# Quantum Newton’s method for solving system of nonlinear algebraic equations

Cheng Xue  CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei, Anhui 230026, P. R. China  Yu-Chun Wu
<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, Anhui 230026, P. R. China  Institute
of Artificial Intelligence, Hefei Comprehensive National Science Center Hefei,
Anhui 230026, P. R. China  Guo-Ping Guo  CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei, Anhui
230026, P. R. China  Institute of Artificial Intelligence, Hefei Comprehensive
National Science Center Hefei, Anhui 230026, P. R. China  Origin Quantum
Computing, Hefei, Anhui 230026, P. R. China

###### Abstract

While quantum computing provides an exponential advantage in solving system of
linear equations, there is little work to solve system of nonlinear equations
with quantum computing. We propose quantum Newton’s method (QNM) for solving
$N$-dimensional system of nonlinear equations based on Newton’s method. In
QNM, we solve the system of linear equations in each iteration of Newton’s
method with quantum linear system solver. We use a specific quantum data
structure and $l_{\infty}$ tomography with sample error $\epsilon_{s}$ to
implement the classical-quantum data conversion process between the two
iterations of QNM, thereby constructing the whole process of QNM. The
complexity of QNM in each iteration is $O(\log^{4}N/\epsilon_{s}^{2})$.
Through numerical simulation, we find that when $\epsilon_{s}>>1/\sqrt{N}$,
QNM is still effective, so the complexity of QNM is sublinear with $N$, which
provides quantum advantage compared with the optimal classical algorithm.

##  I Introduction

The system of nonlinear equations appears in many fields, such as nonlinear
finite element analysisReddy (2004), nonlinear dynamicsThompson and Stewart
(2002), nonlinear programmingMangasarian (1994), etc. In general, the
analytical solutions of system of nonlinear equations cannot be obtained
effectively. Numerical methods are often used to solve system of nonlinear
equations. Newton’s method is a basic method for solving system of nonlinear
equationsRheinboldt (1974); Burden  _et al._ (2016). However, the complexity
of solving a system of nonlinear equations with Newton’s method grows
polynomially with the dimension of the equationsBurden  _et al._ (2016), using
Newton’s method to solve high-dimensional nonlinear equations is intractable.
It is important to develop more efficient algorithms for solving system of
nonlinear equations.

Quantum computing provides a promising way to speed up the solution of system
of nonlinear equations. In recent years there have been many quantum
algorithms developed to solve various equations, such as system of linear
equationsHarrow  _et al._ (2009); Childs  _et al._ (2017); Subaşı  _et al._
(2019), system of linear differential equationsClader  _et al._ (2013); Berry
(2014); Montanaro and Pallister (2016); Berry  _et al._ (2017); Xin  _et al._
(2020); Cao  _et al._ (2013); Costa  _et al._ (2019); Fillion-Gourdeau  _et
al._ (2017); Engel  _et al._ (2019); Arrazola  _et al._ (2019); Linden  _et
al._ (2020); Childs and Liu (2020); Childs  _et al._ (2020) and nonlinear
differential equationsLeyton and Osborne (2008); Lubasch  _et al._ (2020); Liu
_et al._ (2020); Lloyd  _et al._ (2020).

However, there is little work for the solution of system of nonlinear
equations. Qian $et\ al$.Qian  _et al._ (2019) proposed a quantum algorithm
for solving system of nonlinear equations, their work is based on Grover
algorithm and only has square acceleration. Rebentrost $et\ al$.Rebentrost
_et al._ (2019) proposed a quantum Newton’s method for constrained polynomial
optimization, the complexity of their method grows exponentially as iteration
times increases.

In our work, we propose a quantum Newton’s method (QNM) for the solution of
system of nonlinear equations. QNM is an iteration method, in each iteration,
we solve a system of linear equations with quantum linear system solver
(QLSS). Between two iterations, there are some classical-quantum data
conversions. We realize the conversions by using a quantum data structure and
$l_{\infty}$ tomographyKerenidis  _et al._ (2019) with sample error
$\epsilon_{s}$.

The complexity of QNM in each iteration is $O(\log^{4}N/\epsilon_{s}^{2})$.
Through numerical simulation, we found that when $\epsilon_{s}>>1/\sqrt{N}$,
QNM is still effective, so the complexity of QNM is sublinear with $N$.
Compared with the best classical algorithm, QNM has a significant
acceleration. The complexity of QNM grows linearly with iteration times, which
is better than the work proposed in Rebentrost  _et al._ (2019).

This paper is organized as follows. We introduce Newton’s method in Sec. II.
Then the details of QNM are discussed in Sec. III and we analyze the
complexity of QNM in Sec. IV. Sec. V gives some applications of QNM and some
numerical simulation to verify the effectiveness of QNM. Finally, we summarize
our work and propose some future research directions in Sec. VI.

##  II Newton’s Method

Newton’s method is an iterative method for solving system of nonlinear
equations. It obtains solution of the system of nonlinear equations by
iteratively solving system of linear equations.

The system of nonlinear equations is defined as:

$\bm{F}(\bm{x})=\bm{0},$ (1)

where

$\bm{F}(\bm{x})=\begin{bmatrix}f_{1}(\bm{x})\\\ \vdots\\\
f_{N}(\bm{x})\end{bmatrix},\bm{x}=\begin{bmatrix}x_{1}\\\ \vdots\\\
x_{N}\end{bmatrix},\bm{0}=\begin{bmatrix}0\\\ \vdots\\\ 0\end{bmatrix},$ (2)

each $f_{i}(\bm{x})$ is a real-valued nonlinear function that maps
$\bm{x}\in\mathbb{R}^{N}$ to $\mathbb{R}$. We use $\bm{x}^{*}$ to represent
the solution of Eq.(1). The Jacobi matrix of $\bm{F}(\bm{x})$ is defined as:

$\bm{F}^{\prime}(\bm{x})=(\partial_{j}f_{i}(\bm{x}))_{N\times
N}=(a_{ij})_{N\times N},$ (3)

The process of Newton’s method is shown in Algorithm $\bm{1}$.

Algorithm 1 Newton’s method

Input: $\bm{F}(\bm{x})$, initial vector $\bm{x}^{0}$ and accuracy error
$\epsilon$.

Output: Solution $\bm{x}^{*}$.

1: Assuming $k$ iterations have been performed, $\bm{x}^{k}$ and
$F(\bm{x}^{k})$ have been found. Compute Jacobi matrix
$\bm{F}^{\prime}(\bm{x}^{k})=A_{k}$, and define
$\bm{b}_{k}=-\bm{F}(\bm{x}^{k})$;

2: Solve the system linear of equations:  $A_{k}\Delta\bm{x}^{k}=\bm{b}^{k}.$
(4)

3: Compute $\bm{x}^{k+1}=\bm{x}^{k}+\Delta\bm{x}^{k}$ and
$\bm{F}(\bm{x}^{k+1})$.

4: If $||\bm{F}(\bm{x}^{k+1})||\leq\epsilon$, we have
$\bm{x}^{*}=\bm{x}^{k+1}$ and go to step 5. Otherwise, $k+1\to
k,\bm{x}^{k+1}\to\bm{x}^{k},\bm{F}(\bm{x}^{k+1})\to\bm{F}(\bm{x}^{k})$, go to
step 1.

5: Return $\bm{x}^{*}$.

6: End.

##  III QNM

In our work, we only consider sparse system of nonlinear equations, the Jacobi
matrix $\bm{F}^{\prime}(\bm{x})$ is a $d$-sparse matrix, which means
$\bm{F}^{\prime}(\bm{x})$ has at most $d$ non-zero elements in any row or
column. Throughout this paper, $d$ represents the sparsity of
$\bm{F}^{\prime}(\bm{x})$. In QNM, we need some oracles of Eq.(1). Suppose we
are given oracles $O_{f1}$ and $O_{f2}$ that provide the information of
Eq.(1), $O_{f1}$ and $O_{f2}$ are defined as:

$O_{f1}|i\rangle|j\rangle=|i,f(i,j)\rangle,i=1,2,...N,j=1,2,...d$ (5)
$O_{f2}|i\rangle|\bm{x}^{(i)}\rangle|0\rangle=|i\rangle|\bm{x}^{(i)}\rangle|f_{i}(\bm{x}^{(i)})\rangle,i=1,2,...N.$
(6)

where $f(i,j)$ is the subscript of the $j$-th variable in $f_{i}(\bm{x})$, and
$\bm{x}^{(i)}$ represents the related $x_{j}$ in $f_{i}(\bm{x})$. The
construction of $O_{f1}$ and $O_{f2}$ are related to specific problems. In
many specific problems, $O_{f1}$ and $O_{f2}$ can be implemented with quantum
arithmetic. In Sec.V.1, we give some specific examples.

QNM is a quantum version of Newton’s method. In each iteration of Newton’s
method, we solve a system of linear equations represented by Eq.(4). In QNM,
we implement the process of solving the system of linear equations with QLSS,
and give the conversion process of classical-quantum data during each
iteration. The overall block diagram of QNM is shown in Fig.1. Each iteration
of QNM is divided into three steps: preprocessing, processing and
postprocessing. The details of these three steps are introduced in Sec.III.2,
Sec.III.3 and Sec.III.4. During the execution of these 3 steps, we use a
quantum data structure $M_{F}$. Let us introduce $M_{F}$ first.

Figure 1: Overall block of quantum Newton’s method.

Figure 2: Structure of a 4-dimensional $M_{F}$.

###  III.1 Quantum data structure

In QNM, we use $M_{F}$ to implement some sub-processes. The structure of
$M_{F}$ is inspired by previous workKerenidis and Prakash (2016); Giovannetti
_et al._ (2008a, b). $M_{F}$ is a binary tree structure. In each leaf node, we
store $x_{i}$ and $f_{i}(\bm{x})$, In other nodes, we store $d_{ijx}$ and
$d_{ijy}$, which are defined as:

$\displaystyle d_{ijx}=\sqrt{\sum_{k=j\times 2^{n-i}}^{(j+1)\times
2^{n-i}-1}{x_{k}^{2}}},d_{ijy}=\sqrt{\sum_{k=j\times 2^{n-i}}^{(j+1)\times
2^{n-i}-1}{f_{k}^{2}(\bm{x})}}$ $\displaystyle i=0,...,n-1,\
j=0,...,2^{n-i}-1,$ (7)

where $i,j$ represent the $j$-th node of the $i$-th layer, $n=\lceil\log
N\rceil$. In the root node, we store $d_{00x}=||\bm{x}||$ and
$d_{00y}=||\bm{F}(\bm{x})||$. Fig. 2 shows the structure of $M_{F}$ with
dimension $N=4$. Given an $N$-dimensional $M_{F}$, it has such
propertiesKerenidis and Prakash (2016); Giovannetti  _et al._ (2008a, b):

  * (1)

Extract $x_{i}$ or $f_{i}(\bm{x})$ in $O(\log^{2}N)$ time complexity, which
are represented with $O_{M1}$ and $O_{M2}$:

$\displaystyle O_{M1}|i\rangle|0\rangle=|i\rangle|x_{i}\rangle,$
$\displaystyle O_{M2}|i\rangle|0\rangle=|i\rangle|f_{i}(\bm{x})\rangle,$
$\displaystyle i=0,1,...,N-1.$ (8)

  * (2)

Extract the norm of part of $\bm{x}$ or $\bm{F}(\bm{x})$ in $O(\log^{2}N)$
time complexity, which are represented with $O_{M3}$ and $O_{M4}$:

$\displaystyle
O_{M3}|i\rangle|j\rangle|0\rangle=|i\rangle|j\rangle|d_{ijx}\rangle,$
$\displaystyle
O_{M4}|i\rangle|j\rangle|0\rangle=|i\rangle|j\rangle|d_{ijy}\rangle,$
$\displaystyle i=0,1,...,n,\ j=0,1,...,2^{i}-1.$ (9)

  * (3)

The time complexity of Updating the whole $M_{F}$ is $O(M\log^{2}N)$, where
$M$ is the sum of the number of $x_{i}$ and $f_{j}(\bm{x})$ to be updated.

###  III.2 Preprocessing

Now we introduce three steps of each iteration in QNM. In preprocessing step,
we build operations to extract the information of the system of linear
equations in this iteration, which is defined in Eq.(10):

$A\Delta\bm{x}=-|\bm{b}\rangle,$ (10)

$A$ and $|\bm{b}\rangle$ are defined as:

$A=\frac{\bm{F}^{\prime}(\bm{x})}{||\bm{F}^{\prime}(\bm{x})||_{max}},$ (11)
$|\bm{b}\rangle=\frac{1}{C_{b}}\sum_{i}{f_{i}(\bm{x})|i\rangle},$ (12)

where $C_{b}$ is the normalization constant of $\bm{F}(\bm{x})$,
$||\bm{F}^{\prime}(\bm{x})||_{max}:=\max_{j,k}|F^{\prime}(\bm{x})_{jk}|$. We
assume that an upper bound of $||\bm{F}^{\prime}(\bm{x})||_{max}$ can be
estimated and we regard the upper bound as
$||\bm{F}^{\prime}(\bm{x})||_{max}$.

We build $O_{b}$ to prepare $|\bm{b}\rangle$ and $O_{A1}$, $O_{A2}$ to extract
the information of matrix $A$. $O_{b}$ is defined as:

$O_{b}|\bm{0}\rangle=|\bm{b}\rangle.$ (13)

We use the method proposed in Grover and Rudolph (2002) to build $O_{b}$. In
$O_{b}$’s construction process, we query $O_{M4}$ $O(\log N)$ times. $O_{A1}$
and $O_{A2}$ are defined as:

$O_{A1}|j,l\rangle=|j,h(j,l)\rangle,$ (14) $O_{A2}|j,k,z\rangle=|j,k,z\oplus
A_{jk}\rangle,$ (15)

where $h(j,l)$ is the column number of A’s $l$-th non-zero element. It is easy
to see that $O_{A1}$ and $O_{f1}$ are the same. We only need to build
$O_{A2}$. For simplicity, we set $|z\rangle=|0\rangle$, $O_{A2}$ is rewritten
as:

$O_{A2}|j,k,0\rangle=|j,k,||\bm{F}^{\prime}(\bm{x})||_{max}^{-1}\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}\rangle$
(16)

To build $O_{A2}$, we need a quantum circuit to compute
$\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}$. We use finite difference
method (FDM) to approximate $\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}$.
For example, Eq.(17) gives an approximation of
$\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}$.

$\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}\approx\frac{f_{j}(\bm{x}+\Delta
x_{k})-f_{j}(\bm{x})}{\Delta x_{k}}$ (17)

For each $f_{j}(\bm{x})$, we extract $\bm{x}^{(j)}$ by querying $O_{f1}$ and
$O_{M1}$ $d$ times. Next we use $O_{f2}$ to compute $|f_{j}(\bm{x})\rangle$.
We can also change $|\bm{x}^{(j)}\rangle$ to $|\bm{x}^{(j)}+\Delta
x_{k}\rangle$ and compute $|f_{j}(\bm{x}^{(j)}+\Delta x_{k})\rangle$. Finally
we compute $A_{jk}$ and uncompute the ancilla qubits we used. The whole
process is shown as follows:

$\displaystyle|j,k\rangle|0\rangle|0\rangle$
$\displaystyle\xrightarrow{O_{f1},\ O_{M1}}$
$\displaystyle|j,k\rangle|\bm{x}^{(j)}\rangle|0\rangle$
$\displaystyle\xrightarrow{O_{f2},\ FDM}$
$\displaystyle|j,k\rangle|\bm{x}^{(j)}\rangle|\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}\rangle$
$\displaystyle\xrightarrow{uncompute\quad\bm{x}^{(j)}}$
$\displaystyle|j,k\rangle|0\rangle|\frac{\partial{f_{j}(\bm{x})}}{\partial{x_{k}}}\rangle$
$\displaystyle\to$ $\displaystyle|j,k\rangle|0\rangle|A_{jk}\rangle$

###  III.3 Processing

In processing step, we use QLSS proposed in Childs  _et al._ (2017) to solve
Eq.(10). We input $O_{b}$, $O_{A1}$ and $O_{A2}$ and get
$|\Delta\bm{x}\rangle$, the normalized solution of Eq.(10). The success rate
of the QLSS is

$p=\frac{||A^{-1}|b\rangle||_{2}^{2}}{\alpha^{2}},$ (18)

$\alpha$ is defined as:

$\alpha:=\frac{4}{d}\sum_{j=0}^{j_{0}}\frac{\sum_{i=j+1}^{c}\begin{pmatrix}2c\\\
c+i\end{pmatrix}}{2^{2c}},$ (19)

where $j_{0}=\sqrt{c\log(4c/\epsilon)}$ and
$c=\kappa^{2}\log(\kappa/\epsilon)$Childs  _et al._ (2017). $p$ can be
estimated with quantum amplitude estimation algorithmBrassard  _et al._
(2000).

We execute QLSS multiple times and use $l_{\infty}$ tomographyKerenidis  _et
al._ (2019) to sample output state $|\Delta\bm{x}\rangle$ and get the sampled
state $|\widetilde{\Delta\bm{x}\rangle}$. The $l_{\infty}$ tomography is shown
in theorem 1, the sample times is $O(\frac{\log N}{\epsilon_{s}^{2}})$, where
$\epsilon_{s}$ represents the sample error. We also compute the normalization
constant of $\Delta\bm{x}$: $C_{\Delta\bm{x}}=||\Delta\bm{x}||_{2}$. Combine
Eq.(10), Eq.(11), Eq.(12) and Eq.(18), we have

$C_{\Delta\bm{x}}=\frac{\alpha
C_{b}\sqrt{p}}{||\bm{F}^{\prime}(\bm{x})||_{max}}.$ (20)

######  Theorem 1.

($l_{\infty}$ tomographyKerenidis  _et al._ (2019)). Given access to unitary
$U$ such that $U|0\rangle=|x\rangle$ and its controlled version in time
$T(U)$, there is a tomography algorithm with time complexity $O(T(U)\frac{\log
N}{\epsilon_{s}^{2}})$ that produces unit vector $\widetilde{X}\in R^{N}$ such
that $||\widetilde{X}-x||_{\infty}\leq\epsilon_{s}$ with probability at least
$(1-1/poly(N))$.

###  III.4 Postprocessing

In postprocessing step, we use the sampled state
$|\widetilde{\Delta\bm{x}\rangle}$ to update $M_{F}$ and determine whether to
stop the iteration.

We first update the root node of $M_{F}$: $x_{i}$ and $f_{i}(\bm{x})$. The
update of $x_{i}$ is:

$|\bm{x}\rangle\to|\bm{x}+C_{\Delta\bm{x}}\widetilde{\Delta\bm{x}}\rangle\approx|\bm{x}-\bm{F}^{\prime-1}(\bm{x})\bm{F}(\bm{x})\rangle$
(21)

From $|\widetilde{\Delta\bm{x}\rangle}$ we compute the changed $f_{i}(\bm{x})$
and update the changed $f_{i}(\bm{x})$. we use $N_{x}$, $N_{f}$ to represent
the number of changed $x_{i}$ and $f_{i}(\bm{x})$ respectively. $N_{x}$,
$N_{f}$ satisfy:

$N_{x}<\frac{\log N}{\epsilon_{s}^{2}},\ N_{f}<d\frac{\log
N}{\epsilon_{s}^{2}}.$ (22)

Then the update complexity of the whole $M_{F}$ is
$O(d\frac{\log^{3}N}{\epsilon_{s}^{2}})$.

The iteration cutoff condition is $||\bm{F}(\bm{x})||<\epsilon$. At the end of
each iteration, we use $O_{M4}$ introduced in Eq.((2)) to compute
$||\bm{F}(\bm{x})||$, we set $i=0,\ j=0$ and have

$O_{M4}|0\rangle|0\rangle|0\rangle=|0\rangle|0\rangle|||\bm{F}(\bm{x})||\rangle,$
(23)

If $||\bm{F}(\bm{x})||<\epsilon$, we stop iteration, otherwise, we execute the
next iteration until the iteration stops.

Up to now, we have introduced the whole process of QNM. The process is also
shown in Algorithm $\bm{2}$. The output of QNM is the updated $M_{F}$, it
saves the solution of the system of nonlinear equations.

Algorithm 2 Quantum Newton’s method

Input: initial $M_{F}$,$O_{f1}$, $O_{f2}$, sample error $\epsilon_{s}$ and
accuracy error $\epsilon$.

Output: $M_{F}$.

1: Construct $O_{b}$, $O_{A1}$ and $O_{A2}$.

2: Input $O_{b}$, $O_{A1}$ and $O_{A2}$ into QLSS introduced in Sec.III.3,
execute QLSS $O(\frac{\log N}{\epsilon_{s}^{2}})$ times and get
$|\widetilde{\Delta\bm{x}}\rangle$ with $l_{\infty}$ tomography.

3: Compute $C_{\Delta\bm{x}}$ with Eq.(20).

4: Update $M_{F}$ with the process decribed in Sec.III.4.

5: Compute $||\bm{F}(\bm{x})||$, if $||\bm{F}(\bm{x})||\leq\epsilon$, turn to
step 6. Otherwise, turn to step 1.

6: Return $M_{F}$.

7: End.

##  IV Complexity

In this section, we analyze the query complexity and time complexity of QNM.
First, we analyze the query complexity.

###  IV.1 Query complexity

In QNM, we use $O_{f1}$, $O_{f2}$ and $M_{F}$ to build other operations. Now
we analyze the query complexity of $O_{f1}$, $O_{f2}$ and $M_{F}$.

In $O_{b}$’s construction process, we query $M_{F}$ $O(\log N)$ timesGrover
and Rudolph (2002). In $O_{A2}$’s construction process introduced in
Sec.III.2, we query $O_{f1}$, $M_{F}$ $O(d)$ times and query $O_{f2}$ $O(1)$
times.

In QLSS, improved by variable-time amplitude amplificationAmbainis (2010), the
query complexity of $O_{A1}$, $O_{A2}$ and $O_{b}$ is
$O(d\kappa\text{poly}(\log(d\kappa/\epsilon)))$Childs  _et al._ (2017),
$\kappa$ represents the condition number of matrix $A$. As shown in Theorem.1,
the sampling complexity is $O(\frac{\log N}{\epsilon_{s}^{2}})$. In
postprocessing step, we query $M_{F}$ $O(1)$ times and we can omit it.
Therefore in each iteration, the query complexity of $M_{F}$ is

$O((d+\log N)\frac{d\kappa\log
N}{\epsilon_{s}^{2}}\text{poly}(\log(d\kappa/\epsilon))),$ (24)

the query complexity of $O_{f1}$ is

$O(\frac{d^{2}\kappa\log
N}{\epsilon_{s}^{2}}\text{poly}(\log(d\kappa/\epsilon))),$ (25)

and the query complexity of $O_{f2}$ is

$O(\frac{d\kappa\log
N}{\epsilon_{s}^{2}}\text{poly}(\log(d\kappa/\epsilon))).$ (26)

###  IV.2 Time Complexity

Next, we analyze the time complexity of QNM. Here we regard the time
complexity of $O_{f1}$ and $O_{f2}$ as $O(1)$. In construction process of
$O_{b}$, we query $O_{M4}$ $\log N$ times and perform $\log N$ quantum
arithmetic operations, the time complexity of quantum arithmetic is
$O(poly(\log(1/\epsilon)))$Mitarai  _et al._ (2019), therefore the time
complexity of $O_{b}$ is:

$\displaystyle T(O_{b})$ $\displaystyle=O(\log
N\times(poly(\log(1/\epsilon))+\log^{2}N))$ (27)

$O_{A2}$ is built with $O_{f1}$, $O_{f2}$, $O_{M1}$ and quantum arithmetic,
the time complexity of $O_{A2}$ is

$T(O_{A2})=O(d\log^{2}N+poly(\log(1/\epsilon)))$ (28)

Then the time complexity of QLSS isChilds  _et al._ (2017)

$\displaystyle T(QLSS)$ $\displaystyle=$ $\displaystyle
O(d\kappa^{2}\text{poly}(\log(d\kappa/\epsilon))(T(O_{b})+T(O_{A1})+T(O_{A2})))$
$\displaystyle=$ $\displaystyle
O(d\kappa^{2}\text{poly}(\log(d\kappa/\epsilon)(\log^{3}N+d\log^{2}N))$ (29)

The $\kappa$-dependence of $T(QLSS)$ can be improved from quadratic to nearly
linear with variable-time amplitude amplificationAmbainis (2010); Childs  _et
al._ (2017), then

$T(QLSS)=O(d\kappa\text{poly}(\log(d\kappa/\epsilon)(\log^{3}N+d\log^{2}N))$
(30)

In $l_{\infty}$ tomography, we execute QLSS $O(\frac{\log
N}{\epsilon_{s}^{2}})$ times. Finally, the time complexity of updating $M_{F}$
is $O(\frac{d\log^{3}N}{\epsilon_{s}^{2}})$. In summary, the time complexity
of one iteration is

$T_{q}=O\left(\frac{\log^{3}N}{\epsilon_{s}^{2}}d\kappa\text{poly}(\log(d\kappa/\epsilon)(\log
N+d)\right)$ (31)

In each iteration of classical Newton’s method, we need to solve a system of
linear equations, and one of the best algorithms to solve the system of linear
equations is conjugate gradient algorithmShewchuk (1994), the complexity of
conjugate gradient algorithm is $O(Nd\kappa\log(1/\epsilon))$. the complexity
of other processes in classical Newton’s method can be ignored. Therefore the
time complexity of one iteration of classical Newton’s method is

$T_{c}=O(Nd\kappa\log(1/\epsilon))$ (32)

Compare Eq.(31) and Eq.(32), we find that the $\kappa$-dependence and
$\epsilon$-dependence are nearly the same. The difference is mainly in $N$,
$d$ and $\epsilon_{s}$. In general, most of the real systems are sparse, $d$
has little effect on complexity, we can ignore $d$. The complexity of QNM
contains $\epsilon_{s}$. We do some numerical simulations in Sec.V and find
when $\epsilon_{s}>>1/\sqrt{N}$, QNM is still effective, then the time
complexity of QNM is sublinear with $N$. Therefore, compared with classical
Newton’s method, QNM provides a quantum advantage.

##  V Application and Numerical Simulation

QNM can be applied to many fields, including computational fluid dynamics,
nonlinear dynamics, economics, etc., to solve nonlinear problems in these
fields. A general solution process is divided into the following steps: (1)
Given a specific nonlinear problem, such as a nonlinear differential equation;
(2) Discretize the problem by numerical methods, such as FDM, FEM, etc., to
obtain the system of nonlinear equations to be solved. (3) Use QNM to solve
the system of nonlinear equations and obtain the solution of the problem.

Compared with classical Newton’s method, we have an $l_{\infty}$ tomography in
each iteration and update $\bm{x}$ with sampled state, it will affect
iteration speed or precision of QNM. To prove the effectiveness of QNM, we do
some numerical simulations to test the influence of sampling error
$\epsilon_{s}$ on QNM.

We choose two problems: (1) Nonlinear first-order diffusion problem; (2) Beam
lateral vibration problem. We use FDM to discretize the two problems and use
QNM to solve the systems of nonlinear equations obtained by FDM. We first
introduce the details of these two problems.

###  V.1 Problems and difference format

####  V.1.1 Nonlinear first-order diffusion problem

The first problem is solving a nonlinear first-order diffusion problem with
our method, the problem is defined as:

$\begin{cases}\frac{\partial u(x,t)}{\partial t}=g(u,x,t)\frac{\partial
u(x,t)}{\partial x}+f(x,t),-1<x<1,t>1,\\\ g(u,x,t)=-1+x\sin(u(x,t)),\\\
f(x,t)=2e^{2t}[x^{2}-1+x-x^{2}\sin(e^{2t}(e^{2}-1))]\end{cases}$ (33)

The initial condition and boundary condition are:

$\begin{cases}u(x,0)=x^{2}-1,-1<x<1;\\\ u(-1,t)=0,t>0.\end{cases}$ (34)

We discrete the equation in a such way:

$\begin{cases}x_{i}=-1+(i+1)h,i=0,1,...,N_{1}-1,h=\frac{2}{N_{1}},\\\
t_{j}=(j+1)s,j=0,1,...,N_{2}-1,s=\frac{1}{N_{2}}.\end{cases}$ (35)

We define $u_{i,j}=u(x_{i},t_{j})$, $u_{i,j,x}$ and $u_{i,j,t}$ are written
as:

$\begin{cases}u_{i,j,x}=\frac{u_{i,j}-u_{i-1,j}}{h},\\\
u_{i,j,t}=\frac{u_{i,j}-u_{i,j-1}}{s}.\end{cases}$ (36)

Then we have the following system of nonlinear equations:

$\begin{cases}u_{i,j,t}+(1-x_{i}\sin(u_{i,j}))u_{i,j,x}-f(x_{i},t_{j})=0,\\\
i=0,1,...,N_{1}-1;\ j=0,1,...,N_{2}-1.\\\
u_{-1,j}=0,u_{i,-1}=x_{i}^{2}-1.\end{cases}$ (37)

The Jacobi Matrix of Eq.(37) is an $N_{1}\times N_{2}$-dimensional sparse
square matrix and the position of non-zero element is regular, the position of
non-zero element of $m=N_{1}\times j+i$ row is represented as:

$S(m)=\begin{cases}\\{m\\},\ i=0,j=0,\\\ \\{m-1,m\\},\ i\neq 0,j=0,\\\
\\{m-N_{1},m\\},\ i=0,j\neq 0,\\\ \\{m-N_{1},m-1,m\\},\ i\neq 0,j\neq
0.\end{cases}$ (38)

The oracles $O_{f1}$ and $O_{f2}$ of Eq.(37) can be constructed by realizing
Eq.(38) and Eq.(37) with some quantum arithmetic circuits. The gate complexity
is $O(\log N+\text{poly}(\log(1/\epsilon)))$Nielsen and Chuang (2002). When
considering the complexity of $O_{f1}$ and $O_{f2}$ in the overall time
complexity of our method, we can find that the complexity of $O_{f1}$ and
$O_{f2}$ can be ignored, and the overall time complexity does not change.

####  V.1.2 Beam lateral vibration problem

The second problem is a beam lateral vibration problem, the problem is defined
as:

$\begin{cases}g(x)\frac{\partial^{4}\omega}{\partial x^{4}}+2\frac{\partial
g}{\partial x}\frac{\partial^{3}\omega}{\partial
x^{3}}+\frac{\partial^{2}g}{\partial x^{2}}\frac{\partial^{2}\omega}{\partial
x^{2}}+\mu(x)\frac{\partial^{2}\omega}{\partial t^{2}}\\\ +G(\omega)=f(x,t),\
-4\leq x\leq 4,0\leq t\leq 2,\\\ \mu(x)=e^{-2x^{2}}+1,g(x)=2e^{-2x^{2}}+1,\\\
G(\omega)=(1+2e^{-2x^{2}})\omega+(5+e^{-3x^{2}})\omega^{3},\end{cases}$ (39)

where $f(x,t)$ is determined by analytical solution
$\omega(x,t)=e^{-x^{2}-t^{2}}$. The initial condition and the boundary
condition are:

$\begin{cases}\omega(x,0)=e^{-x^{2}},\omega_{t}(x,0)=0,\\\ \omega(\pm
4,t)=0,\omega_{x}(\pm 4,t)=0.\end{cases}$ (40)

We discrete the equation in a such way:

$\begin{cases}x_{i}=-4+(i+1)h,i=0,1,...,N_{1}-1,h=\frac{8}{N_{1}},\\\
t_{j}=(j+1)s,j=0,1,...,N_{2}-1,s=\frac{2}{N_{2}}.\end{cases}$ (41)

$\omega_{i,j}$ represents $\omega(x_{i},t_{j})$, the difference format is:

$\begin{cases}\omega_{i,j,xx}=\frac{\omega_{i+1,j}-2\omega_{i,j}+\omega_{i-1,j}}{h^{2}},\\\
\omega_{i,j,tt}=\frac{\omega_{i,j}-2\psi_{i,j-1}+\omega_{i,j-2}}{s^{2}},\\\
\omega_{i,j,xxx}=\frac{\omega_{i+2,j}-2\omega_{i+1,j}+2\omega_{i-1,j}-\psi_{i-2,j}}{2h^{3}},\\\
\omega_{i,j,xxxx}=\frac{\omega_{i+2,j}-4\omega_{i+1,j}+\omega_{i,j}-4\omega_{i-1,j}+\omega_{i-2,j}}{h^{4}}.\end{cases}$
(42)

The system of nonlinear equations is represented as:

$\begin{cases}g(x_{i})\omega_{i,j,xxxx}+2g_{x}(x_{i})\omega_{i,j,xxx}+g_{x}x(x_{i})\omega_{i,j,xx}\\\
+\mu(x_{i})\omega_{i,j,tt}+G(\omega_{i,j})-f(x_{i},t_{j})=0,\\\
i=0,1,...,N_{1}-1;\ j=0,1,...,N_{2}-1.\end{cases}$ (43)

and based on the boundary condition, we have

$\begin{cases}\omega_{-1,j}=\omega_{N_{1},j}=0,\\\
\omega_{-2,j}=\omega_{0,j},\omega_{N_{1}+1,j}=\omega_{N_{1}-1,j},\\\
\omega_{i,-1}=e^{-x_{i}^{2}},\omega_{i,-2}=\omega_{i,0}.\end{cases}$ (44)

Combine Eq.(43) and Eq.(44), we have an $N_{1}\times N_{2}$-dimensional system
of nonlinear equations. Similar to the nonlinear first-order diffusion problem
described in Sec.V.1.1, the oracles $O_{f1}$ and $O_{f2}$ of this problem can
also be constructed by realizing Eq.(44) and Eq.(43) with some quantum
arithmetic circuits.

###  V.2 Numerical simulations

Now we show the details of our numerical simulations. We use C++ to implement
our numerical simulations. In our simulation, we use the LU decomposition
linear solverSaad (2003) in the C++ $Eigen$ template library to solve the
linear system in each iteration of QNM and get $\Delta\bm{x}$, then we execute
Algorithm $\bm{3}$ in Kerenidis  _et al._ (2019) to get the sampled
$\Delta\bm{x}$ and update $\bm{x}$ with the sampled $\Delta\bm{x}$. We set the
difference points of the two dimensions as $N_{1}=200$, $N_{2}=200$, then we
have $N=40000$-dimensional system of nonlinear equations.

The numerical simulations of the two problems are shown in Fig.3. From the
numerical simulations, we get such results: (1) When
$\epsilon_{s}=1/\sqrt{N}=0.005$, QNM converges for the two problems. When
$\epsilon_{s}>1/\sqrt{N}$, QNM also converges, the upper bound of
$\epsilon_{s}$ when QNM converges for the two problems is $0.05$ and $0.1$
respectively. (2) Sample error $\epsilon_{s}$ slows down the iteration speed
and increases iteration times. Compared with $N$, iteration times has little
influence on the complexity of QNM, so we can ignore the influence on
iteration times. (3) When $\epsilon_{s}$ is relatively large,
$||\bm{F}(\bm{x})||$ cannot converge to the result that the sampling error is
0, so QNM can only get an approximate solution.

We also test the effects of $\epsilon_{s}$ on convergent $||\bm{F}(\bm{x})||$.
In Fig.4, we repeat QNM 20 times to get the statistical results of the effects
of $\epsilon_{s}$ on convergent $||\bm{F}(\bm{x})||$. We get the following
results: (1) As $\epsilon_{s}$ increases, the convergent $||\bm{F}(\bm{x})||$
also increases. In Fig.4(a), there is an interval in which $\epsilon_{s}$
increases and convergent $||\bm{F}(\bm{x})||$ decreases, but the overall trend
that convergent $||\bm{F}(\bm{x})||$ increases with the increase of
$\epsilon_{s}$ remains unchanged. (2) In both problems, convergent
$||\bm{F}(\bm{x})||<<\epsilon_{s}$, so even if we take a relatively large
$\epsilon_{s}$, we can still get a result error much lower than
$\epsilon_{s}$.

From these numerical simulations, we can see that when
$\epsilon_{s}>>1/\sqrt{N}$, QNM is still effective, the convergent
$||\bm{F}(\bm{x})||$ is much smaller than $\epsilon_{s}$.

(a)

(b)

Figure 3: Numerical Results. Panels (a),(b) represent the influence of sample
error $\epsilon_{s}$ on the convergence process of nonlinear first-order
diffusion problem and beam lateral vibration problem respectively.

(a)

(b)

Figure 4: Relationship between $\epsilon_{s}$ and convergent
$||\bm{F}(\bm{x})||$. Panels (a),(b) represent nonlinear first-order diffusion
problem and beam lateral vibration problem respectively.

##  VI Conclusion

In this paper, we developed QNM for solving system of nonlinear equations. We
give the details of the QNM, analyze the query complexity and time complexity
of QNM. We also discuss the application of QNM do some numerical simulations
to study the influence of $\epsilon_{s}$ on QNM. We find when
$\epsilon_{s}>>1/\sqrt{N}$, QNM is still effective, so the complexity of QNM
is sublinear with $N$. Compared with optimal classical algorithm, QNM provides
a quantum advantage. We also notice that when $\epsilon_{s}<1/\sqrt{N}$, the
complexity of QNM is superlinear with $N$, which is worse than classical
Newton’s method.

QNM has the potential to provide quantum acceleration in many fields, such as
nonlinear finite element analysisReddy (2004), nonlinear dynamicsThompson and
Stewart (2002), nonlinear programmingMangasarian (1994) and so on. In the
future, we will develop the application of QNM in various practical problems,
test the impact of $\epsilon_{s}$ on QNM in various specific problems.

## Acknowledgement

This work was supported by the National Key Research and Development Program
of China (Grant No. 2016YFA0301700), the National Natural Science Foundation
of China (Grants Nos. 11625419), the Strategic Priority Research Program of
the Chinese Academy of Sciences (Grant No. XDB24030600), and the Anhui
Initiative in Quantum Information Technologies (Grants No. AHY080000).

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# Weakly cofiniteness of local cohomology modules

Moharram Aghapournahr  Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and
$I\in\Phi$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that
$\dim M\leq 1$, then the $R$-module $\operatorname{Ext}^{i}_{R}(R/I,M)$ is
weakly Laskerian, for all $i\geq 0$, if and only if the $R$-module
$\operatorname{Ext}^{i}_{R}(R/I,M)$ is weakly Laskerian, for $i=0,1$. Let
$t\in\mathbb{N}_{0}$ be an integer and $M$ an $R$-module such that
$\operatorname{Ext}^{i}_{R}(R/I,M)$ is weakly Laskerian for all $i\leq t+1$.
We prove that if the $R$-module $\operatorname{H}^{i}_{\Phi}(M)$ is ${\rm
FD_{\leq 1}}$ for all $i<t$, then $\operatorname{H}^{i}_{\Phi}(M)$ is
$\Phi$-weakly cofinite for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or
minimax) submodule $N$ of $\operatorname{H}^{t}_{\Phi}(M)$, the $R$-modules
$\operatorname{Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)$ and
$\operatorname{Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)$ are weakly
Laskerian. Let $N$ be a finitely generated $R$-module. We also prove that
$\operatorname{Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ and ${\rm
Tor}^{R}_{j}(N,H^{i}_{\Phi}(M))$ are $\Phi$-weakly cofinite for all $i$ and
$j$ whenever $M$ is weakly Laskerian and $\operatorname{H}^{i}_{\Phi}(M)$ is
${\rm FD_{\leq 1}}$ for all $i$. Similar results are true for ordinary local
cohomology modules and local cohomology modules defined by a pair of ideals.

###### Key words and phrases:

Local cohomology, ${\rm FD_{\leq n}}$ modules, weakly cofinite modules,
$ETH$-weakly cofinite modules

###### 2010 Mathematics Subject Classification:

13D45, 13E05, 14B15.

##  1\. Introduction

Throughout this paper $R$ is a commutative Noetherian ring with non-zero
identity and $I$ an ideal of $R$. For an $R$-module $M$, the $i^{th}$ local
cohomology module $M$ with respect to ideal $I$ is defined as

$\operatorname{H}^{i}_{I}(M)\cong\underset{n}{\varinjlim}\operatorname{Ext}^{i}_{R}(R/{I}^{n},M).$

Grothendieck in [19] posed the following conjecture:

Conjecture 1.1. Let $M$ be a finitely generated $R$–module and $I$ an ideal of
$R$. Then $\operatorname{Hom}_{R}(R/I,\operatorname{H}^{i}_{I}(M))$ is finite
for all $i\geq 0$.

This conjecture is not true in general as Hartshorne showed in [21], but some
authors proved that for some number $t$, the module
$\operatorname{Hom}_{R}(R/I,H^{t}_{I}(M))$ is finite under some conditions.
See [5, Theorem 3.3], [15, Theorem 6.3.9], [16, Theorem 2.1], [6, Theorem
2.6], [7, Theorem 2.3] and [2, Theorem 3.4]. Hartshorne also defined a module
$M$ to be _$I$ –cofinite_ if
$\operatorname{Supp}_{R}(M)\subseteq\operatorname{V}(I)$ and
$\operatorname{Ext}^{i}_{R}(R/I,M)$ is finitely generated for all $i\geq 0$
and posed the following question:

Question 1.2. _Let $M$ be a finite $R$–module and $I$ be an ideal of $R$. When
are $\operatorname{H}^{i}_{I}(M)$ $I$–cofinite for all $i\geq 0$?_

This question was studied by several authors in [21, 22, 14, 25, 30, 28, 6]
and [2].

There are some generalizations of the theory of ordinary local cohomology
modules. The following is introduced by Bijan-Zadeh in [10].

Let $\Phi$ be a non-empty set of ideals of $R$. We call $\Phi$ a system of
ideals of $R$ if, whenever $I_{1},I_{2}\in\Phi$, then there is an ideal
$J\in\Phi$ such that $J\subseteq I_{1}{I_{2}}$. For such a system, for every
$R$-module $M$, one can define

$\operatorname{\Gamma}_{\Phi}(M)=\\{~{}x\in M\mid Ix=0\textmd{\ \ for
some}~{}I\in\Phi\\}.$

Then $\operatorname{\Gamma}_{\Phi}(-)$ is a functor from $\mathscr{C}(R)$ to
itself (where $\mathscr{C}(R)$ denotes the category of all $R$-modules and all
$R$-homomorphisms). The functor $\operatorname{\Gamma}_{\Phi}(-)$ is additive,
covariant, $R$-linear and left exact. In [11],
$\operatorname{\Gamma}_{\Phi}(-)$ is denoted by $L_{\Phi}(-)$ and is called
the “general local cohomology functor with respect to $\Phi$”. For each $i\geq
0$, the $i$-th right derived functor of $\operatorname{\Gamma}_{\Phi}(-)$ is
denoted by $\operatorname{H}_{\Phi}^{i}(-)$. The functor
$\operatorname{H}_{\Phi}^{i}(-)$ and
$\underset{I\in\Phi}{\varinjlim}\operatorname{H}_{I}^{i}(-)$ (from
$\mathscr{C}(R)$ to itself) are naturally equivalent (see [10]). For an ideal
$I$ of $R$, if $\Phi=\\{I^{n}|n\in\mathbb{N}_{0}\\}$, then the functor
$\operatorname{H}_{\Phi}^{i}(-)$ coincides with the ordinary local cohomology
functor $\operatorname{H}_{I}^{i}(-)$. It is shown that, the study of torsion
theory over $R$ is equivalent to study the general local cohomology theory
(see [11]).

As a special case of [35, Definition 2.1] and generalization of FSF modules
(see [23, Definition 2.1]), in [2, Definition 2.1] the author of present paper
and Bahmanpour introduced the class of ${\rm FD_{\leq n}}$ modules. A module
$M$ is said to be ${\rm FD_{\leq n}}$ module, if there exists a finitely
generated submodule $N$ of $M$ such that $\dim M/N\leq n$. For more details
about properties of this class see [2, Lemma 2.3]. Note that the class of
${\rm FD_{\leq-1}}$ is the same as finitely generated $R$-modules. Recall that
a module $M$ is a _minimax_ module if there is a finitely generated submodule
$N$ of $M$ such that the quotient module $M/N$ is Artinian. Minimax modules
have been studied by Zöschinger in [37]. Recall too that an $R$-module $M$ is
called _weakly Laskerian_ if $\operatorname{Ass}_{R}(M/N)$ is a finite set for
each submodule $N$ of $M$. The class of weakly Laskerian modules was
introduced in [17] by Divaani-Aazar and Mafi. They also as a generalization of
cofinite modules with respect to an ideal in [18] defined an $R$-module $M$ to
be _weakly cofinite with respect to ideal $I$ of $R$_ or $I$-weakly cofinite
if $\operatorname{Supp}_{R}(M)\subseteq\operatorname{V}(I)$ and
$\operatorname{Ext}^{i}_{R}(R/I,M)$ is weakly Laskerian for all $i\geq 0$. In
[3, Definition 4.2] the author of present paper and Bahmanpour introduced the
concept of $\Phi$-weakly cofiniteness of general local cohomology modules. The
general local cohomology module $\operatorname{H}_{\Phi}^{j}(M)$ is defined to
be $\Phi$-weakly cofinite if there exists an ideal $I\in\Phi$ such that
$\operatorname{Ext}^{i}_{R}(R/I,\operatorname{H}_{\Phi}^{j}(M))$ is weakly
Laskerian, for all $i,j\geq 0$.

Recently many authors studied the weakly Laskerianness and weakly cofiniteness
of local cohomology modules and answered the Conjecture 1.1 and Question 1.2
in the class of weakly Laskerian modules in some cases (see [17, 18, 4, 36,
31, 33, 2, 8]). The purpose of this note is to make a suitable generalization
of Conjecture 1.1 and Question 1.2 in terms of weakly Laskerian modules
instead of finitely generated modules for general local cohomology modules. In
this direction in Section 2, we generalize [2, Theorem 3.4 and Corollaries 3.5
and 3.6] and [3, Theorem 2.9 and Corollaries 2.10]. More precisely, we shall
show that:

Theorem 1.3. (See Theorem 2.12) Let $R$ be a Noetherian ring and $I\in\Phi$ an
ideal of $R$. Let $t\in\mathbb{N}_{0}$ be an integer and $M$ an $R$-module
such that $\operatorname{Ext}^{i}_{R}(R/I,M)$ are weakly Laskerian for all
$i\leq t+1$. Let the $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are ${\rm
FD_{\leq 1}}$ $R$-modules for all $i<t$. Then, the following conditions hold:

  * (i)

The $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are $I$-$ETH$-weakly cofinite
(in particular $\Phi$-weakly cofinite) for all $i<t$.

  * (ii)

For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of
$\operatorname{H}^{t}_{\Phi}(M)$, the $R$-modules

${\rm Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)$

are weakly Laskerian.

Corrolary 1.4. (See Corollary 2.14) Let $R$ be a Noetherian ring and
$I\in\Phi$ an ideal of $R$. Let $M$ be an $I$-$ETH$-weakly cofinite $R$-module
such that the $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are ${\rm FD_{\leq
1}}$ $R$-modules for all $i$. Then,

  * (i)

the $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are $I$-$ETH$-weakly cofinite
(in particular, $\Phi$-weakly cofinite) for all $i$.

  * (ii)

for any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$
of $\operatorname{H}^{i}_{\Phi}(M)$, the $R$-module
$\operatorname{H}^{i}_{\Phi}(M)/N$ is $I$-$ETH$-weakly cofinite (in
particular, $\Phi$-weakly cofinite).

Hartshorne also asked the following question:

Question 1.5. _Whether the category $\mathscr{M}(R,I)_{cof}$ of $I$-cofinite
modules forms an Abelian subcategory of the category of all $R$-modules? That
is, if $f:M\longrightarrow N$ is an $R$-module homomorphism of $I$-cofinite
modules, are $\operatorname{Ker}f$ and $\operatorname{Coker}f$ $I$-cofinite?_

With respect to this question, Hartshorne showed that if $I$ is a prime ideal
of dimension one in a complete regular local ring $R$, then the answer to his
question is positive. On the other hand, in [14], Delfino and Marley extended
this result to arbitrary complete local rings. Recently, Kawasaki [24]
generalized the Delfino and Marley’s result for an arbitrary ideal $I$ of
dimension one in a local ring $R$. Finally, Melkersson in [27] completely have
removed local assumption on $R$. More recently, in [9] (resp. [2]) it is shown
that Hartshorne’s question is true for the category of all $I$-cofinite
$R$-modules $M$ with $\dim M\leq 1$ (resp. the class of $I$\- cofinite ${\rm
FD_{\leq 1}}$ modules), for all ideals $I$ in a commutative Noetherian ring
$R$. Also in [8] it is proved that the same question is true for the category
of all $I$-weakly cofinite $R$-modules $M$ with $\dim M\leq 1$ for all ideals
$I$ in $R$. In this direction we introduced the concept of $I$-$ETH$-weakly
cofinite or $ETH$-weakly cofinte modules with respect to $I$ in Definition
2.1. One of the main results of this section is to prove that the class of
$I$-$ETH$-weakly cofinite and ${\rm FD_{\leq
1}}$($\mathscr{FD}^{1}(R,I)_{ethwcof}$) modules are Abelian category (see
Theorem 2.19). Using this fact we prove the following corollary:

Corrolary 1.6. (See Corollary 2.21) Let $\Phi$ be a system of ideals of a
Noetherian ring $R$, $M$ a non-zero $I$-$ETH$-weakly cofinite $R$-module such
that $\operatorname{H}^{i}_{\Phi}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules for
all $i\geq 0$. Then for each finite $R$-module $N$, the $R$-modules ${\rm
Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ and ${\rm
Tor}^{R}_{j}(N,\operatorname{H}^{i}_{\Phi}(M))$ are $\Phi$-weakly cofinite and
${\rm FD_{\leq 1}}$ $R$-modules for all $i\geq 0$ and $j\geq 0$.

In Section 3 we prove that similar corollaries are true for local cohomology
modules defined by a pair of ideals because it is a special case of local
cohomology with respect to a system of ideals.

Throughout this paper, $R$ will always be a commutative Noetherian ring with
non-zero identity and $I$ will be an ideal of $R$. We denote
$\\{\mathfrak{p}\in{\rm Spec}\,R:\,\mathfrak{p}\supseteq I\\}$ by $V(I)$. For
any unexplained notation and terminology we refer the reader to [12] and [13].

##  2\. $ETH$-weakly cofinite modules with respect to an ideal

The definitions of $ETH$-cofinite module and weakly cofinite module with
respect to an ideal ([1, Definitions 2.2] and [18, Definition 2.4]), motivate
the following definition.

######  Definition 2.1.

An $R$-module $M$ (not necessary $I$-torsion) is called $ETH$-weakly cofinite
with respect to an ideal $I$ of $R$ or $I$-$ETH$-weakly cofinite if
$\operatorname{Ext}^{i}_{R}(R/I,M)$ is a weakly Laskerian $R$-module for all
$i$.

######  Remark 2.2.

Let $I$ be an ideal of $R$.

  * (i)

All weakly Laskerian $R$-modules, $ETH$-cofinite and weakly cofinite
$R$-modules with respect to ideal $I$ are $I$-$ETH$-weakly cofinite.

  * (ii)

Suppose $M$ is an $I$-torsion module, then $M$ is $I$-$ETH$-weakly cofinite if
and only if it is $I$-weakly cofinite module.

We claim that the class of $ETH$-weakly cofinite modules with respect to an
ideal is strictly larger than the class of $ETH$-cofinite and weakly cofinite
modules with respect to the same ideal. To do this, see the following
examples.

######  Example 2.3.

(i) Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$. Let
$M=R\oplus E(R/\mathfrak{m})$. It is easy to see that $M$ is an
$\mathfrak{m}$-$ETH$-weakly cofinite $R$-module that is not
$\mathfrak{m}$-cofinite.

(ii) Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$. Let
$M=R\oplus(\underset{i\in\mathbb{N}}{\oplus}R/\mathfrak{m})$. It is easy to
see that $M$ is an $\mathfrak{m}$-$ETH$-weakly cofinite $R$-module that is not
$\mathfrak{m}$-weakly cofinite.

Hajkarimi in [20, Definition 2.1] introduced the class of weakly Artinian
modules as below:

######  Definition 2.4.

An $R$-module $M$ is said to be weakly Artinian if its injective envelope, can
be written as
$\operatorname{E}_{R}(M)=\oplus_{i=1}^{k}\mu^{0}({\mathfrak{m}}_{i},M)\operatorname{E}_{R}(R/{\mathfrak{m}}_{i})$
where ${\mathfrak{m}}_{1},\dots,{\mathfrak{m}}_{k}$ are maximal ideals of $R$.

By [20, Lemma 2.3 (a) and (c)] the class of weakly Artinian $R$-modules is a
Serre subcategory of the category of $R$-modules and an $R$-module $M$ is
Artinian if and only if it is weakly Artinian and $\mu^{0}({\mathfrak{m}},M)$
is finite for all $\mathfrak{m}\in\operatorname{Ass}_{R}(M)$.

The following lemma represent the other equivalent condition for a module to
be weakly Artinian.

######  Lemma 2.5.

Let $M$ be an $R$-module. Then the following statements are equivalent:

  * (i)

$M$ is weakly Artinian.

  * (ii)

$\operatorname{Ass}_{R}(M)$ consists of finitely many maximal ideals.

  * (iii)

$\operatorname{Supp}_{R}(M)$ consists of finitely many maximal ideals.

  * (iv)

$\operatorname{Ass}_{R}(M)=\operatorname{Supp}_{R}(M)$ and it consists of
finitely many maximal ideals.

  * (v)

$M$ is weakly Laskerin and
$\operatorname{Ass}_{R}(M)\subseteq\operatorname{Max}(R)$.

###### Proof.

See [20, Lemma 2.3 (b)]. ∎

######  Lemma 2.6.

Let $I$ be an ideal of a Noetherian ring $R$ and $M$ be an $R$-module such
that $\operatorname{Supp}_{R}(M)\subseteq\operatorname{Max}(R)$. Then the
following statements are equivalent:

  * (i)

$M$ is $I$-$ETH$-weakly cofinite.

  * (ii)

The $R$-module $\operatorname{Hom}_{R}(R/I,M)$ is weakly Laskerian.

###### Proof.

(i)$\Longrightarrow$(ii) follows by definition.

In order to prove (ii)$\Longrightarrow$(i) note that

$\operatorname{Hom}_{R}(R/I,\operatorname{\Gamma}_{I}(M))\cong\operatorname{Hom}_{R}(R/I,M).$

Since
$\operatorname{Supp}_{R}(\operatorname{\Gamma}_{I}(M))\subseteq\operatorname{Max}(R)$,
it is easy to see that
$\operatorname{Hom}_{R}(R/I,\operatorname{\Gamma}_{I}(M))$ is a weakly
Artinian $R$-module and so by [20, Lemma 2.8] $\operatorname{\Gamma}_{I}(M)$
is also a weakly Artinian $R$-module. On the other hand by [12, Theorem 6.1.2]
$\operatorname{H}^{i}_{I}(M)=0$ for all $i\geq 1$. Since any weakly Artinian
$R$-module is weakly Laskerian, therefore $\operatorname{H}^{i}_{I}(M)$ is
$I$-weakly cofinite for all $i\geq 0$. Now by [28, Corollary 3.10], it follows
that $\operatorname{Ext}^{i}_{R}(R/I,M)$ are weakly Laskerian for all $i\geq
0$, as required. ∎

The following Lemma is well-known for $I$-cofinite modules.

######  Lemma 2.7.

If $0\longrightarrow N\longrightarrow L\longrightarrow T\longrightarrow 0$ is
exact and two of the modules in the sequence are $I$-$ETH$-weakly cofinite,
then so is the third one.

######  Theorem 2.8.

Let $I$ be an ideal of a Noetherian ring $R$ and $M$ be an ${\rm FD_{\leq 0}}$
(or minimax) $R$-module. Then the following statements are equivalent:

  * (i)

$M$ is $I$-$ETH$-weakly cofinite.

  * (ii)

The $R$-module $\operatorname{Hom}_{R}(R/I,M)$ is weakly Laskerian.

###### Proof.

By definition there is a finitely generated submodule $N$ of $M$ such that
${\rm dim}(M/N)\leq 0$. Also, the exact sequence

$0\rightarrow N\rightarrow M\rightarrow M/N\rightarrow 0,\,\,\,\,\,\,(*)$

induces the following exact sequence

$0\longrightarrow{\rm Hom}_{R}(R/I,N)\longrightarrow{\rm
Hom}_{R}(R/I,M)\longrightarrow{\rm Hom}_{R}(R/I,M/N)$ $\longrightarrow{\rm
Ext}^{1}_{R}(R/I,N).$

Whence, it follows that the $R$-modules $\operatorname{Hom}_{R}(R/I,M/N)$ is
weakly Laskerian. Therefore, in view of Lemma 2.6, the $R$-module $M/N$ is
$I$-$ETH$-weakly cofinite. Now it follows from the exact sequence $(*)$ and
Lemma 2.7 that $M$ is $I$-$ETH$-weakly cofinite. ∎

We are now ready to state and prove the first main theorem of this section.
The following theorem is a generalization of [8, Proposition 3.2]. In fact, we
remove $I$-torsion condition from this theorem. Note that $I$ is not dimension
one too.

######  Lemma 2.9.

Let $R$ be a Noetherian ring and $I$ be an ideal of $R$ (not necessary
dimension one). Let $M$ be a non-zero $R$-module (not necessary $I$-torsion)
such that $\dim M\leq 1$. Then the following conditions are equivalent:

  * (i)

$M$ is $I$-$ETH$-weakly cofinite.

  * (ii)

$\operatorname{H}^{i}_{I}(M)$ are $I$-weakly cofinite for all $i$.

  * (iii)

The $R$-modules $\operatorname{Hom}_{R}(R/I,M)$ and
$\operatorname{Ext}^{1}_{R}(R/I,M)$ are weakly Laskerian.

###### Proof.

(iii)$\Longrightarrow$(ii) Using the exact sequence

$0\rightarrow\operatorname{\Gamma}_{I}(M)\rightarrow M\rightarrow
M/\operatorname{\Gamma}_{I}(M)\rightarrow 0,$

we get the exact sequence

$0\longrightarrow\operatorname{Hom}_{R}(R/I,\operatorname{\Gamma}_{I}(M))\longrightarrow\operatorname{Hom}_{R}(R/I,M)\longrightarrow\operatorname{Hom}_{R}(R/I,M/\operatorname{\Gamma}_{I}(M))\longrightarrow\operatorname{Ext}^{1}_{R}(R/I,\operatorname{\Gamma}_{I}(M))\longrightarrow\operatorname{Ext}^{1}_{R}(R/I,M).$

Since $\operatorname{Hom}_{R}(R/I,M/\operatorname{\Gamma}_{I}(M))=0$, it
follows that the $R$-modules

$\operatorname{Hom}_{R}(R/I,\operatorname{\Gamma}_{I}(M))$ and
$\operatorname{Ext}^{1}_{R}(R/I,\operatorname{\Gamma}_{I}(M))$

are weakly Laskerian, and so in view of [8, Proposition 3.2] the $R$-module
$\operatorname{\Gamma}_{I}(M)$ is $I$-weakly cofinite. Now as the $R$-module
$\operatorname{Ext}^{1}_{R}(R/I,M)$ is weakly Laskerian, it follows from [4,
Theorem 4.1 (c)] that the $R$-module
$\operatorname{Hom}_{R}(R/I,\operatorname{H}^{1}_{I}(M))$ is weakly Laskeran.
If
$\mathfrak{p}\in\operatorname{Supp}_{R}(\operatorname{H}^{1}_{I}(M))\subseteq\operatorname{Supp}_{R}(M)$,
then

$\operatorname{H}^{1}_{IR_{\mathfrak{p}}}(M_{\mathfrak{p}})\cong\operatorname{H}^{1}_{I}(M)_{\mathfrak{p}}\neq
0.$

Since $\dim M\leq 1$, it is easy to see that $\dim R/\mathfrak{p}=0$ or $\dim
R/\mathfrak{p}=1$. If $\dim R/\mathfrak{p}=1$ then $M_{\mathfrak{p}}$ is a
zero dimensional $R_{\mathfrak{p}}$-module that implies
$\operatorname{H}^{1}_{IR_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ by using
Grothendieck vanishing theorem [12, Theorem 6.1.2] which is a contradiction.
Thus $\dim R/\mathfrak{p}=0$ and so $\mathfrak{p}$ is a maximal ideal. So we
have the following inclusion

$\operatorname{Supp}_{R}(\operatorname{Hom}_{R}(R/I,\operatorname{H}^{1}_{I}(M)))\subseteq\operatorname{Supp}_{R}(\operatorname{H}^{1}_{I}(M))\subseteq\operatorname{Max}R.$

By Lemma 2.5 (v), it is easy to see that the $R$-module
$\operatorname{Hom}_{R}(R/I,\operatorname{H}^{1}_{I}(M))$ is weakly Artinian
and so by [20, Lemma 2.8] the $R$-module $\operatorname{H}^{1}_{I}(M)$ is
weakly Artinian. Since any weakly Artinian module is weakly Laskerian,
therefore in view of [12, Theorem 6.1.2] the $R$-module
$\operatorname{H}^{i}_{I}(M)$ is $I$-weakly cofinite for all $i\geq 0$.

(i)$\Longrightarrow$(ii) by [28, Corollary 3.10], it follows that
$\operatorname{Ext}^{i}_{R}(R/I,M)$ are weakly Artinian for all $i\geq 0$, as
required.

(i)$\Longrightarrow$(iii) It is obviously true. ∎

The following theorem is a generalization of [2, Theorem 3.1] that in what
follows the next theorem plays an important role.

######  Theorem 2.10.

Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. Let $M$ be an ${\rm
FD_{\leq 1}}$ $R$-module. Then $M$ is $I$-$ETH$-weakly cofinite if and only if
$\operatorname{Hom}_{R}(R/I,M)$ and $\operatorname{Ext}^{1}_{R}(R/I,M)$ are
weakly Lakerian.

###### Proof.

By definition there is a finitely generated submodule $N$ of $M$ such that
${\rm dim}(M/N)\leq 1$. Also, the exact sequence

$0\rightarrow N\rightarrow M\rightarrow M/N\rightarrow 0,\,\,\,\,\,\,(*)$

induces the following exact sequence

$0\longrightarrow{\rm Hom}_{R}(R/I,N)\longrightarrow{\rm
Hom}_{R}(R/I,M)\longrightarrow{\rm Hom}_{R}(R/I,M/N)$ $\longrightarrow{\rm
Ext}^{1}_{R}(R/I,N)\longrightarrow{\rm Ext}^{1}_{R}(R/I,M)\longrightarrow{\rm
Ext}^{1}_{R}(R/I,M/N)\longrightarrow{\rm Ext}^{2}_{R}(R/I,N).$

Whence, it follows that the $R$-modules $\operatorname{Hom}_{R}(R/I,M/N)$ and
$\operatorname{Ext}^{1}_{R}(R/I,M/N)$ are weakly Laskerian. Therefore, in view
of Proposition 2.9, the $R$-module $M/N$ is $I$-$ETH$-weakly cofinite. Now it
follows from the exact sequence $(*)$ and Lemma 2.7 that $M$ is
$I$-$ETH$-weakly cofinite. ∎

The following lemma is needed in the proof of second main result of this
paper.

######  Lemma 2.11.

Let $I$ be an ideal of a Noetherian ring $R$, $M$ a non-zero $R$-module and
$t\in\mathbb{N}_{0}$. Suppose that the $R$-module
$\operatorname{H}^{i}_{\Phi}(M)$ is $I$-$ETH$-weakly cofinite for all
$i=0,...,t-1$, and the $R$-modules ${\rm Ext}^{t}_{R}(R/I,M)$ and ${\rm
Ext}^{t+1}_{R}(R/I,M)$ are weakly Laskerian. Then the $R$-modules ${\rm
Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M))$ and ${\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M))$ are weakly Laskerian.

###### Proof.

We use induction on $t$. The exact sequence

$0\longrightarrow\operatorname{\Gamma}_{\Phi}(M)\longrightarrow
M\longrightarrow M/\operatorname{\Gamma}_{\Phi}(M)\longrightarrow
0,\,\,\,\,\,\,(*)$

induces the following exact sequence:

$0\longrightarrow{\operatorname{Hom}}_{R}(R/I,\operatorname{\Gamma}_{\Phi}(M))\longrightarrow{\operatorname{Hom}}_{R}(R/I,M)\longrightarrow{\operatorname{Hom}}_{R}(R/I,M/\operatorname{\Gamma}_{\Phi}(M))$
$\longrightarrow{\operatorname{Ext}}^{1}_{R}(R/I,\operatorname{\Gamma}_{\Phi}(M))\longrightarrow{\operatorname{Ext}}^{1}_{R}(R/I,M).$

Since ${\operatorname{Hom}}_{R}(R/I,M/\operatorname{\Gamma}_{\Phi}(M))=0$ so
${\rm Hom}_{R}(R/I,\operatorname{\Gamma}_{\Phi}(M))$ and ${\rm
Ext}^{1}_{R}(R/I,\operatorname{\Gamma}_{\Phi}(M))$ are weakly Laskerian.
Assume inductively that $t>0$ and that we have established the result for non-
negative integers smaller than $t$. By applying the functor
${\operatorname{Hom}}_{R}(R/I,-)$ to the exact sequence $(*)$, we can deduce
that ${\operatorname{Ext}}_{R}^{j}(R/I,M/\operatorname{\Gamma}_{\Phi}(M))$ is
weakly Laskerian for $j=t,t+1$. On the other hand,
$\operatorname{H}_{I}^{0}(M/\operatorname{\Gamma}_{\Phi}(M))=0$ and
$\operatorname{H}_{\Phi}^{j}(M/\operatorname{\Gamma}_{\Phi}(M))\cong\operatorname{H}_{\Phi}^{j}(M)$
for all $j>0$. Therefore we may assume that
$\operatorname{\Gamma}_{\Phi}(M)=0$. Let $E$ be an injective hull of $M$ and
put $N=E/M$. Then
${\operatorname{Hom}}_{R}(R/I,E)=0=\operatorname{\Gamma}_{\Phi}(E)$. Hence
${\operatorname{Ext}}_{R}^{j}(R/I,N)\cong{\operatorname{Ext}}_{R}^{j+1}(R/I,M)$
and $\operatorname{H}_{\Phi}^{j}(N)\cong\operatorname{H}_{\Phi}^{j+1}(M)$ for
all $j\geq 0$. Now, the induction hypothesis yields that ${\rm
Hom}_{R}(R/I,\operatorname{H}_{\Phi}^{t-1}(N))$ and ${\rm
Ext}^{1}_{R}(R/I,\operatorname{H}_{\Phi}^{t-1}(N))$ are weakly Laskerian and
so ${\rm Hom}_{R}(R/I,\operatorname{H}_{\Phi}^{t}(M))$ and ${\rm
Ext}^{1}_{R}(R/I,\operatorname{H}_{\Phi}^{t}(M))$ are weakly Laskerian, as
required. ∎

We are now ready to state and prove the following main results (Theorem 2.12
and the Corollaries 2.13, 2.14, 2.15, 2.17) which are extension of [2, Theorem
3.4 and Corollaries 3.5 and 3.6], Bahmanpour-Naghipour’s results in [6, 7] in
terms of weakly Laskerian modules, Hong Quy’s result in [23], Divaani-Aazar
and Mafi’s result in [17, Corrolary 2.7], [36, Theorem 2.13] and [31,
Corollaries 2.6 and 2.7].

######  Theorem 2.12.

Let $R$ be a Noetherian ring and $I\in\Phi$ an ideal of $R$. Let
$t\in\mathbb{N}_{0}$ be an integer and $M$ an $R$-module such that
$\operatorname{Ext}^{i}_{R}(R/I,M)$ are weakly Laskerian for all $i\leq t+1$.
Let the $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are ${\rm FD_{\leq 1}}$
$R$-modules for all $i<t$. Then, the following conditions hold:

  * (i)

The $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are $I$-$ETH$-weakly cofinite
(in particular $\Phi$-weakly cofinite) for all $i<t$.

  * (ii)

For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of
$\operatorname{H}^{t}_{\Phi}(M)$, the $R$-modules

${\rm Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)$

are weakly Laskerian. In particular the sets

$\operatorname{Ass}_{R}({\rm
Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N))\,\,\,{\rm
and}\,\,\,\operatorname{Ass}_{R}({\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N))$

are finite sets.

###### Proof.

(i) We proceed by induction on $t$. In the case $t=0$ there is nothing to
prove. So, let $t>0$ and the result has been proved for smaller values of $t$.
By the inductive assumption, $\operatorname{H}^{i}_{\Phi}(M)$ is
$I$-$ETH$-weakly cofinite for $i=0,1,...,t-2$. Hence by Lemma 2.11 and
assumption, ${\rm Hom}_{R}(R/I,\operatorname{H}^{t-1}_{\Phi}(M))\,\,\,{\rm
and}\,\,\,{\rm Ext}^{1}_{R}(R/I,\operatorname{H}^{t-1}_{\Phi}(M))$ are weakly
Laskerian. Therefore by Theorem 2.10, $\operatorname{H}^{i}_{\Phi}(M)$ is
$I$-$ETH$-weakly cofinite (in particular, $\Phi$-weakly cofinite) for all
$i<t$. This completes the inductive step.
(ii) In view of (i) and Lemma 2.11, ${\rm
Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M))\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M))$ are weakly Laskerian. On the
other hand, according to Lemma 2.8, $N$ is $I$-$ETH$-weakly cofinite. Now, the
exact sequence

$0\longrightarrow
N\longrightarrow\operatorname{H}^{t}_{\Phi}(M)\longrightarrow\operatorname{H}^{t}_{\Phi}(M)/N\longrightarrow
0$

induces the following exact sequence,

${\rm Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M))\longrightarrow{\rm
Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)\longrightarrow{\rm
Ext}^{1}_{R}(R/I,N)\longrightarrow$ ${\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M))\longrightarrow{\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)\longrightarrow{\rm
Ext}^{2}_{R}(R/I,N).$

Consequently

${\rm Hom}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,\operatorname{H}^{t}_{\Phi}(M)/N)$

are weakly Laskerian, as required.∎

######  Corollary 2.13.

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $t\in\mathbb{N}_{0}$
be an integer and $M$ an $R$-module such that
$\operatorname{Ext}^{i}_{R}(R/I,M)$ are weakly Laskerian for all $i\leq t+1$.
Let the $R$-modules $H^{i}_{I}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules for all
$i<t$. Then, the following conditions hold:

(i) The $R$-modules $H^{i}_{I}(M)$ are $I$-weakly cofinite for all $i<t$.

(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $H^{t}_{I}(M)$,
the $R$-modules

${\rm Hom}_{R}(R/I,H^{t}_{I}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)$

are weakly Laskerian. In particular the set
$\operatorname{Ass}_{R}(H^{t}_{I}(M)/N)$ is finite.

The following corollaries answer to Hartshorne’s question (i.e., Question 1.2)
in terms of weakly cofiniteness.

######  Corollary 2.14.

Let $R$ be a Noetherian ring and $I\in\Phi$ an ideal of $R$. Let $M$ be an
$I$-$ETH$-weakly cofinite $R$-module such that the $R$-modules
$\operatorname{H}^{i}_{\Phi}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules for all
$i$. Then,

  * (i)

the $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are $I$-$ETH$-weakly cofinite
(in particular, $\Phi$-weakly cofinite) for all $i$.

  * (ii)

for any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$
of $\operatorname{H}^{i}_{\Phi}(M)$, the $R$-module
$\operatorname{H}^{i}_{\Phi}(M)/N$ is $I$-$ETH$-weakly cofinite (in
particular, $\Phi$-weakly cofinite).

###### Proof.

(i) Clear.
(ii) In view of (i) the $R$-module $\operatorname{H}^{i}_{\Phi}(M)$ is
$I$-$ETH$-weakly cofinite for all $i$. Hence the $R$-module ${\rm
Hom}_{R}(R/I,N)$ is weakly Laskerian, and so it follows from Lemma 2.8 that
$N$ is $I$-$ETH$-weakly cofinite. Now, the exact sequence

$0\longrightarrow
N\longrightarrow\operatorname{H}^{i}_{\Phi}(M)\longrightarrow\operatorname{H}^{i}_{\Phi}(M)/N\longrightarrow
0,$

and Lemma 2.7 implies that the $R$-module $\operatorname{H}^{i}_{\Phi}(M)/N$
is $I$-$ETH$-weakly cofinite.∎

######  Corollary 2.15.

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be an
$I$-$ETH$-weakly cofinite $R$-module such that $R$-modules $H^{i}_{I}(M)$ are
${\rm FD_{\leq 1}}$ $R$-modules for all $i$. Then,

(i) The $R$-modules $H^{i}_{I}(M)$ are $I$-weakly cofinite for all $i$.

(ii) For any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule
$N$ of $H^{i}_{I}(M)$, the $R$-module $H^{i}_{I}(M)/N$ is $I$-weakly cofinite.

######  Corollary 2.16.

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be an $R$-module
such that the $R$-modules $H^{i}_{I}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules
for all $i$. Then, the following conditions are equivalent:

(i) The $R$-module $M$ is an $I$-$ETH$-weakly cofinite.

(ii) The $R$-modules $H^{i}_{I}(M)$ are $I$-weakly cofinite for all $i$.

###### Proof.

(i)$\Rightarrow$(ii) Follows by Corollary 2.15.

(ii)$\Rightarrow$(i) It follows by [28, Proposition 3.9].∎

The following corollary is a generalization of [6, Corollary 2.7] in terms of
weakly cofiniteness.

######  Corollary 2.17.

Let $R$ be a Noetherian ring and $I\in\Phi$ an ideal of $R$. Let $M$ be an
$I$-$ETH$-weakly cofinite $R$-module such that $\dim M/IM\leq 1$ (e.g., $\dim
R/I\leq 1$) for all $I\in\Phi$. Then,

  * (i)

the $R$-modules $\operatorname{H}^{i}_{\Phi}(M)$ are $I$-$ETH$-weakly cofinite
(in particular, $\Phi$-weakly cofinite) for all $i$.

  * (ii)

for any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$
of $\operatorname{H}^{i}_{\Phi}(M)$, the $R$-module
$\operatorname{H}^{i}_{\Phi}(M)/N$ is $I$-$ETH$-weakly cofinite (in
particular, $\Phi$-weakly cofinite).

###### Proof.

(i) Since by [10, Lemma 2.1],

$\operatorname{H}^{i}_{\Phi}(M)\cong\underset{I\in\Phi}{\varinjlim}\operatorname{H}^{i}_{I}(M),$

it is easy to see that
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\underset{I\in\Phi}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{I}(M))$
and therefore

$\operatorname{dimSupp}\operatorname{H}^{i}_{\Phi}(M)\leq\sup\\{\operatorname{dimSupp}\operatorname{H}^{i}_{I}(M)|I\in\Phi\\}\leq
1,$

thus $\operatorname{H}^{i}_{\Phi}(M)$ is ${\rm FD_{\leq 1}}$ $R$-module and
the assertion follows by Corollary 2.14 (i).
(ii) Proof is the same as 2.14 (ii). ∎

######  Corollary 2.18.

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be an $R$-module
such that $\dim M/IM\leq 1$ (e.g., $\dim R/I\leq 1$). Then, the following
conditions are equivalent:

(i) The $R$-modules $M$ is an $I$-$ETH$-weakly cofinite.

(ii) The $R$-modules $H^{i}_{I}(M)$ are $I$-weakly cofinite for all $i$.

###### Proof.

(i)$\Rightarrow$(ii) Follows by Corollary 2.17.

(ii)$\Rightarrow$(i) It follows by [28, Proposition 3.9].∎

One of the main results of this section is to prove that for an arbitrary
ideal $I$ of a Noetherian ring $R$, the category of
$\mathscr{FD}^{1}(R,I)_{ethwcof}$ modules is Abelian category.

######  Theorem 2.19.

Let $I$ be an ideal of a Noetherian ring $R$. Let
$\mathscr{FD}^{1}(R,I)_{ethwcof}$ denote the category of $I$-$ETH$-weakly
cofinite and ${\rm FD_{\leq 1}}$ $R$-modules. Then
$\mathscr{FD}^{1}(R,I)_{ethwcof}$ is an Abelian category.

###### Proof.

Let $M,N\in\mathscr{FD}^{1}(R,I)_{ethwcof}$ and let $f:M\longrightarrow N$ be
an $R$-homomorphism. Since by [2, Lemma 2.3 (v)] the class of ${\rm FD_{\leq
1}}$ is a Serre subcategory of the category of $R$-modules, it is enough to
show that the $R$-modules $\operatorname{Ker}f$ and $\operatorname{Coker}f$
are $I$-$ETH$-weakly cofinite.

To this end, the exact sequence

$0\longrightarrow\operatorname{Ker}f\longrightarrow
M\longrightarrow\operatorname{Im}f\longrightarrow 0,$

induces an exact sequence

$0\longrightarrow{\rm Hom}_{R}(R/I,\operatorname{Ker}f)\longrightarrow{\rm
Hom}_{R}(R/I,M)\longrightarrow{\rm Hom}_{R}(R/I,\operatorname{Im}f)$
$\longrightarrow{\rm
Ext}^{1}_{R}(R/I,\operatorname{Ker}f)\longrightarrow\operatorname{Ext}^{1}_{R}(R/I,M),$

that implies the $R$-modules $\operatorname{Hom}_{R}(R/I,\operatorname{Ker}f)$
and $\operatorname{Ext}^{1}_{R}(R/I,\operatorname{Ker}f)$ are weakly
Laskerian. Since $\operatorname{Ker}f$ is ${\rm FD_{\leq 1}}$ $R$-module,
therefore it follows from Theorem 2.10 that $\operatorname{Ker}f$ is
$I$-$ETH$-weakly cofinite. Now, the assertion follows from the following exact
sequences

$0\longrightarrow\operatorname{Ker}f\longrightarrow
M\longrightarrow\operatorname{Im}f\longrightarrow 0,$

and

$0\longrightarrow\operatorname{Im}f\longrightarrow
N\longrightarrow\operatorname{Coker}f\longrightarrow 0.$

∎

The following corollaries are generalization of [2, Corolaries 3.8 and 3.9].

######  Corollary 2.20.

Let $R$ be a Noetherian ring and $I$ a proper ideal of $R$. Let $M$ is a non-
zero $I$-$ETH$-weakly cofinite ${\rm FD_{\leq 1}}$ $R$-module. Then, the
R-modules ${\rm Ext}^{i}_{R}(N,M)$ and ${\rm Tor}^{R}_{i}(N,M)$ are
$I$-$ETH$-weakly cofinite ${\rm FD_{\leq 1}}$ $R$-modules, for all finitely
generated $R$-modules $N$ and all integers $i\geq 0$.

###### Proof.

Since $N$ is finitely generated it follows that $N$ has a free resolution of
finitely generated free modules. Now the assertion follows using Theorem 2.19
and computing the modules ${\rm Tor}_{i}^{R}(N,M)$ and ${\rm
Ext}^{i}_{R}(N,M)$, by this free resolution. ∎

######  Corollary 2.21.

Let $\Phi$ be a system of ideals of a Noetherian ring $R$, $M$ a non-zero
$I$-$ETH$-weakly cofinite $R$-module such that
$\operatorname{H}^{i}_{\Phi}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules for all
$i\geq 0$. Then for each finite $R$-module $N$, the $R$-modules ${\rm
Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ and ${\rm
Tor}^{R}_{j}(N,\operatorname{H}^{i}_{\Phi}(M))$ are $\Phi$-weakly cofinite and
${\rm FD_{\leq 1}}$ $R$-modules for all $i\geq 0$ and $j\geq 0$.

###### Proof.

Apply Corollaries 2.20 and 2.14.∎

######  Corollary 2.22.

Let $\Phi$ be a system of ideals of a Noetherian ring $R$, $M$ a non-zero
$I$-$ETH$-weakly cofinite $R$-module such that $\dim M/IM\leq 1$ (e.g., $\dim
R/I\leq 1$) for all $I\in\Phi$. Then for each finite $R$-module $N$, the
$R$-modules ${\rm Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ and ${\rm
Tor}^{R}_{j}(N,\operatorname{H}^{i}_{\Phi}(M))$ are $\Phi$-weakly cofinite and
${\rm FD_{\leq 1}}$ $R$-modules for all $i\geq 0$ and $j\geq 0$.

###### Proof.

By proof of Corollary 2.17 (i)
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\underset{I\in\Phi}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{I}(M))$
and therefore

$\operatorname{dimSupp}\operatorname{H}^{i}_{\Phi}(M)\leq\sup\\{\operatorname{dimSupp}\operatorname{H}^{i}_{I}(M)|I\in\Phi\\}\leq
1,$

thus $\operatorname{H}^{i}_{\Phi}(M)$ is ${\rm FD_{\leq 1}}$ $R$-module and
the assertion follows by Corollary 2.21.∎

######  Corollary 2.23.

Let $I$ be an ideal of a Noetherian ring $R$, $M$ a non-zero $I$-$ETH$-weakly
cofinite $R$-module such that $\dim M/IM\leq 1$ (e.g., $\dim R/I\leq 1$). Then
for each finite $R$-module $N$, the $R$-modules ${\rm
Ext}^{j}_{R}(N,\operatorname{H}^{i}_{I}(M))$ and ${\rm
Tor}^{R}_{j}(N,\operatorname{H}^{i}_{I}(M))$ are $I$-weakly cofinite and ${\rm
FD_{\leq 1}}$ $R$-modules for all $i\geq 0$ and $j\geq 0$.

##  3\. Weakly cofiniteness of local cohomology defined by a pair of ideals

As a special case of general local cohomology and generalization of ordinary
local cohomology modules, R. Takahashi, Y. Yoshino, and T. Yoshizawa [32],
introduced local cohomology modules with respect to a pair of ideals. The
$(I,J)$-torsion submodule $\Gamma_{I,J}(M)$ of $M$ is a submodule of $M$
consists of all elements $x$ of $M$ with Supp$(Rx)\subseteq W(I,J)$, in which

$W(I,J)=\\{~{}\mathfrak{p}\in\textmd{Spec}(R)\mid
I^{n}\subseteq\mathfrak{p}+J\textmd{\ \ for an integer}\ n\geq 1\\}.$

For an integer $i$, the $i$-th local cohomology functor
$\operatorname{H}^{i}_{I,J}$ with respect to $(I,J)$ is the $i$-th right
derived functor of $\Gamma_{I,J}$. The $R$-module
$\operatorname{H}^{i}_{I,J}(M)$ is called the $i$-th local cohomology module
of $M$ with respect to $(I,J)$. In the case $J=0$,
$\operatorname{H}^{i}_{I,J}(-)$ coincides with the ordinary local cohomology
functor $\operatorname{H}^{i}_{I}(-)$. Also, we are concerned with the
following set of ideals of $R$:

$\tilde{W}(I,J)=\\{~{}\mathfrak{a}\trianglelefteq R\mid
I^{n}\subseteq\mathfrak{a}+J\textmd{\ \ for an integer}\ n\geq 0\\}.$

The definition of weakly cofinite module with respect to an ideal ([18,
Definition 2.4]) motivates the following definition.

######  Definition 3.1.

An $R$-module $M$ is called $(I,J)$-weakly cofinite if
$\operatorname{Supp}_{R}(M)\subseteq W(I,J)$ and
$\operatorname{Ext}^{i}_{R}(R/I,M)$ is a weakly Laskerian $R$-module, for all
$i\geq 0$.

######  Remark 3.2.

Let $I$ and $J$ be two ideals of $R$. Replacing $\Phi$ by $\tilde{W}(I,J)$,
$\operatorname{H}^{i}_{\Phi}(M)$ by $\operatorname{H}^{i}_{I,J}(M)$ and
$\Phi$-weakly cofinite module by $(I,J)$-weakly cofinite module, the Theorem
2.12 and Corollaries 2.14, 2.17, 2.21 and 2.22 are true for local cohomology
modules defined by a pair of ideals. Because by [32, Definition 3.1 and
Theorem 3.2], it is easy to see that the local cohomology modules defined by a
pair of ideals is a special case of local cohomology modules with respect to a
system of ideals.

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