{n+1}_{R}(R/I,M)$
such that $E_{\infty}^{0,n+1}\cong
\Ext^{n+1}_{R}(R/I,M)/\varphi^{1}H^{n+1}$.
 Since \linebreak$\sharp \Ass_R(\Ext^{n+1}_{R}(R/I,M))<\infty $ we have $\sharp \Ass_R
(E_{\infty}^{0,n+1})<\infty$, as desired

$(\Rightarrow)$ Now, assume that $\Ass_R
(\Hom_{R}(R/I,H^{n+1}_{I,J}(M)))<\infty$ and consider the exact
sequence
$$0\rightarrow \Ker d_{2}^{0,n+1}\rightarrow
E_{2}^{0,n+1}\xrightarrow{d_{2}^{0,n+1}} \Im
d_{2}^{0,n+1}\rightarrow 0.$$ Since $\Ker
d_{2}^{0,n+1}=E_{3}^{0,n+1}$ and $\sharp \Supp_R (E_{3}^{0,n+1})<
\infty$, in view of Lemma \ref{*}, we have \linebreak $\sharp \Ass_R
(\Im d_{2 }^{0,n+1})< \infty$. Now, using the exact sequence
$$0\rightarrow \Im d_{2}^{0,n+1}\rightarrow
E_{2}^{2,n}\xrightarrow{d_{2}^{2,n}} E_{2}^{4,n-1}$$ and the fact
that $ E_{2}^{4,n-1}=\Ext^{4}_{R}(R/I,H^{n-1}_{I,J}(M))$ has finite
support, we have $\sharp \Ass_R (E_{2}^{2,n})<\infty$, as desired.

\end{proof}
\begin{thm}\label{assnd}
Let $n$ be a non-negative integer and $M$ be an $R$-module of
dimension $d$, such that \emph{$\Ass_R(\Ext^{n+d} _{R}(R/I,M))$} and
\emph{$\Supp_R (\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$} are finite for
all $i \geq n+1$ and all $j<d$. Then \emph{$\Ass_R(\Ext^{n}
_{R}(R/I,H^{d}_{I,J}(M)))$} is finite.
\end{thm}
\begin{proof}
The method of the proof is similar to the Theorem \ref{ass2},
considering \cite[4.7]{TAK}.

%Considering the spectral sequences of Lemma \ref{**} we have to show
%that $\Ass(E_{2}^{n,d})$ is a finite set. using similar arguments as
%used in Theorem \ref{ass1}, one can see that it is enough to show
%that there exists an integer $k$ such that
%$\Ass(E_{\infty}^{n,d})\cong \Ass(E_{k}^{n,d})$ is finite. There
%exists a filtration


%$0=\varphi^{n+d+1}H^{n+d+1}\subseteq
%\varphi^{n+d}H^{n+d}\subseteq... \subseteq
%\varphi^{1}H^{n+d}\subseteq\varphi^{0}H^{n+d}=
%\Ext^{n+d}_{R}(R/I,M)$ of submodules of $\Ext^{n+d}_{R}(R/I,M)$ such
%that $E_{\infty}^{i,n+d-i}\cong
%\varphi^{i}H^{n+d}/\varphi^{i+1}H^{n+d}$ for all $i=0,...,n+d$.
%Using the fact that $\sharp \Supp(E_{2}^{i,j})< \infty$ for all
%$i\geq n+2$, one can see that $\Supp(\varphi^{i}H^{n+d})$ is a
%finite set for all $i=n+1,...,n+d$. Also, $\sharp
%\Ass(\varphi^{n}H^{n+d})<\infty$. Now, since $E_{k}^{n,d}\cong
%E_{\infty}^{n,d}\cong \varphi^{n}H^{n+d}/\varphi^{n+1}H^{n+d}$,
%using Lemma \ref{*}, we have $\sharp \Ass(E_{\infty}^{n,d})<\infty$,
%and the result follows.
\end{proof}

In the rest of this paper, we study "the grade" of prime ideals
$\fp\in \Ass_R (H^{t}_{I,J}(M))$ on $M$.

For an $R$-module $M$ and an ideal $\fa$ of $R$,   the grade of
$\fa$ on $M$ is defined by
$$\underset{M}{\grade\fa} := \inf \{i\in
\mathbb{N}_{0} : H^{i}_{\fa}(M)\neq 0\},$$
 if this infimum exists,
and $\infty$ otherwise. If $M$ is a finite $R$-module and $\fa M\neq
M$, this definition coincides with the length of a maximal
$M$-sequence in $\fa$ (cf. \cite[6.2.7]{B-SH}).

Also, we shall use the following notations introduced in \cite{TAK},
in which $W(I,J)$ is closed under specialization, but not
necessarily a closed subset of $\Spec(R)$. \[W(I,J):= \{\fp\in
\Spec(R): I^{n}\subseteq \fp + J \mbox{ for some integer }
n\geq1\},\] and

\[\widetilde{W}(I,J):= \{\fa: \fa \mbox{  is an ideal of } R \mbox{and} I^{n}\subseteq
\fa+J \mbox{ for some integer } n\geq 1\}.\]

The following lemma can be proved using \cite[3.2]{TAK}.



\begin{lem}\label{supp}
For any non-negative integer $i$ and $R$-module $M$,

$(i)$ \emph{$\Supp_R(H^{i}_{I,J}(M))\subseteq \underset{\fa \in
\widetilde{W}(I,J)}{\bigcup}\Supp(H^{i}_{\fa}(M))$}.

$(ii)$ \emph{$\Supp_R(H^{i}_{I,J}(M))\subseteq \Supp_R(M) \cap
W(I,J)$}.

\end{lem}

 The following theorem was proved in \cite[3.6]{T-T} under the hypothesis that $M$ is finite. But the proof is
not correct. Here we bring an extension and another proof of this
theorem.
\begin{thm}\label{tal}
Let $M$ be an $R$-module and $t=inf\{i\in \mathbb{N}_{0}:
H^{i}_{I,J}(M)\neq 0 \}$ be a non-negative integer. Then for all
\emph{$\fp\in \Ass_R (H^{t}_{I,J}(M))$},
\emph{$\underset{M}{\grade\fp=} t$.}
\end{thm}

\begin{proof}
We use induction on $t$. Let $t = 0$ and $\fp\in \Ass_R
(\Gamma_{I,J}(M))$. Then $\fp =(0:_{R}x)$ for some $x\in
\Gamma_{I,J}(M)$. Hence $x\in\Gamma_{\fp}(M)$ and so
$\Gamma_{\fp}(M)\neq 0$.

 Now suppose that $t>0$ and the case $t - 1$
is settled. Let $\fp \in \Ass(H^{t}_{I,J}(M))$ and consider the
exact sequence $0\rightarrow M\rightarrow E\rightarrow L\rightarrow
0$, where $E=E_{R}(M)$ is the injective envelope of $M$. Therefore,
using \cite[2.2]{T-T}, $H^{i}_{I,J}(L)\cong H^{i+1}_{I,J}(M)$ for
all $i\geq 0$ and we get
$$inf\{i\in \mathbb{N}_{0}: H^{i}_{I,J}(L)\neq 0\}= inf\{i\in
\mathbb{N}_{0}: H^{i}_{I,J}(M)\neq 0 \}-1=t-1$$ and that
$\fp\in\Ass_R(H^{t-1}_{I,J}(L))$ . Thus, by inductive hypothesis,
$\underset{L}{\grade\fp=} t-1$. Now, consider the long exact
sequence
$$H^{i-1}_{\fp}(M)\rightarrow H^{i-1}_{\fp}(E)\rightarrow
H^{i-1}_{\fp}(L)\rightarrow H^{i}_{\fp}(M).$$ If $t>1$, then
$H^{i}_{\fp}(M)\cong H^{i-1}_{\fp}(L)=0$ for all $i<t$ and
$H^{t}_{\fp}(M)\cong H^{t-1}_{\fp}(L)\neq 0$. Thus
$\underset{M}{\grade\fp=} t$.

Let $t=1$. Then $\Gamma_{\fp}(L)\neq 0$. By the above exact
sequence, it is enough to show that $\Gamma_{\fp}(E)=0$. On the
contrary, assume that $\Gamma_{\fp}(E)\neq 0$. Then there exists a
non-zero element $x\in E$ and $n\in \mathbb{N}$ such that
$\fp^{n}x=0$. We may assume that $\fp^{n}x=0$ and $\fp^{n-1}x\neq
0$. So, there exists $r\in \fp^{n-1}$ such that $rx\neq 0$. Thus
$\fp\subseteq (0:_{R}rx)$. On the other hand, by Lemma \ref{supp},
$$\fp\in \Ass_R(H^{1}_{I,J}(M))\subseteq \Supp_R
(H^{1}_{I,J}(M))\subseteq \underset{\fa \in
\widetilde{W}(I,J)}{\bigcup}\Supp_R(H^{1}_{\fa}(M)).$$ So that there
exists $\fa \in \widetilde{W}(I,J)$ such that $\fa\subseteq \fp$.
Let $m\in\mathbb{N}$ with $I^{m}\subseteq \fa +J\subseteq
\fp+J\subseteq (0:_{R}rx)+J$. Hence $rx\in \Gamma_{I,J}(M)$ which
contradicts with hypothesis and the choice of $rx$. Therefore
$\Gamma_{\fp}(E)=0$ and so $\underset{M}{\grade\fp=} 1$.
\end{proof}



% ------------------------------------------------------------------------

%\subsection*{Acknowledgment}


\begin{thebibliography}{1}
\bibitem{AGH-AM}

 M. Aghapournahr, Kh. Ahmadi-Amoli and M. Y. Sadeghi, \textit{The concepts of depth of a pair of ideals $(I,J)$ on modules and
$(I,J)$-Cohen-Macaulay modules}, arXiv:1301.1015v1 [math. AC] 6 Jan
2013.



\bibitem{B-S-SH}
K. Borna Lorestani, P. Sahandi and T. Sharif, \textit{A note on the
Associated Primes of Local Cohomology Modules},  Comm. Alg.,
{\bf34}(9) (2006)   3409-3412.

\bibitem{b}
M. Brodmann, {\em Asymptotic behaviour of cohomology: tameness,
supports and associated primes}, S. Ghorpade, H. Srinivasan, J.
Verma (Eds.), "Commutative Algebra and Algebraic Geometry"
Proceedings, Joint International Meeting of the AMS and the IMS on
Commutative Algebra and Algebraic Geometry, Bangalore/India,
December 17-20, 2003, Contemporary Mathematics Vol {\bf 390} (2005)
31-61.

\bibitem{bl}
 M. P. Brodmann and A. Lashgari Faghani, \textit{A finiteness result for associated primes
of local cohomology modules}, Proc. Amer. Math. Soc., {\bf 128}(10)
(2000) 2851- 2853.


\bibitem{B-SH}
 M. P. Brodmann and R. Y. Sharp, \textit{Local cohomology: An algebraic
introduction with geometric applications}, Cambridge University
Press, 1998.

\bibitem{chu1}
 L. Chu, \textit{Top local cohomology modules with respect to a pair of
ideals}, Proc. Amer. Math. Soc., \textbf{139}  (2011)  777-782.


\bibitem{chu2}
 L. Chu and Q. Wang, \textit{Some results on local cohomology modules
defined by a pair of ideals}, J. Math. Kyoto Univ.,\textbf{ 49}
(2009) , 193-200.

\bibitem{HART}
R. Hartshorne, \textit{Affine duality and cofiniteness}, Invent.
Math., {\bf9} (1970)  145-164.

\bibitem{HU}
C. Huneke, , Free resolutions in commutative algebra and algebraic
geometry, Res. Notes Math. 2, Jones and Bartlett, Boston, MA.,
(1992)  93-108.

\bibitem{ROT}
J. Rotman, \textit{An introduction to Homological Algebra}, Academic
Press, Second Edition, (2009).



\bibitem{sch}
P. Schenzel, \textit{Explicit computations around the
Lichtenbaum�-Hartshorne vanishing theorem}, Manuscripta Math. 78 (1)
(1993) 57�68.

\bibitem{si}
A. Singh, \textit{P-torsion elements in local cohomology modules}.
(English summary) Math. Res. Lett., {\bf7}  (2000)    165-176.

\bibitem{TAK}

 R. Takahashi, Y. Yoshino and T. Yoshizawa, \textit{Local cohomology
based on a nonclosed support defined by a pair of ideals}, J. Pure
Appl. Algebra., {\bf213}  (2009)  582-600.

\bibitem{T-T}


A. Tehranian and A. Pour Eshmanan Talemi, \textit{Cofinitness of
local cohomology based on a non-closed spport defiend by a pair of
ideals}, Bull. Iranian Math. Soc., {\bf36}(2) (2010) 145-155.




\end{thebibliography}

% ------------------------------------------------------------------------
\end{document}

1502.04978.extracted_bib
{AGH-AM M. Aghapournahr, Kh. Ahmadi-Amoli and M. Y. Sadeghi, The concepts of depth of a pair of ideals $(I,J)$ on modules and $(I,J)$-Cohen-Macaulay modules, arXiv:1301.1015v1 [math. AC] 6 Jan 2013.
1502.04978.extracted_bib
{B-S-SH K. Borna Lorestani, P. Sahandi and T. Sharif, A note on the Associated Primes of Local Cohomology Modules, Comm. Alg., {34(9) (2006) 3409-3412.
1502.04978.extracted_bib
{b M. Brodmann, {\em Asymptotic behaviour of cohomology: tameness, supports and associated primes, S. Ghorpade, H. Srinivasan, J. Verma (Eds.), "Commutative Algebra and Algebraic Geometry" Proceedings, Joint International Meeting of the AMS and the IMS on Commutative Algebra and Algebraic Geometry, Bangalore/India, December 17-20, 2003, Contemporary Mathematics Vol {390 (2005) 31-61.
1502.04978.extracted_bib
{bl M. P. Brodmann and A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., {128(10) (2000) 2851- 2853.
1502.04978.extracted_bib
{B-SH M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge University Press, 1998.
1502.04978.extracted_bib
{chu1 L. Chu, Top local cohomology modules with respect to a pair of ideals, Proc. Amer. Math. Soc., 139 (2011) 777-782.
1502.04978.extracted_bib
{chu2 L. Chu and Q. Wang, Some results on local cohomology modules defined by a pair of ideals, J. Math. Kyoto Univ., 49 (2009) , 193-200.
1502.04978.extracted_bib
{HART R. Hartshorne, Affine duality and cofiniteness, Invent. Math., {9 (1970) 145-164.
1502.04978.extracted_bib
{HU C. Huneke, , Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 2, Jones and Bartlett, Boston, MA., (1992) 93-108.
1502.04978.extracted_bib
{ROT J. Rotman, An introduction to Homological Algebra, Academic Press, Second Edition, (2009).
1502.04978.extracted_bib
{sch P. Schenzel, Explicit computations around the Lichtenbaum�-Hartshorne vanishing theorem, Manuscripta Math. 78 (1) (1993) 57�68.
1502.04978.extracted_bib
{si A. Singh, P-torsion elements in local cohomology modules. (English summary) Math. Res. Lett., {7 (2000) 165-176.
1502.04978.extracted_bib
{TAK R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra., {213 (2009) 582-600.
1502.04978.extracted_bib
{T-T A. Tehranian and A. Pour Eshmanan Talemi, Cofinitness of local cohomology based on a non-closed spport defiend by a pair of ideals, Bull. Iranian Math. Soc., {36(2) (2010) 145-155.