# Quantum corrections to nonlinear ion acoustic wave with Landau damping

Abhik Mukherjee1, Anirban Bose2, M.S. Janaki1 1Saha Institute of Nuclear
Physics,
Calcutta, INDIA
2Serampore College, West Bengal, India

###### Abstract

Quantum corrections to nonlinear ion acoustic wave with Landau damping have
been computed using Wigner equation approach. The dynamical equation governing
the time development of nonlinear ion acoustic wave with semiclassical quantum
corrections is shown to have the form of higher KdV equation which has higher
order nonlinear terms coming from quantum corrections, with the usual
classical and quantum corrected Landau damping integral terms. The
conservation of total number of ions is shown from the evolution equation. The
decay rate of KdV solitary wave amplitude due to presence of Landau damping
terms has been calculated assuming the Landau damping parameter
$\alpha_{1}=\sqrt{{m_{e}}/{m_{i}}}$ to be of the same order of the quantum
parameter $Q={\hbar^{2}}/({24m^{2}c^{2}_{s}L^{2}})$. The amplitude is shown to
decay very slowly with time as determined by the quantum factor $Q$.

##  I Introduction

The study of plasmas, is in general limited to the domain of classical physics
where temperature is high and particle density is low. In recent years, the
study of plasmas such as dense astrophysical plasmas Astro , laser plasmas
Laser  as well as miniature electronic devices that are under extreme physical
conditions requires Electronics1 ,Electronics2  quantum mechanical effects to
be taken into account. In such systems, the scale length becomes comparable to
the particle de Broglie wavelength rendering classical transport models
unsuitable and quantum mechanical effects to be relevant. In broad aspect
there are mainly two approaches to model quantum plasmas which are quantum
hydrodynamic approachHaas1 ,Manfredi  and quantum kinetic approachBonitz  i.e,
Wigner equation approach. The plasma fluid equations with the inclusion of
quantum diffraction and statistical pressure effects give rise to new physical
phenomena in the context of linear and nonlinear waves and instabilities.
HaasHaas2  et al. have examined quantum quasilinear plasma turbulence using
quasilinear equation derived from Wigner-Poisson system.

The quantum fluid equations being macroscopic in nature are relatively simple
and are easily accessible for nonlinear calculations. However, working with
such macroscopic models leads to loss of understanding in the situations where
single particle effects like Landau damping are important and which can be
explored by moving into a kinetic picture. The kinetic description of plasma
possessing quantum mechanical features is provided by the Wigner equation that
can be considered as the quantum analogue of the Vlasov equation. It describes
the evolution of the quantum mechanical phase space distribution function
given by the Wigner-Moyal distribution and can be a useful tool to look into
the microscopic nature of the system. The Wigner function is called quasi-
distribution as it can have negative values although its velocity moments give
rise to various physical variables such as density, current etc.
GardnerGardner  derived the full three-dimensional quantum hydrodynamic (QHD)
model for the first time by a moment expansion of the Wigner-Boltzmann
equation.

So far nonlinear problems like KdV equation and BGK modes have been tackled
successfully in classical plasma. Recently, Lange et al.Lange  have provided a
generalization of the classical BGK modes by obtaining a solution of the
stationary Wigner-Poisson equation. In this work we have attempted to look
into the quantum KdV problem in the semi-classical limit. For a classical
plasma OttSudan1  Ott and Sudan have modeled nonlinear ion acoustic wave in a
kinetic picture taking the mass of electron into account. They obtained a KdV
equation together with a Landau damping term as an evolution equation for the
ion acoustic wave. In order to explore the quantum corrections to the
nonlinear evolution of an ion acoustic wave in presence of Landau damping
terms we have to replace the Vlasov equation by the Wigner equation. In this
article we have tried to investigate, in the semiclassical limit, the quantum
corrections to nonlinear ion acoustic wave with Landau damping. We have
derived a higher order KdV equation which has higher order nonlinear quantum
corrections with the usual classical Landau damping term and a term containing
the quantum corrections due to Landau damping as the dynamical evolution
equation. The equation converges to the same equation as derived by Ott and
Sudan in the classical limit i.e, when $\hbar$ tends to zero. The equation
shows some features like conservation of total ion number , decay of initial
waveform due to Landau damping etc. In the next stage we have carried out the
perturbative approach of Bogoliubov and Mitropolsky to get the decay nature of
KdV solitary wave amplitude. For this purpose we have assumed the Landau
damping parameter $\alpha_{1}$ to be of the order of the quantum factor $Q$.
The procedure reveals that the amplitude decays inversely with the square of
time depending on the factor $Q$.

The paper is organized in the following manner. In section-II the derivation
of the evolution equation of ion acoustic wave with the Landau damping term
and the quantum corrections is given. Some relevant properties of this higher
order kdV equation are discussed in section-III. Subsection III-A discusses
the conservation of total number of ions. The Bogoliubov- Mitropolsky
perturbation approach with the condition $\alpha_{1}\approx Q$ and the decay
nature of the KdV solitary wave is given in subsection III-B. The conclusive
remarks are given in section-IV.

##  II Derivation of the dynamical equation

The Wigner distribution function is a function of the phase-space variables
(x, v) and time, which, is given by N single particle wave function
$\psi_{\alpha}(x,t)$ each characterized by a probability $P_{\alpha}$
satisfying $\sum_{\alpha=1}^{N}P_{\alpha}=1$.

It is given as,

$f(x,v,t)=\sum_{\alpha=1}^{N}\frac{m}{2\pi\hbar}P_{\alpha}\int_{-\infty}^{\infty}\psi_{\alpha}^{*}(x+\lambda/2,t)\psi_{\alpha}(x-\lambda/2,t)e^{\frac{imv\lambda}{\hbar}}d\lambda,$
(1)

where $m$ is the mass of the particle. The Wigner function follows the
following evolution equation called the Wigner equation

$\displaystyle\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial
x}+\frac{em}{2i\pi\hbar^{2}}\iint[\phi(x+\lambda/2)-\phi(x-\lambda/2)]f(x,v^{\prime},t)e^{\frac{im\lambda(v-v^{\prime})}{\hbar}}d\lambda
dv^{\prime}=0,$ (2)

where $\hbar,\phi$ are the reduced Planck’s constant and self- consistent
electrostatic potential. Considering semi-classical limit, we develop the
integral upto $O(\hbar^{2})$ and neglect all higher order terms containing
$\hbar$ to obtain

$\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial
x}+\frac{e}{m}\frac{\partial\phi}{\partial x}\frac{\partial f}{\partial
v}-(\frac{e\hbar^{2}}{24m^{3}})\frac{\partial^{3}\phi}{\partial
x^{3}}\frac{\partial^{3}f}{\partial v^{3}}=0$ (3)

We can see from (3) that the Vlasov equation is recovered in the limit
$\hbar\rightarrow 0$.

In our work, we consider a situation where ions are cold ($T_{i}=0$) and
electrons have finite temperature and the quantum effects are relevant for
electrons only. Therefore, we consider the usual fluid equations for
describing the dynamics of ions and the Wigner equation for describing the
electrons.

Hence in this case the relevant normalized system of one-dimensional equations
are -

$\frac{\partial n}{\partial t}+\frac{\partial(nu)}{\partial x}=0,$ (4)

which is the continuity equation for ions. The momentum conservation equation
for the ions is given by

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial
x}=-\frac{\partial\phi}{\partial x},$ (5)

$(\frac{\lambda_{D}}{L})^{2}\frac{\partial^{2}\phi}{\partial x^{2}}=n_{e}-n,$
(6)

which is the Poisson’s equation appropriate for the description of dispersive
ion acoustic waves. The electron number density if obtained as the velocity
space average of the single particle distribution function $f$

$n_{e}=\int_{-\infty}^{\infty}fdv,$ (7)

that is described by the Wigner equation in the semiclassical limit

$(\frac{m_{e}}{m_{i}})^{\frac{1}{2}}\frac{\partial f}{\partial
t}+v\frac{\partial f}{\partial x}+\frac{\partial\phi}{\partial
x}\frac{\partial f}{\partial v}-Q\frac{\partial^{3}\phi}{\partial
x^{3}}\frac{\partial^{3}f}{\partial v^{3}}=0,,$ (8)

where $n_{e},n,u$ are the electron number density, ion number density and ion
velocity respectively, $\lambda_{D}=\sqrt{{KT_{e}}/{4\pi n_{0}e^{2}}}$ is the
Debye Length, $L$ is the characteristic length for variations of
$n,u,\phi,n_{e},f$ and $Q$ is the quantum parameter
$={\hbar^{2}}/{24m^{2}c^{2}_{s}L^{2}}$.

Here the following normalization scheme has been used :

$\displaystyle
x=\frac{\tilde{x}}{L},t=\frac{c_{0}\tilde{t}}{L},v=\frac{\tilde{v}}{c_{s}},\phi=\frac{e\tilde{\phi}}{K_{B}T_{e}},n=\frac{\tilde{n}}{n_{0}},f=\frac{\tilde{f}}{n_{0}},u=\frac{\tilde{u}}{c_{0}},$
(9)

where $c_{0}$ is the ion acoustic sound speed $=\sqrt{{KT_{e}}/{m_{i}}}$,
$c_{s}$ is the electron thermal velocity $=\sqrt{{KT_{e}}/{m_{e}}}$ , $n_{0}$
is the ambient number density of electrons (ions) and $T_{e}$ is the electron
temperature.

As in case of OttSudan1 , here also three basic parameters enter into the
problem which are parameters due to Landau damping by electrons, measure of
nonlinearity and measure of dispersive effects. In this calculation we do not
neglect the electron to ion mass ratio and since $T_{i}=0$, the Landau damping
is provided solely by electrons. We consider all these three effects i.e.,
Landau damping, nonlinearity and dispersion to be small but of the same order
of magnitude.

1)$\sqrt{({m_{e}}/{m_{i}})}=\alpha_{1}\epsilon$, effect due to Landau damping
by electrons.

2)${\vartriangle n}/{n_{0}}=\alpha_{2}\epsilon$, measure of the strength of
nonlinearity.

3) $({\lambda_{D}}/{L})^{2}=2\alpha_{3}\epsilon$, measure of strength of
dispersive effects.

Here $\epsilon$ is smallness parameter. As is the usual mathematical procedure
we transform our co-ordinates to a moving frame with a stretched time as

$\displaystyle\xi=x-t,\ \tau=\epsilon t,$ (10)

and expand the dependent variables for small nonlinearity as

$\displaystyle n=1+\alpha_{2}\epsilon
n^{(1)}+\alpha_{2}^{2}\epsilon^{2}n^{(2)}+...,$ $\displaystyle
u=\alpha_{2}\epsilon u^{(1)}+\alpha_{2}^{2}\epsilon^{2}u^{(2)}+...,$
$\displaystyle\phi=\alpha_{2}\epsilon\phi^{(1)}+\alpha_{2}^{2}\epsilon^{2}\phi^{(2)}+...,$
$\displaystyle n_{e}=1+\alpha_{2}\epsilon
n_{e}^{(1)}+\alpha_{2}^{2}\epsilon^{2}n_{e}^{(2)}+...,$ $\displaystyle
f=f^{(0)}+\alpha_{2}\epsilon f^{(1)}+\alpha_{2}^{2}\epsilon^{2}f^{(2)}+...$
(11)

Considering semiclassical limit, the form of $f^{(0)}$ is chosen as

$f^{(0)}(v)=\frac{1}{\sqrt{2\pi}}\exp{(\frac{-v^{2}}{2})}$ (12)

Substituting Eqns. (10), (11), (12) in (4)-(8) and equating coefficients of
$\epsilon$, $\epsilon^{2}$ to zero we get first and second order equations
which need to be solved.

###  II.1 $\epsilon$ order calculation:

From Eqns (4)-(6) we get

$\displaystyle\frac{\partial n^{(1)}}{\partial\xi}=\frac{\partial
u^{(1)}}{\partial\xi}=\frac{\partial\phi^{(1)}}{\partial\xi},n^{(1)}=n_{e}^{(1)}$
(13)

From equation (8) we get

$v\frac{\partial
f^{(1)}}{\partial\xi}=v\frac{\partial\phi^{(1)}}{\partial\xi}f^{(0)}+Q\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}(3v-v^{3})f^{(0)},$
(14)

which yields

$\frac{\partial
f^{(1)}}{\partial\xi}=\frac{\partial\phi^{(1)}}{\partial\xi}f^{(0)}+Q\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}(3-v^{2})f^{(0)}+\lambda(\xi,\tau)\delta(v),$
(15)

where $\delta(v)$ is the Dirac delta function and $\lambda(\xi,\tau)$ is an
arbitrary function of $\xi,\tau$. Here also the problem of non-uniqueness
arises as in case of OttSudan1 ,AnupB  which can be removed by taking a $\tau$
derivative term from higher $\epsilon$ order. Thus, we write

$(\alpha_{1}\epsilon^{2})\frac{\partial
f_{\epsilon}^{(1)}}{\partial\tau}+v\frac{\partial
f_{\epsilon}^{(1)}}{\partial\xi}=v\frac{\partial\phi^{(1)}}{\partial\xi}f^{(0)}+Q\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}(3v-v^{3})f^{(0)},$
(16)

where the first term of (16) has been taken from order $\epsilon^{3}$
equation. Once $f_{\epsilon}^{(1)}$ is known, $f^{(1)}$ can be determined
uniquely by :

$f^{(1)}=\lim_{\epsilon\to 0}f_{\epsilon}^{(1)}$ (17)

We introduce Fourier transform in $\xi$ and $\tau$ as

$\widehat{f_{\epsilon}^{(1)}}(\omega,k)=\frac{1}{2\pi}\int_{\xi=-\infty}^{\infty}\int_{\tau=0}^{\infty}f_{\epsilon}^{(1)}(\xi,\tau)\exp[{i(\omega\tau-k\xi)}]d\xi
d\tau$ (18)

Now ,

$\widehat{(\frac{\partial
f_{\epsilon}^{(1)}}{\partial\xi})}(\omega,k)=(ik)\widehat{f_{\epsilon}^{(1)}}(\omega,k),$
(19)

and

$\widehat{(\frac{\partial
f_{\epsilon}^{(1)}}{\partial\tau})}(\omega,k)=-(i\omega)\widehat{f_{\epsilon}^{(1)}}(\omega,k)-\frac{1}{2\pi}\int_{\xi=-\infty}^{\infty}\exp[-ik\xi]f_{\epsilon}^{(1)}|_{\tau=0}d\xi,$
(20)

and

$\widehat{\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}}(\omega,k)=(-ik^{3})\widehat{{\phi^{(1)}}}(\omega,k)$
(21)

Now applying these Fourier transforms on (16), letting $\epsilon\rightarrow 0$
and using

$\lim_{\epsilon\to
0}\frac{1}{(kv-\omega\alpha_{1}\epsilon^{2})}=P(\frac{1}{kv})+i\pi\delta(kv)$
(22)

we get,

$\widehat{f^{(1)}}(\omega,k)=\widehat{\phi^{(1)}}(\omega,k)f^{(0)}-Qk^{2}(3-v^{2})\widehat{\phi^{(1)}}(\omega,k)f^{(0)},$
(23)

where $P$ is the principal part of the integral. Taking inverse Fourier
transform we get the form of $f^{(1)}$ as,

$f^{(1)}=\phi^{(1)}f^{(0)}+Q(3-v^{2})\frac{\partial^{2}\phi^{(1)}}{\partial\xi^{2}}f^{(0)}$
(24)

The first term of (24) is same with the classical case whereas the second term
is the quantum correction term. Thus the procedure yields that
$\lambda(\xi,\tau)$ appearing in (15) is zero.

###  II.2 $\epsilon^{2}$ order calculation:

From equations (4)- (6), we can obtain in a straightforward way,

$2\frac{\partial n^{(1)}}{\partial{\tau}}+3\alpha_{2}n^{(1)}\frac{\partial
n^{(1)}}{\partial{\xi}}+2\alpha_{3}\frac{\partial^{3}n^{(1)}}{\partial\xi^{3}}=\alpha_{2}\frac{\partial}{\partial\xi}(n_{e}^{(2)}-\phi^{(2)})$
(25)

From equation (8) we get,

$(\alpha_{1}\epsilon^{2})\frac{\partial
f_{\epsilon}^{(2)}}{\partial{\tau}}+v\frac{\partial
f_{\epsilon}^{(2)}}{\partial{\xi}}-vf^{(0)}\frac{\partial\phi^{(2)}}{\partial\xi}-Q\frac{\partial^{3}\phi^{(2)}}{\partial\xi^{3}}\frac{\partial^{3}f^{(0)}}{\partial
v^{3}}=C(\xi,\tau,v),$ (26)

where the $\tau$ derivative term is taken from $\epsilon^{4}$ order and terms
which are product of quantum term and second order perturbation term are
neglected as small compared to other terms. Here $C(\xi,\tau,v)$ is defined as

$C(\xi,\tau,v)=[C_{a}(\xi,\tau)+C_{b}(\xi,\tau)v+C_{c}(\xi,\tau)v^{2}+C_{d}(\xi,\tau)v^{3}]f^{(0)},$
(27)

where

$C_{a}(\xi,\tau)=(\frac{\alpha_{1}}{\alpha_{2}})[\frac{\partial\phi^{(1)}}{\partial\xi}+3Q\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}]$
(28)

$C_{b}(\xi,\tau)=[\phi^{(1)}\frac{\partial\phi^{(1)}}{\partial\xi}+5Q\frac{\partial^{2}\phi^{(1)}}{\partial\xi^{2}}\frac{\partial\phi_{1}}{\partial\xi}+3Q\phi^{(1)}\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}]$
(29)

$C_{c}(\xi,\tau)=-Q(\frac{\alpha_{1}}{\alpha_{2}})[\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}]$
(30)

$C_{d}(\xi,\tau)=[-Q\frac{\partial^{2}\phi^{(1)}}{\partial\xi^{2}}\frac{\partial\phi_{1}}{\partial\xi}-Q\phi^{(1)}\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}]$
(31)

Introducing Fourier transform in (26) and letting $\epsilon$ tends to zero we
get,

$\displaystyle\widehat{f^{(2)}}(\omega,k)-f^{(0)}\widehat{\phi^{(2)}}(\omega,k)=-i\widehat{C_{a}}[P(\frac{1}{kv})+i\pi\delta(kv)]f^{(0)}-iv\widehat{C_{b}}P(\frac{1}{kv})f^{(0)}$
$\displaystyle-
iv^{2}\widehat{C_{c}}P(\frac{1}{kv})f^{(0)}-iv^{3}\widehat{C_{d}}P(\frac{1}{kv})f^{(0)}$
(32)

Multiplying by $(ik)$ and integrating over v yields

$ik\widehat{n^{(2)}}-ik\widehat{\phi^{(2)}}=i\sqrt{\frac{\pi}{2}}\widehat{C_{a}}sgn(k)+\widehat{C_{b}}+\widehat{C_{d}},$
(33)

where we have used $k\delta(kv)=sgn(k)\delta(v)$.

Now taking inverse Fourier transform of equation (33) we obtain,

$\displaystyle\frac{\partial}{\partial\xi}(n^{(2)}-\phi^{(2)})=C_{b}+C_{d}-\frac{1}{\sqrt{2\pi}}[P\int_{-\infty}^{\infty}(\frac{\alpha_{1}}{\alpha_{2}})\frac{\partial
n^{(1)}}{\partial\xi^{\prime}}\frac{d\xi^{\prime}}{\xi-\xi^{\prime}}+P\int_{-\infty}^{\infty}(\frac{3Q\alpha_{1}}{\alpha_{2}})\frac{\partial^{3}n^{(1)}}{\partial\xi^{\prime
3}}\frac{d\xi^{\prime}}{\xi-\xi^{\prime}}],$ (34)

Now using (25) and (34) we get finally,

$\displaystyle\frac{\partial
n^{(1)}}{\partial\tau}+\alpha_{2}n^{(1)}\frac{\partial
n^{(1)}}{\partial\xi}+\alpha_{3}\frac{\partial^{3}n^{(1)}}{\partial\xi^{3}}-Q\alpha_{2}\frac{\partial}{\partial\xi}[\frac{\partial
n^{(1)}}{\partial\xi}]^{2}-Q\alpha_{2}n^{(1)}\frac{\partial^{3}n^{(1)}}{\partial\xi^{3}}+$
$\displaystyle\frac{\alpha_{1}}{\sqrt{8\pi}}[P\int_{-\infty}^{\infty}\frac{\partial
n^{(1)}}{\partial\xi^{\prime}}\frac{d\xi^{\prime}}{(\xi-\xi^{\prime})}]+\frac{3\alpha_{1}Q}{\sqrt{8\pi}}[P\int_{-\infty}^{\infty}\frac{\partial^{3}n^{(1)}}{\partial\xi^{\prime
3}}\frac{d\xi^{\prime}}{(\xi-\xi^{\prime})}],=0$ (35)

which is the main equation of interest of this work. This equation implies the
evolution equation of motion of nonlinear ion acoustic wave taking into
account the Landau damping effect with quantum corrections arising from
semiclassical kinetic approach i.e, the Wigner equation approach. The fourth
and fifth terms of (35) are nonlinear quantum corrections and the last term of
the LHS is the quantum correction on the Landau damping. We can see that the
equation converges exactly to the equation derived by Ott and Sudan OttSudan1
in the limit $\hbar$ $\rightarrow$ 0\. The equation is like a higher order KdV
equation which have higher order nonlinear quantum correction terms and Landau
damping term with its quantum correction. Due to the nature of the equation we
can show that it conserves total number of particles. The presence of Landau
damping terms also assure that the amplitude of soliton must decay with time.
These relevant facts are derived in the next section.

##  III Some relevant properties

###  III.1 Conservation of ion number

The equation (35) is the higher order KdV equation with Landau damping terms.
Integrating (35)w.r.to $\xi$ and assuming $n^{(1)},{\partial
n^{(1)}}/{\partial\xi},{\partial^{2}n^{(1)}}/{\partial\xi^{2}}=0$ at
$\xi=\pm\infty$ and renaming $n^{(1)}=U$, we can show that

$\frac{\partial}{\partial\tau}\int_{-\infty}^{\infty}Ud\xi=0$ (36)

Here we have used the fact that

$P\int_{-\infty}^{\infty}\frac{d\xi}{\xi-\xi^{\prime}}=0$ (37)

Hence ion number is conserved.

###  III.2 Decay of solitary wave

Ott and Sudan in their paper OttSudan1 ; OttSudan2  considered $\alpha_{1}$ to
be a small perturbation parameter and used the fact that due to Landau damping
the amplitude of KdV solitary wave will decrease with time. Then using
Bogoliubov- Mitropolsky Bogoliubov  approximation method they found the decay
rate of amplitude, which depends on the small parameter $\alpha_{1}$ . In
(35),we see that there are higher order KdV terms with Landau damping term and
its quantum correction. But since exact Sech- solitary wave solution of a
general higher order KdV equation of above form is possible only when
(coefficient of the term $\frac{\partial U}{\partial\xi}[\frac{\partial
U}{\partial\xi}]^{2}$) = -2 (coefficient of the term
$U\frac{\partial^{3}U}{\partial\xi^{3}}$), which is not present in (35), hence
the exact solitary wave solution of the higher order KdV equation and its
decay due to Landau damping terms cannot be worked out here. Also it can be
seen that (35) contains $2$ small parameters $\alpha_{1}$ and $Q$ where
$\alpha_{2},\alpha_{3}$ are assumed to be $\approx 1$. Hence in the subsequent
part of the work, the quantum correction terms and the Landau damping term are
treated as perturbation term to the KdV equation. But since perturbation with
multiple small parameters will include multiple time scales in the
calculation, hence it will be too complicated to be computed analytically. In
order to simplify the case and find out the nature of decay of the KdV
solitary wave amplitude we will assume that $\alpha_{1}\approx\alpha_{2}Q$.
For example, in the case of hydrogen plasma $\alpha_{1}$ is approximately
0.025 and in QElecHoles1 ; QElecHoles2 , the factor $Q$ is taken to be equal
to be order of $0.01$. Assuming this relation between small parameters we can
consider that the quantum correction to the Landau damping term which appears
as the last term of (35) is $\alpha_{1}\approx$ $Q^{2}$, and hence it can be
neglected as small compared to the other terms.

Now we have to apply the well known method of Bogoliubov and Mitropolsky
Bogoliubov ; OttSudan1 ; OttSudan2  with $\alpha_{2}Q=C$ as small perturbation
parameter. Hence $\alpha_{1}$ can be taken as $\alpha_{1}=\beta C$ where
$\beta$ is any number $\approx$ unity. In order for the perturbation analysis
to be consistent with the condition of validity of (35) it is also required
that $1\gg C\gg\epsilon$. Assuming a new phase co-ordinate to have the form

$\phi(\xi,\tau)=\sqrt{\frac{N(\tau)\alpha_{2}}{12\alpha_{3}}}(\xi-\frac{\alpha_{2}}{3}\int_{0}^{\tau}N(\tau)d\tau),$
(38)

where $N(\tau)$ is assumed to vary slowly with time.

We introduce two time scales following Bogoliubov  as

$t_{0}=\tau,t_{1}=C\tau,$ (39)

and $N=N(C,\tau)$ and shall seek a solution of the form

$U(\phi,C,\tau)=U_{0}(\phi,t_{0},t_{1})+O(C),$ (40)

where (40) is to be valid for long times,i.e., times as large as $\tau\sim
O(1/C)$. In order to find such a solution, valid for long times, we first
expand $u(\phi,\tau,C)$ to $O(C)$:

$U(\phi,\tau,C)=U_{0}(\phi,t_{0},t_{1})+CU_{1}(\phi,t_{0})+O(C^{2})$ (41)

Using (38), (39), (41) in (35) we get an equation containing different powers
of $C$ and equating coefficients of each power of $C$ we get different order
equations which need to be solved.

Since we are interested in the damping of solitary waves, we have the
following initial and boundary conditions:

$\displaystyle U(\phi,0,C)=N_{0}sech^{2}(\phi),$ $\displaystyle
U(\pm{\infty},\tau,C)=0$ (42)

Solving the order unity equation which is

$\rho\frac{\partial U_{0}}{\partial
t_{0}}+\frac{\partial^{3}U_{0}}{\partial\phi^{3}}-4\frac{\partial
U_{0}}{\partial\phi}+\frac{12}{N}U_{0}\frac{\partial U_{0}}{\partial\phi}=0,$
(43)

we get

$U_{0}(\phi,t_{0},t_{1})=N(t_{1})sech^{2}(\phi),$ (44)

where $\rho={24\sqrt{3\alpha_{3}}}/{(N\alpha_{2})\sqrt{N\alpha_{2}}}$ and
$N(t_{1})$ is an arbitrary function of $t_{1}$ except for the initial
condition $N(0)=N_{0}$. Hence $U_{0}$ doesn’t depend on $t_{0}$.

The order $C$ equation is

$\frac{\partial U_{1}}{\partial t_{0}}+L[U_{1}]=M[U_{0}],$ (45)

where

$\displaystyle M[U_{0}]=-\frac{\partial U_{0}}{\partial
t_{1}}-\frac{\phi}{2N}\frac{\partial
U_{0}}{\partial\phi}\frac{dN}{dt_{1}}+\frac{1}{(\rho\alpha_{3})}[\frac{\partial^{3}U_{0}}{\partial\phi^{3}}U_{0}+2\frac{\partial
U_{0}}{\partial\phi}\frac{\partial^{2}U_{0}}{\partial\phi^{2}}]-\frac{\beta}{\sqrt{8\pi}}[P\int_{-\infty}^{\infty}\sqrt{\frac{N(\tau)\alpha_{2}}{12\alpha_{3}}}\frac{\partial
U_{0}}{\partial\phi^{\prime}}\frac{d\xi^{\prime}}{\xi-\xi^{\prime}}],$ (46)
$\displaystyle
L[U_{1}]=\frac{1}{\rho}\frac{\partial^{3}U_{1}}{\partial\phi^{3}}-\frac{4}{\rho}\frac{\partial
U_{1}}{\partial\phi}+\frac{12}{(N\rho)}\frac{\partial(U_{0}U_{1})}{\partial\phi}$
(47)

Again the boundary and initial conditions are

$\displaystyle U_{1}(\pm\infty,t_{0})=0,U_{1}(\phi,0)=0$ (48)

In order that (41) to be valid for times as large as $\tau\sim O(1/C)$ it is
required that $U_{1}(\phi,t_{0})$ does not behave secularly with $t_{0}$. To
eliminate secular behavior of $U_{1}$ it is necessary that $M[U_{0}]$ be
orthogonal to all solutions, $g(\phi)$, of $L^{+}[g]=0$ which satisfy
(48)[i.e, $g(\pm\infty)=0$], where $L^{+}$ is the operator adjoint to $L$
given by,

$L^{+}=-\frac{1}{\rho}\frac{\partial^{3}}{\partial\phi^{3}}+\frac{4}{\rho}\frac{\partial}{\partial\phi}-\frac{12}{\rho}sech^{2}(\phi)\frac{\partial}{\partial\phi}.$
(49)

The only solution of $L^{+}[g]=0$, $g(\pm\infty)=0$, is
$g(\phi)=sech^{2}(\phi)$.

Thus,

$\int_{-\infty}^{\infty}sech^{2}(\phi)M[U_{0}]d\phi=0$ (50)

In order to evaluate this integral we have to consider term by term of (46).
The first 2 terms of $M[U_{0}]$ together give $-{dN}/{dt_{1}}$ after
integration. The third and fourth terms which come from the nonlinear quantum
correction terms give zero after integration due to the odd nature of the
integrand. Finally the last term, i.e the Landau damping term gives
$(2.92)\frac{-\beta({\alpha_{2}}}{/}{\sqrt{96\pi\alpha_{3}}})N\sqrt{N}$, where
we have used that

$P\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}sech^{2}(\phi)\frac{\partial(sech^{2}(\phi^{\prime}))}{\partial\phi^{\prime}}d\phi\frac{d\phi^{\prime}}{(\phi-\phi^{\prime})}=(24/\pi^{2})\zeta(3)=2.92$
(51)

Thus we get a first order differential equation in $N$, solving which we get

$N=\frac{N(0)}{[1+(\frac{1}{2}\beta_{1}\alpha_{2}QN(0)^{(\frac{1}{2})})\tau]^{2}},$
(52)

where $\beta_{1}=(2.92)\beta\sqrt{{\alpha_{2}}/{96\pi\alpha_{3}}}$.

From eqn (52) we see that the decay law of amplitude depends on the quantum
factor $Q$. A full numerical computation of (35) could reveal the total
dynamical nature of the solution.

Figure 1: Decay of soliton amplitude with time when when
$Q=0.01,N(0)=1,\alpha_{3}=1,\alpha_{2}=6,$ and $\beta=1$

##  IV Concluding remarks:

In this work we have extended the methodology of the work of Ott and Sudan to
include the semiclassical quantum effects to obtain a new evolution equation
in the context of a nonlinear ion acoustic wave. This equation is of the form
of a higher order KdV equation having higher order nonlinear terms as quantum
corrections, together with a classical Landau damping term as well as quantum
contribution coming from resonant particle effects.

Using the fluid equations for ions and the classical kinetic Vlasov equation
for electrons, Ott and Sudan obtained a KdV equation with a Landau damping
term as the evolution equation for the nonlinear ion acoustic wave. In order
to introduce the quantum corrections, the classical Vlasov equation is
replaced by an appropriate quantum analog i.e, the Wigner equation. In a
similiar approach using the Wigner equation in place of the Vlasov equation
gives rise to our higher order KdV equation with Landau damping terms. The
equation exactly converges to the equation done in OttSudan1  when $\hbar$
tends to zero i.e, in the classical limit. The mathematical nature of the
equation shows that it conserves the total number of ions. The importance of
the higher order KdV equation derived here, lies in the fact that its solution
would give the quantum modification of the KdV solitary wave. But
unfortunately, exact solitary wave solutions of this equation cannot be
obtained. Since there are two small parameters in the equation, $\alpha_{1}$
and $Q$, we treat the quantum corrections as well as the Landau damping terms
as perturbation to the KdV equation. In order to carry out the Bogoliubov and
Mitropolsky approximation technique, multiple time scales stretched by these
small parameters have to be introduced. Such a technique is too complicated to
comprehend analytically. Hence in order to get a useful analytical result , we
have assumed $\alpha_{1}\approx Q$. Hence, the quantum correction to Landau
damping term turns out to be of the order of $Q^{2}$ and therefore neglected.

In the perturbative approach, the contribution to the decay rate coming from
the nonlinear quantum correction terms turns out to be zero because of the odd
nature of the integrand. The final contribution to the decay of solitary wave
amplitude comes from the classical Landau terms, whose coefficient, due to the
perturbation scheme, turns out to be of the order of $Q$. The amplitude is
shown to decay inversely with the square of time depending on the quantum
factor $Q$. In our final equation of decay rate no terms come from the quantum
correction, i.e quantum nonlinear part goes to zero when the integration over
$\phi$ is performed and the quantum Landau damping terms being of order
$Q^{2}$ are neglected. This is due to our chosen scheme, and application of
perturbation scheme with multiple time scales could give rise to solutions
with more appropriate dependance on quantum effects. But the importance of the
equation cannot be turned down and could be the initiator of numerical
computation that would reveal the entire dynamical nature of the solution with
the inclusion of quantum mechanical effect.

## References

  * (1) Y. D. Jung, Phys. Plasma 8, 3842 (2001)
  * (2) D. Kremp, Th. Bornath, M. Bonitz, and M. Schlanges, Phys. Rev. E 60, 4725 (1999)
  * (3) N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Phys. Rev. B 39, 7720 (1989)
  * (4) A. A. G. Driskill- Smith, D.G.Hasko and H.Ahmed, Appl. Phys. Lett 75, 2845 (1999)
  * (5) F. Haas, Quantum Plasmas- An hydrodynamic approach, Springer, New York (2011)
  * (6) G. Manfredi and F. Haas, Phys. Rev. B 64, 075316 (2001).
  * (7) M. Bonitz, AIP Conf. Proc.1421, 135 (2012)
  * (8) F. Haas, B. Eliasson, P. K. Shukla, and G. Manfredi, Phys. Rev. E 78, 056407 (2008).
  * (9) C.L Gardner, SIAM. J.Appl. Math 54, 2(409).
  * (10) H. Lange, B. Toomire and P.F. Zweifel, Trans. theor. Stat .Phys. 25(6), 713 (1996).
  * (11) E. Ott and R.N. Sudan, Phys. Fluids 12, 11(1969).
  * (12) A. Bandyopadhyay and K.P. Das, Phys. Plasma, 9, 2 (2002).
  * (13) A. Luque, H. Schamel, and R. Fedele Phys. Lett. A 324, 185-192 (2004)
  * (14) D. Jovanovic, R. Fedele, Phys. Lett. A 364, 304-312 (2007)
  * (15) Bogoliubov N. N, Mitropolsky Y. A, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach Science Publishers, Inc, New York, 1961)
  * (16) Ott E, Sudan R. N 1970 Phys. Fluids 13 6
  * (17) M. Salimullah, M. Jamil, I. Zeba, Ch. Uzma, and H. A. Shah, Phys. Plasma. 16, 034503(2009)
  * (18) H. Ren, Z. Wu, J. Cao, and P. K. Chu, J. Phys. A. 41, 115501(2008).


# A NOTE ON WEAKLY PROREGULAR SEQUENCES AND LOCAL COHOMOLOGY

Ryoya Ando  Department of Mathematics, Faculty of Science and Technology,
Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this note, we give an elementary proof of the result given by Schenzel that
there are functorial isomorphisms between local cohomology groups and Čech
cohomology groups, by using weakly proregular sequences. In [Schenzel], he
used notions of derived category theory in his proof, but we do not use them
in this paper.

###### 2020 Mathematics Subject Classification:

13D03 (Primary) 13D45 (Secondary)

##  1\. Introduction

In this note, we assume all rings are commutative with identity element. Let
$A$ denote a ring and $I$ an ideal of $A$. The functor $\Gamma_{I}$ is defined
by

$\Gamma_{I}(M)=\left\\{x\in M\mathrel{}\middle|\mathrel{}\text{$I^{n}x=0$ for
some $n\geq 0$}\right\\}$

for an $A$-module $M$. Here $H^{i}_{I}(-)$ denote the local cohomology
functors, defined as the right derived functors of $\Gamma_{I}(-)$. In other
words, let $J^{\bullet}$ be an injective resolution of the $A$-module $M$,
then $H^{i}_{I}(M)\cong H^{i}(\Gamma_{I}(J^{\bullet}))$. In Noetherian cases,
the local cohomology can be written by using Čech cohomology. Let
$\underline{a}=a_{1},\dots,a_{r}$ be a sequence of elements of $A$, and
$I=(a_{1},\dots,a_{r})$. $\check{H}^{i}(\underline{a},M)$ denote the Čech
cohomology (see 3.1). It is well-known that there are isomorphisms;

$H^{i}_{I}(M)\cong\check{H}^{i}(\underline{a},M)$

for any $A$-module $M$ if $A$ is a Noetherian ring and
$I=(a_{1},\dots,a_{r})$, see for example [BrunsHerzog, Theorem 3.5.6.].

This result was generalised by [Schenzel]. For arbitrary ring $A$, he showed
that formula $(\ast)$ is true for any $A$-module $M$ if and only if
$\underline{a}$ is a weakly proregular sequence. Let $H_{i}(\underline{a})$ be
the Koszul homology of the sequence $\underline{a}$. A system of elements
$\underline{a}=a_{1},\dots,a_{r}$ is called a weakly proregular sequence if
for any $1\leq i\leq d$ and for each $n>0$ there is an $m\geq n$ such that the
natural map;

$H_{i}(\underline{a}^{m})\to H_{i}(\underline{a}^{n})$

is zero map, note that $\underline{a}^{n}$ is the sequence defined by
$a_{1}^{n},\dots,a_{r}^{n}$.

The goal of this note is the explanation of the following result without using
notions of derived category theory.

######  Theorem 1.1 (5.3).

Let $A$ be a ring, $\underline{a}=a_{1},\dots,a_{r}$ a system of elements of
$A$ and $I=(a_{1},\dots,a_{r})$. $\underline{a}$ is a weakly proregular
sequence if and only if for any $i$ and $A$-module $M$,
$H^{i}_{I}(M)\cong\check{H}^{i}(\underline{a},M)$.

In section 2, we summarise the $\delta$-functors. It is also mentioned in
[Hartshorne, Chap.3.1] without proof. And in section 3 we resume the
definition of Čech cohomology in commutative algebra. In section 4, we present
the theory of weakly proregular sequences, following [Schenzel, Sect.2].
Finally, We introduce the main results in section 5.

##  2\. $\delta$-functors

######  Definition 2.1.

Let $\mathscr{A},\mathscr{B}$ be an Abelian categories, and $\mathscr{A}$ has
enough injectives. Let $F\colon\mathscr{A}\to\mathscr{B}$ be an additive left
exact functor. $I^{\bullet}$ denote an injective resolution of
$A\in\mathscr{A}$. The functor;

$R^{i}F\colon\mathscr{A}\to\mathscr{B};A\mapsto H^{i}(F(I^{\bullet}))$

is called the right derived functor of $F$.

Note that derived functors are independent up to natural transformation of the
selection of injective resolution. The following is a characteristic property
of the derived functor.

######  Proposition 2.2.

Let $\mathscr{A},\mathscr{B}$ be Abelian categories, and assume $\mathscr{A}$
has enough injectives. Let $F\colon\mathscr{A}\to\mathscr{B}$ be additive left
exact functor. Then

  1. (1)

$R^{0}F\cong F$ (as functors).

  2. (2)

For any exact sequence in $\mathscr{A}$;

${0}$${A_{1}}$${A_{2}}$${A_{3}}$${0}$$\scriptstyle{f}$$\scriptstyle{g}$

for each $i\geq 0$, there are connecting morphisms $\delta^{i}\colon
R^{i}F(A_{3})\to R^{i+1}F(A_{1})$ such that;

${0{}}$${{}F(A_{1}){}}$${{}F(A_{2}){}}$${{}F(A_{3}){}}$${{}{}\cdots}$${{}R^{i}F(A_{1}){}}$${{}R^{i}F(A_{2}){}}$${{}R^{i}F(A_{3}){}}$${{}\cdots}$$\scriptstyle{F(f)}$$\scriptstyle{F(g)}$$\scriptstyle{\delta^{0}}$$\scriptstyle{\delta^{i-1}}$$\scriptstyle{R^{i}F(f)}$$\scriptstyle{R^{i}F(g)}$$\scriptstyle{\delta^{i}}$

is exact sequence in $\mathscr{B}$.

  3. (3)

Given a commutative diagram in $\mathscr{A}$ of the form (where the rows are
exact);

${0{}}$${{}A_{1}{}}$${{}A_{2}{}}$${{}A_{3}{}}$${{}0}$${0{}}$${{}B_{1}{}}$${{}B_{2}{}}$${{}B_{3}{}}$${{}0}$$\scriptstyle{\gamma}$$\scriptstyle{f}$$\scriptstyle{\beta}$$\scriptstyle{g}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

for any $i\geq 0$ the following diagram;

${R^{i}F(A_{3}){}}$${{}R^{i+1}F(A_{1})}$${R^{i}F(B_{3}){}}$${{}R^{i+1}F(B_{1})}$$\scriptstyle{R^{i}F(\alpha)}$$\scriptstyle{R^{i+1}F(\gamma)}$

is commutative in $\mathscr{B}$.

  4. (4)

For each injective object $I\in\mathscr{A}$, for any $i>0$, $R^{i}F(I)=0$.

See a textbook on homological algebra for the proof of this proposition. The
$\delta$-functor can be thought of as an extract of the above property.

######  Definition 2.3.

Let $\mathscr{A},\mathscr{B}$ be an Abelian categories. Families of additive
functors $T^{\bullet}\coloneqq\\{T^{i}\\}$ are called $\delta$-functors when
the following conditions hold;

  1. (1)

For any exact sequence in $\mathscr{A}$;

${0}$${A_{1}}$${A_{2}}$${A_{3}}$${0}$$\scriptstyle{f}$$\scriptstyle{g}$

for each $i\geq 0$, there are connecting morphism $\delta^{i}\colon
T^{i}(A_{3})\to T^{i+1}(A_{1})$ such that;

${0{}}$${{}T^{0}(A_{1}){}}$${{}T^{0}(A_{2}){}}$${{}T^{0}(A_{3}){}}$${{}{}\cdots}$${{}T^{i}(A_{1}){}}$${{}T^{i}(A_{2}){}}$${{}T^{i}(A_{3}){}}$${{}\cdots}$$\scriptstyle{T^{0}(f)}$$\scriptstyle{T^{0}(g)}$$\scriptstyle{\delta^{0}}$$\scriptstyle{\delta^{i-1}}$$\scriptstyle{T^{i}(f)}$$\scriptstyle{T^{i}(g)}$$\scriptstyle{\delta^{i}}$

is exact sequence in $\mathscr{B}$.

  2. (2)

Given a commutative diagram in $\mathscr{A}$ of the form (where the rows are
exact);

${0{}}$${{}A_{1}{}}$${{}A_{2}{}}$${{}A_{3}{}}$${{}0}$${0{}}$${{}B_{1}{}}$${{}B_{2}{}}$${{}B_{3}{}}$${{}0}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\scriptstyle{\beta}$$\scriptstyle{g}$$\scriptstyle{\gamma}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

then for any $i\geq 0$ the following diagram;

${T^{i}(A_{3}){}}$${{}T^{i+1}(A_{1})}$${T^{i}(B_{3}){}}$${{}T^{i+1}(B_{1})}$$\scriptstyle{T^{i}(\gamma)}$$\scriptstyle{T^{i+1}(\alpha)}$

is commutative in $\mathscr{B}$.

Let us define that two $\delta$-functors are isomorphic in the following way.
Let $T^{\bullet},U^{\bullet}$ be a $\delta$-functor. Families of natural
transformation $\theta^{\bullet}=\\{\theta^{i}\colon T^{i}\Rightarrow
U^{i}\\}$ is called morphism of $\delta$-fanctors when for each exact sequence
;

${0}$${A_{1}}$${A_{2}}$${A_{3}}$${0}$

in $\mathscr{A}$, the following diagram;

${T^{i}(A_{3}){}{}}$${{}T^{i+1}(A_{1}){}}$${U^{i}(A_{3}){}}$${{}U^{i+1}(A_{1})}$$\scriptstyle{\theta^{i}_{A_{3}}}$$\scriptstyle{\delta^{i}_{T}}$$\scriptstyle{\theta^{i+1}_{A_{1}}}$$\scriptstyle{\delta^{i}_{U}}$

is commutative. An isomorphism is a morphism which has a two-sided inverse.

######  Definition 2.4.

Let $\mathscr{A},\mathscr{B}$ be Abelian categories. The $\delta$-functor
$T^{\bullet}$ is called universal when for each $\delta$-functor $U^{\bullet}$
and natural transformation $\theta\colon T^{0}\Rightarrow U^{0}$, there is a
unique morphism of $\delta$-functors $\theta^{\bullet}\colon T^{\bullet}\to
U^{\bullet}$ such that $\theta^{0}=\theta$.

By definition, two universal $\delta$-functors such that $T^{0}=U^{0}$ are
isomorphic up to unique isomorphism. So for each additive functor
$F\colon\mathscr{A}\to\mathscr{B}$, a universal $\delta$-functor $T^{\bullet}$
with $T^{0}=F$ is unique up to unique isomorphism if it exists. A universal
$\delta$-functor $T^{\bullet}$ with such a property is called a right
satellite functor of $F$.

The following property gives a condition for the $\delta$-functor to be
universal, which shows that the derived functor is so if $\mathscr{A}$ has
enough injectives.

######  Definition-Proposition 2.5.

Let $\mathscr{A},\mathscr{B}$ be Abelian categories, $F$ an additive functor.
$F$ said to be effaceable if for each $A\in\mathscr{A}$, there are
$M\in\mathscr{A}$ and injection (monomorphism) $u\colon A\to M$ such that
$F(u)=0$. A $\delta$-functor $T^{\bullet}$ is universal if for each $i>0$,
$T^{i}$ is effaceable.

###### Proof.

Let $U^{\bullet}$ be $\delta$-functor and $\theta\colon T^{0}\Rightarrow
U^{0}$ natural transformation. We show that there exists uniquely a morphism
of $\delta$-functor $\theta^{\bullet}\colon T^{\bullet}\to U^{\bullet}$ such
that $\theta^{0}=\theta$. We construct it inductively. For any
$A\in\mathscr{A}$, there is injection $u\colon A\to M$ such that $T^{1}(u)=0$
since $T^{1}$ is effaceable. Let $C$ be the cokernel of $u$. We consider the
long exact sequences induced by
${0}$${A}$${M}$${C}$${0}$$\scriptstyle{u}$$\scriptstyle{\pi}$. So we get the
following commutative diagram.

${T^{0}(M){}{}}$${{}{}T^{0}(C){}}$${{}T^{1}(A){}}$${{}0}$${U^{0}(M){}}$${{}U^{0}(C){}}$${{}U^{1}(A)}$$\scriptstyle{\theta_{M}}$$\scriptstyle{T^{0}(\pi)}$$\scriptstyle{\theta_{C}}$$\scriptstyle{\delta^{0}_{T}}$$\scriptstyle{T^{1}(u)=0}$$\scriptstyle{U^{0}(\pi)}$$\scriptstyle{\delta^{0}_{U}}$

Now
$\theta^{1}_{A,u}\coloneqq\delta^{0}_{U}\circ\theta_{C}\circ(\delta^{0}_{T})^{-1}\colon
T^{1}(A)\to U^{1}(A)$ is well-defined since the rows are exact. We show
$\theta^{1}_{A,u}$ is independent of the choice of $u$. Let $u^{\prime}\colon
A\to M^{\prime}$ be an injection such that $T^{1}(u^{\prime})=0$.
$M\sqcup_{A}M^{\prime}$ denote co-fibre products of $M$ and $M^{\prime}$ on
$A$. Then we get an injection $u^{\prime\prime}\colon A\to M\sqcup M^{\prime}$
such that $T^{1}(u^{\prime\prime})=0$. Let $C^{\prime\prime}$ be the cokernel
of $u^{\prime\prime}$. Then the following diagram;

${T^{0}(C)}$${T^{1}(A)}$${0}$${T^{0}(C^{\prime\prime})}$${T^{1}(A)}$${0}$${U^{0}(C)}$${U^{1}(A)}$${U^{0}(C^{\prime\prime})}$${U^{1}(A)}$$\scriptstyle{\theta_{C^{\prime\prime}}}$$\scriptstyle{\theta_{A,u}^{1}}$$\scriptstyle{\theta_{C^{\prime\prime}}}$$\scriptstyle{\theta_{A,u^{\prime\prime}}^{1}}$

is commutative. So we have $\theta_{A,u}^{1}=\theta_{A,u^{\prime\prime}}^{1}$,
similarly we obtain
$\theta_{A,u^{\prime}}^{1}=\theta_{A,u^{\prime\prime}}^{1}$, then we get
$\theta_{A,u}^{1}=\theta_{A,u^{\prime}}^{1}$. So $\theta_{A}^{1}$ is
independent of the choice of $u$.

Secondly, we show that for each $f\in\operatorname{Hom}_{\mathscr{A}}(A,B)$
the following diagram;

${T^{1}(A){}{}}$${{}T^{1}(B){}}$${U^{1}(A){}}$${{}U^{1}(B)}$$\scriptstyle{\theta_{A}^{1}}$$\scriptstyle{T^{1}(f)}$$\scriptstyle{\theta_{B}^{1}}$$\scriptstyle{U^{1}(f)}$

is commutative to prove $\theta^{1}$ is a natural transformation. For any
injections $u\colon A\to M$ and $v\colon B\to N$ with $T^{1}(u)=T^{1}(v)=0$,
we take co-fibre product;

${A{}{}}$${{}M{}}$${N{}}$${{}M\sqcup_{A}N}$$\scriptstyle{v\circ
f}$$\scriptstyle{u}$$\scriptstyle{u^{\prime}}$

then $u^{\prime}$ is injective. So we have an injection $u^{\prime}\circ
v\colon B\to M\sqcup N$ with $T^{1}(u^{\prime}\circ v)=0$. Then we replace $N$
by $M\sqcup N$ and get the following commutative diagram with exact rows;

${0{}}$${{}A{}{}}$${{}M{}{}}$${{}C{}}$${{}0}$${0{}}$${{}B{}}$${{}N{}}$${{}C^{\prime}{}}$${{}0}$$\scriptstyle{f}$

So $\theta^{1}$ is a natural transformation since the following diagram;

${T^{0}(C)}$${T^{1}(A)}$${0}$${T^{0}(C^{\prime})}$${T^{1}(B)}$${0}$${U^{0}(C)}$${U^{1}(A)}$${U^{0}(C^{\prime})}$${U^{1}(B)}$$\scriptstyle{T^{1}(f)}$$\scriptstyle{\theta_{A}^{1}}$$\scriptstyle{U^{1}(f)}$$\scriptstyle{\theta_{B}^{1}}$

is commutative.

Finally, we show that $\theta_{A}^{1}$ is commutative with the connecting
morphism. Let ${0}$${A_{1}}$${A_{2}}$${A_{3}}$${0}$ be the short exact
sequence in $\mathscr{A}$, we use the same method as above for the injection
$u\colon A_{1}\to M$ with $T^{1}(u)=0$ so that each row of the following
commutative diagram is exact.

${0{}}$${{}A{}}$${{}A_{2}{}{}}$${{}A_{3}{}{}}$${{}0}$${0{}}$${{}A_{1}{}}$${{}M{}}$${{}C{}}$${{}0}$

We consider the following diagram.

${T^{0}(A_{3})}$${T^{1}(A_{1})}$${U^{0}(A_{3})}$${U^{1}(A_{1})}$${T^{0}(C)}$${T^{1}(A_{1})}$${U^{0}(C)}$${U^{1}(A_{1})}$$\scriptstyle{\theta_{A_{3}}}$$\scriptstyle{\delta}$$\scriptstyle{\theta_{A_{1}}^{1}}$$\scriptstyle{\delta}$$\scriptstyle{\theta_{C}}$$\scriptstyle{\delta}$$\scriptstyle{\theta_{A_{1}}^{1}}$$\scriptstyle{\delta}$

The desired commutativity of $\theta_{A_{1}}^{1},\theta_{A_{3}}$ and $\delta$
follows from commutativity of other squares, which follow from the
construction of $\theta_{A_{1}}^{1}$ and the facts that $T^{\bullet}$ and
$U^{\bullet}$ are $\delta$-functor and that $\theta$ is a natural
transformation.

This shows that $\theta^{1}$ is a natural transformation commutative with
connecting morphism, and its uniqueness can be seen from its construction (the
universality of cokernel). In this way $\theta^{1}$ to $\theta^{2}$ can be
created and continued inductively. ∎

######  Corollary 2.6.

Let $\mathscr{A},\mathscr{B}$ be Abelian categories, and $\mathscr{A}$ has
enough injectives. Let $T^{\bullet}\colon\mathscr{A}\to\mathscr{B}$ be a
universal $\delta$-functor, then $T^{0}$ is left-exact and for each $i\geq 0$
there is a natural isomorphism $T^{i}\cong R^{i}T^{0}$.

###### Proof.

$T^{0}$ is left exact by the definition of $\delta$-functor, so there are
right derived functors $R^{i}T^{0}$. For each $i>0$, $R^{i}T^{0}$ is
effaceable by (4) of 2.2, then $R^{\bullet}T^{0}$ is a universal
$\delta$-functor. Now $R^{0}T^{0}=T^{0}$, so there is a unique isomorphism
$R^{\bullet}T^{0}\cong T^{\bullet}$ by universality. ∎

##  3\. Čech cohomology and Koszul comoplex

In this section, we review the Čech cohomology of rings and modules. Let $A$
be a ring and fix a sequence $a_{1},\dots,a_{r}$ of elements of $A$. For each
$I=\\{j_{1},\dots,j_{i}\\}\subset\\{1,\dots,r\\}~{}(j_{1}<\dots<j_{i})$, let
$a_{I}=a_{j_{1}}\dots a_{j_{i}}$. $e_{1},\dots,e_{r}$ denotes the standard
basis of $A^{r}$ and let $e_{I}=e_{j_{1}}\wedge\dots\wedge e_{j_{i}}$.

######  Definition 3.1.

Let $A$ be a ring, $\underline{a}=a_{1},\dots,a_{r}\in A$. For each $1\leq
i\leq r$, $C^{i}(\underline{a})$ is the complex defined by following
equations;

$C^{i}(\underline{a})=\sum_{\\#I=i}A_{a_{I}}e_{I},$ $d^{i}\colon
C^{i}(\underline{a})\to
C^{i+1}(\underline{a});e_{I}\mapsto\sum_{j=1}^{n}e_{I}\wedge e_{j}.$

And for $i=0$, defined by $C^{0}(\underline{a})=A,d^{0}\colon
a\mapsto\sum_{j=1}^{r}ae_{j}$ so $C^{\bullet}(\underline{a})$ form a complex.
It called Čech complex. $\check{H}^{i}(\underline{a})$ denote the cohomology
of this complex and it is called Čech cohomology.

For $A$-module $M$, we define $C^{\bullet}(\underline{a},M)\coloneqq
C^{\bullet}(\underline{a})\otimes M$. Here $\check{H}^{i}(\underline{a},M)$
denote the $i$-th cohomology of $C^{i}(\underline{a},M)$. This form complex,
denoted by $\check{H}^{\bullet}(\underline{a},M)$.

######  Proposition 3.2.

Let $A$ be a ring, for each $a_{1},\dots,a_{r}\in A$,
$\check{H}^{\bullet}(\underline{a},-)$ is a $\delta$-functor.

###### Proof.

Consider an exact sequence ${0}$${M_{1}}$${M_{2}}$${M_{3}}$${0}$ of
$A$-modules. Since
$C^{\bullet}(\underline{a},M)=C^{\bullet}(\underline{a})\otimes M$ and each
component of the Čech complex is a flat $A$-modules, the following sequence of
complexes is exact;

${0}$${C^{\bullet}(\underline{a},M_{1})}$${C^{\bullet}(\underline{a},M_{2})}$${C^{\bullet}(\underline{a},M_{3})}$${0}$

then there are connection morphisms. So $\check{H}^{\bullet}(\underline{a},-)$
is $\delta$-functor. ∎

For the result we want, we need to look at the relationship between Čech
complex and Koszul complex.

For $\underline{a}=a_{1},\dots,a_{r}\in A$, let $\\{e_{i}\\}$ be the standard
basis of a free $A$-module $A^{r}$. $f\colon A^{r}\to A;e_{i}\mapsto a_{i}$
induces a chain complex $K_{\bullet}(\underline{a})$. In other words
$K_{\bullet}(\underline{a})$ is the complex defined by the following equation;

$K_{i}(\underline{a})=\bigwedge^{i}A^{r},$ $d_{i}\colon
K_{i}(\underline{a})\to K_{i-1}(\underline{a});x_{1}\wedge\dots\wedge
x_{i}\mapsto\sum_{j=1}^{i}(-1)^{j+1}f(x_{j})x_{1}\wedge\dots\wedge\widehat{x}_{j}\wedge\dots\wedge
x_{i}.$

Note that $K_{\bullet}(\underline{a})$ does not depend on the order of $a_{i}$
.

We get a co-chain complex $K^{\bullet}(\underline{a})$ from a contravariant
functor $\operatorname{Hom}(-,A)$;

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$K^{\bullet}(\underline{a})$ is called a Koszul complex. For each $A$-module
$M$, $K^{\bullet}(\underline{a},M)=K^{\bullet}(\underline{a})\otimes M$. Here
$H^{i}(\underline{a},M)$ denote the cohomology of Koszul complex.

######  Lemma 3.3.

Let $A$ be a ring and $\underline{a}=a_{1},\dots,a_{r}\in A$. For each $1\leq
i\leq r$;

$\varphi^{i}\colon K^{i}(\underline{a})\to
C^{i}(\underline{a});(e_{I})^{\ast}\mapsto(1/a_{I})e_{I}$

is a morphism of complexes.

###### Proof.

Let $\delta^{i}$ be the derivative in the Koszul complex. Then;

$\delta^{i}(e_{I}^{\ast})(e_{J})=\begin{cases}a_{j}&(j\not\in
I,J=I\cup\\{j\\})\\\ 0&(\text{otherwise})\end{cases}$

So;

$\varphi^{i+1}\circ\delta^{i}(e_{I}^{\ast})=\sum_{j\not\in
I}\frac{a_{j}}{a_{I}a_{j}}e_{I}\wedge e_{j}=\sum_{j\not\in
I}\frac{1}{a_{I}}e_{I}\wedge e_{j}$

is equal to $d^{i}\circ\varphi^{i}(e_{I}^{\ast})$. ∎

For any pair $n\leq m$;

$\varphi_{mn}^{\bullet}\colon K^{\bullet}(\underline{a}^{n})\to
K^{\bullet}(\underline{a}^{m});(e_{I})^{\ast}\mapsto(a_{I})^{m-n}(e_{I})^{\ast}$

then $\\{K^{\bullet}(\underline{a}^{n})\\}_{n\in\mathbb{N}}$ is an inductive
system.

######  Proposition 3.4.

Let $A$ be a ring, $a_{1},\dots,a_{r}\in A$. Then;

$\varinjlim K^{\bullet}(\underline{a}^{n})\cong C^{\bullet}(\underline{a}).$

###### Proof.

We define $\varphi_{n}^{\bullet}\colon K^{\bullet}(\underline{a}^{n})\to
C^{\bullet}(\underline{a}^{n})=C^{\bullet}(\underline{a})$ in the same way as
above lemma. Then
$\varphi_{m}^{\bullet}\circ\varphi_{nm}^{\bullet}=\varphi_{n}^{\bullet}$ where
$n\leq m$. So we have $\varphi\colon\varinjlim
K^{\bullet}(\underline{a}^{n})\to C^{\bullet}(\underline{a})$. Each element of
$C^{i}(\underline{a})$ is represented finite sum of
$(b_{I}/a_{I}^{n_{I}})e_{I}$, so it can be displayed as
$\sum(1/a_{I}^{n})b_{I}e_{I}$ by taking the maximum of $n$ and replacing
$b_{I}$. Then it is image of $\sum(b_{I}e_{I})\in K^{i}(\underline{a}^{n})$ ,
so $\varphi$ is surjective.

Secondly, we show $\varphi$ is injective. Assume $\varphi^{i}_{n}(x)=0$ for
$x\in K^{i}(\underline{a}^{n})$. If $x=\sum b_{I}e_{I}^{\ast}$ then
$\varphi^{i}_{n}(x)=\sum(b_{I}/a_{I}^{n})e_{I}^{\ast}=0$, so
$b_{I}/a_{I}^{n}=0$ in $A_{a_{I}^{n}}$. Therefore if we take a sufficiently
large $l$, $a_{I}^{l}b_{I}=0$. So $\varphi_{nm}^{l}(x)=0$ by increasing $l$ if
necessary, then $\varphi$ is injective. ∎

Since the functor of taking the inductive limit is exact, the following
Corollary follows.

######  Corollary 3.5.

Let $A$ be a ring, $a_{1},\dots,a_{r}\in A$. For each $A$-module $M$;

$\check{H}^{i}(\underline{a},M)\cong\varinjlim H^{i}(\underline{a}^{n},M).$

##  4\. Weakly proregular sequences

######  Definition 4.1.

Let $\mathscr{A}$ be an Abelian category, $(X_{n},\varphi_{mn})$ a projective
system in $\mathscr{A}$. $(X_{n})$ is said to be essentially zero or pro-zero
if for each $n$, there is $m\geq n$ such that $\varphi_{mn}\colon X_{m}\to
X_{n}$ is zero map.

Obviously, if $(X_{n})$ is essentially zero then $\varprojlim X_{n}=0$.

######  Proposition 4.2.

Let $\mathscr{A}$ be an Abelian category. We consider a exact sequence of
projective system in $\mathscr{A}$;

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Then $(Y_{n})$ is essentially zero if and only if the other two are
essentially zero.

###### Proof.

$(\Longrightarrow)$ Trivial. $(\Longleftarrow)$ For each $n$, there is $m\geq
n$ such that $X_{m}\to X_{n}$ since $(X_{n})$ is essentially zero. Similarly
there is $l\geq m$ such that $Z_{l}\to Z_{m}$ is zero map, then we have the
following commutative diagram with the rows are exact;

${0{}}$${{}X_{l}{}{}}$${{}Y_{l}{}{}}$${{}Z_{l}{}{}}$${{}0}$${0{}}$${{}X_{m}{}{}}$${{}Y_{m}{}{}}$${{}Z_{m}{}{}}$${{}0}$${0{}}$${{}X_{n}{}}$${{}Y_{n}{}}$${{}Z_{n}{}}$${{}0}$$\scriptstyle{f_{l}}$$\scriptstyle{\varphi_{lm}}$$\scriptstyle{g_{l}}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{f_{m}}$$\scriptstyle{\varphi_{mn}}$$\scriptstyle{g_{m}}$$\scriptstyle{f_{n}}$$\scriptstyle{g_{n}}$

So we get $\varphi_{ln}=\varphi_{mn}\circ\varphi_{lm}=0$ by easy diagram
chasing. ∎

We use same symbols as before for the Koszul and Čech complexes. Note that
$(K_{i}(\underline{a}^{n}))_{n\in\mathbb{N}}$ is a projective system defined
by $K_{i}(\underline{a}^{m})\to K_{i}(\underline{a}^{n});e_{I}\mapsto
a_{I}^{m-n}e_{I}~{}(m\geq n)$.

######  Definition 4.3.

Let $A$ be a ring. $a_{1},\dots,a_{r}\in A$ is called a weakly proregular
sequence if for each $1\leq i\leq r$, the projective system
$\\{H_{i}(\underline{a}^{n})\\}$ is essentially zero.

The property of being weakly proregular does not depend on the order by the
definition.

######  Proposition 4.4.

Let $A$ be a ring, $\underline{a}=a_{1},\dots,a_{r}\in A$. $\underline{a}$ is
a weakly proregular sequence if and only if
$\check{H}^{\bullet}(\underline{a},-)$ is an effaceable $\delta$-functor.

###### Proof.

Assume that $\underline{a}$ is a weakly proregular sequence. Let $I$ be an
injective module. Now there is a isomorphism
$H^{i}(\underline{a}^{n},I)\cong\operatorname{Hom}(H_{i}(\underline{a}^{n}),I)$
since
$K^{\bullet}(\underline{a}^{n},I)=\operatorname{Hom}(K_{\bullet}(\underline{a}^{n}),I)$
and $\operatorname{Hom}(-,I)$ is an exact functor. For each $n\geq 0$, there
is $m\geq n$ such that $H_{i}(\underline{a}^{m})\to H_{i}(\underline{a}^{n})$
is zero map since $H_{i}(\underline{a}^{n})$ is essentially zero. So
$\check{H}^{i}(\underline{a},I)=\varinjlim H^{i}(\underline{a}^{n},I)=0$.

Secondly, assume that $\check{H}^{\bullet}(\underline{a},-)$ is effaceable
$\delta$-functor. For each $n\geq 0$, we have injective module $I$ and
injection $\varepsilon\colon H_{i}(\underline{a}^{n})\to I$. Then there is
$m\geq n$ such that;

${H_{i}(\underline{a}^{m}){}}$${{}H_{i}(\underline{a}^{n}){}}$${{}I}$$\scriptstyle{\varepsilon}$

is zero map by $\varepsilon\in H^{i}(\underline{a}^{n},I)$ and $\varinjlim
H^{i}(\underline{a}^{n},I)=0$. ∎

Then Čech cohomology is the derived functor of
$\check{H}^{0}(\underline{a},-)$ if $\underline{a}$ is a weakly proregular
sequence. So the next question of interest is when is a sequence weakly
proregular?

######  Definition 4.5.

Let $A$ be a ring, $\underline{a}=a_{1},\dots,a_{r}\in A$. $\underline{a}$ is
called proregular sequence if for each $1\leq i\leq r$ and $n>0$, there is
$m\geq n$ such that
$((a_{1}^{m},\dots,a_{i-1}^{m}):a_{i}^{m}A)\subset((a_{1}^{n},\dots,a_{i-1}^{n}):a_{i}^{m-n}A)$.

Note that a regular sequence is proregular and this phenomenon is
characteristic of non-Noetherian rings.

######  Proposition 4.6.

Let $A$ be a Noetherian ring. For each $a_{1},\dots,a_{r}\in A$ is a
proregular sequence.

###### Proof.

Let
$J_{m}=((a_{1}^{m},\dots,a_{i-1}^{m}):a_{i}^{m}A),I_{n,m}=((a_{1}^{n},\dots,a_{i-1}^{n}):a_{i}^{m-n}A)$.
Now $\\{I_{n,m}\\}_{m\geq n}$ is an ascending chain of ideals, then there is
$m_{0}\geq n$ such that for each $m\geq m_{0},I_{n,m_{0}}=I_{n,m}$. Let
$m=m_{0}+n$, then for each $a\in J_{m_{0}}$,
$aa_{i}^{m-n}=aa_{i}^{m_{0}}\in(a_{1}^{m_{0}},\dots,a_{i-1}^{m_{0}})\subset(a_{1}^{n},\dots,a_{i-1}^{n})$.
So $a\in I_{n,m}=I_{n,m_{0}}$. ∎

######  Proposition 4.7.

Let $A$ be a ring. A proregular sequence is weakly proregular.

###### Proof.

We use induction on $r$. When $r=1$, let $a\in A$ be proregular. Then for each
$n>0$, there is $m\geq n$ such that
$\operatorname{Ann}a^{m}\subset\operatorname{Ann}a^{m-n}$. So $(H_{1}(a^{n}))$
is essentially zero since $H_{1}(a^{n})=\operatorname{Ann}a^{n}$. Now we
assume that claim up to $r-1$. The exact sequence of complexes;

${0{}}$${{}{K_{\bullet}(a_{1}^{n},\dots,a_{r-1}^{n})}{}}$${{}{K_{\bullet}(a_{1}^{n},\dots,a_{r}^{n})}}$
${{}K_{\bullet}(a_{1}^{n},\dots,a_{r-1}^{n})(-1){}}$${{}0}$

induces the exact sequence of homology;

$\cdots$${{}H_{i}(a_{1}^{n},\dots,a_{r-1}^{n}){}}$${H_{i}(a_{1}^{n},\dots,a_{r-1}^{n}){}}$${{}H_{i}(a_{1}^{n},\dots,a_{r}^{n}){}}$${{}H_{i-1}(a_{1}^{n},\dots,a_{r-1}^{n}){}}$${H_{i-1}(a_{1}^{n},\dots,a_{r-1}^{n}){}}$$\cdots$$\scriptstyle{(-1)^{i}a_{r}^{n}}$$\scriptstyle{(-1)^{i-1}a_{r}^{n}}$

Then we have the following exact sequence;

${0{}}$${{}{H_{0}(a_{r}^{n},H_{i}(a_{1}^{n},\dots,a_{r-1}^{n}))}{}}$${{}{H_{i}(a_{1}^{n},\dots,a_{r}^{n})}}$
${{}{H_{1}(a_{r}^{n},H_{i-1}(a_{1}^{n},\dots,a_{r-1}^{n}))}{}}$${{}0}$

and this induces the exact sequence of projective systems. The first
projective system is essentially zero by the assumption of induction. Also for
each $i>1$, the third system is essentially zero since
$H_{1}(a_{r}^{n},H_{i-1}(a_{1}^{n},\dots,a_{r-1}^{n}))=\left\\{x\in
H_{i-1}(a_{1}^{n},\dots,a_{r-1}^{n})\mathrel{}\middle|\mathrel{}a_{r}^{n}x=0\right\\}$.
If $i=1$, the system with ;

$H_{1}(a_{r},H_{0}(a_{1}^{n}\dots,a_{r-1}^{n}))=\left\\{x\in
H_{0}(a_{1}^{n},\dots,a_{r-1}^{n})\mathrel{}\middle|\mathrel{}a_{r}^{n}x=0\right\\}$

is essentially zero since $\underline{a}$ is proregular. So this completes the
proof by induction. ∎

######  Corollary 4.8.

Let $A$ be a Noetherian ring. For each $\underline{a}=a_{1},\dots,a_{r}\in A$,
$\underline{a}$ is weakly proregular.

##  5\. Local cohomology

Let $A$ be a ring and $I$ an ideal of $A$. The functor $\Gamma_{I}$ is defined
by;

$\Gamma_{I}(M)=\left\\{x\in M\mathrel{}\middle|\mathrel{}\text{$I^{n}x=0$ for
some $n\geq 0$}\right\\}$

for an $A$-module $M$. Note that
$\Gamma_{I}(M)=\varinjlim\operatorname{Hom}_{A}(A/I^{n},M)$ and this
isomorphism is functorial in $M$. By the definition, $\Gamma_{I}$ is a left
exact functor.

######  Definition 5.1.

Let $A$ be a ring and $I$ an ideal of $A$. $H^{i}_{I}(-)$ denote the derived
functor of $\Gamma_{I}(-)$ and it is called local cohomology.

Note that $H^{i}(M)\cong\varinjlim\operatorname{Ext}^{i}(A/I^{n},M)$. We
summarize the relationship between local cohomology and Čech cohomology.
First, we note that the 0-th parts of each cohomology are naturally
isomorphic.

######  Lemma 5.2.

Let $A$ be a ring, $\underline{a}=a_{1},\dots,a_{r}\in A$, and
$I=(a_{1},\dots,a_{r})$. For each $A$-module $M$;

$\Gamma_{I}(M)\cong\check{H}^{0}(\underline{a},M).$

###### Proof.

Here $\check{H}^{0}(\underline{a},M)$ is the kernel of

$M\to\bigoplus_{i=1}^{r}M_{a_{i}}e_{i};x\mapsto(x/1)e_{i}.$

Then for each $x\in\check{H}^{0}(\underline{a},M)$ and $1\leq i\leq r$, there
is $n_{i}\geq 0$ such that $a_{i}^{n_{i}}x=0$. So we have $x\in\Gamma_{I}(M)$.
Similarly the converse is true, so they are equal as submodules of $M$. ∎

With the preparations we have made above, we can prove the results we have
been aiming for.

######  Theorem 5.3.

Let $A$ be a ring, $\underline{a}=a_{1},\dots,a_{r}\in A$ and
$I=(a_{1},\dots,a_{r})$. $\underline{a}$ is a weakly proregular sequence if
and only if for any $i$ and $A$-module $M$,
$H^{i}_{I}(M)\cong\check{H}^{i}(\underline{a},M)$.

###### Proof.

Assume that $\underline{a}$ is weakly proregular.
$\check{H}^{\bullet}(\underline{a},-)$ is a $\delta$-functor by 3.2. Moreover
$\check{H}^{\bullet}(\underline{a},-)$ is universal by 4.4 and 2.5. So
$H^{i}_{I}(M)\cong\check{H}^{i}(\underline{a},M)$ by above lemma. The converse
is true by 4.4. ∎

### Acknowledgments

I thank Hisanori Ohashi, and Yuya Matsumoto for helpful comments and
discussions.

## References