Transit cosmological models coupled with zero-mass scalar-field with high
redshift in higher derivative theory
Archana Dixit1, Dinesh Chandra Maurya2, Anirudh Pradhan3

1,3Department of Mathematics, Institute of Applied Sciences & Humanities, GLA
University, Mathura-281 406, Uttar Pradesh, India

2Department of Mathematics, Faculty of Engineering & Technology, IASE (Deemed
to be University), Sardarshahar-331 403, Rajsthan, India

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The present study deals with a flat FRW cosmological model filled with perfect
fluid coupled with the zero-mass scalar field in the higher derivative theory
of gravity. We have obtained two types of universe models, the first one is
the accelerating universe (power-law cosmology) and the second one is the
transit phase model (hyperbolic expansion-law). We have obtained various
physical and kinematic parameters and discussed them with observationally
constrained values of $H_{0}$. The transit redshift value is obtained
$z_{t}=0.414$ where the transit model shows signature-flipping and is
consistent with recent observations. In our models, the present values of EoS
parameter $\omega_{0}$ crosses the cosmological constant value
$\omega_{0}=-1$. Also, the present age of the universe is calculated.

PACS No.: 98.80.Jk; 95.36.+x; 98.80.-k

Keywords: Flat FRW-universe; Higher derivative theory of gravity; Zero-mass
scalar field; Redshift

##  1 Introduction

From the recent supernovae observations ( Garnavich et al. [1, 2], Perlmutter
et al. [3, 4, 5], Riess et al. [6] and Schmidt et al. [7]), it was predicted
that the expansion rate of the universe is increasing at present. This
prediction points towards the presence of something (a kind of repulsive
force) in the Universe which is pushing everything farther apart faster than
it did in the early universe. The agent for this repulsive force is attributed
to a mysterious entity present in the universe along with the usual matter and
radiation. This hypothetical unknown exotic physical entity is termed as dark
energy (DE). To know the reason behind this acceleration scenario of the
present universe, there were two ways, first one to modify Einstein’s field
equations and another to add a time-dependent cosmological constant $\Lambda$.
In order of modifying Einstein’s general theory of relativity (GR), one is
obtained by adding $R^{2}$ factor (square of Ricci scalar curvature) in the
Einstein-Hilbert action which generalizes the Einstein’s GR theory of gravity.
Before the fruitful work of Guth, Starobinsky [8] proposed higher derivative
theories of gravity that admit inflation. But the theory come into focus only
after the work of Guth [9], in which, he used a temperature phase transition
mechanism. The good features of the higher-order theories of gravity are very
interesting for a long time. Also, in literature, it is a tradition to get a
perturbation theory by adding suitable counter terms, viz to the cosmological
term $(\Lambda)$ and the $C^{ijkl}C_{ijkl}$, $R^{2}$ to the Einstein’s action
is asymptotically free, well behaved, and formally re-normalizable Stelle
[10].

The quantum field theory states a new approach to inflation and finds the
dynamical history of the expanding universe through GR. Even if, there is a
general belief among researchers that the inflation is a part of cosmological
evolution. Because of this, to solve the various problems of inflation by
reserving its importance, among the researchers have a continuous contest by
presenting different modifications (Linde [11]; Kolb and Turner [12]).
Chimento and Jakubi [13] have presented cosmological models with perfect fluid
in the context of higher derivative theory of gravity. Even if, a perfect
fluid source of matter may explain matter distribution in the universe
satisfactorily, but reason behind the accelerating scenario of the universe
i.e. also, well known by the dark energy (DE) problem, in literature, can not
be explained by perfect fluid, properly. In some literature, it is claim that
the dark energy problem can be pledged by adding a higher derivative terms in
the transport equation (Muller [14]; Hiscock and Salmonson [15]). The theory
known by extended irreversible thermodynamics (EIT) is a fully relativistic
formulation of the theory taking the second derivative terms in the theory, is
developed by Israel and Stewart [16], and Pavon et al. [17], Pavon [18]. Ratra
and Peebles [19] have obtained a new type of cosmological solutions in the
context of GR by using transport equation the theory EIT. Pavon and Zimdahl
[20] are claimed that a viscous fluid source could be the presence of dark
matter that causes the acceleration in expansion of the universe and Zimdahl
et al. [21] have claimed to the accelerating expansion due to a effective
negative force driven by a cosmic anti-friction force. A consistency of
various inflation with an arbitrary potential driving inflation are
investigated by Ram [22]. A perfect fluid FRW cosmological model with a
minimally coupled scalar field have been investigated by Ellis and Madsen
[23]. In the case of minimally and non-minimally coupled scalar scalar field,
a cosmological solution for the scale factor are obtained with the assumption
the kinetic and potential terms for the scalar field are proportional to each
other, Barrow and Saich [24], Barrow and Mimoso [25] and Mimoso and Wands
[26]. Casana et al. [27] have investigated massless DKP field in a Lyra
manifold. Recently, Singh et al. [28] have investigated the cosmological
solutions in the presence of an imperfect fluid and zero-mass scalar field in
higher derivative theory of gravity. Recently, Maurya and Zia [29] have
investigated Brans-Dicke scalar field cosmological model in Lyra’s Geometry.
Some authors [30]-[33] have studied and discussed zero-mass scalar field in
different contexts.

Motivated by above discussion, we have considered a flat FRW universe filled
with perfect fluid coupled with the zero-mass scalar field in higher
derivative theory of gravity and obtained a transit phase model. The out lines
of the paper as follows: Section $1$ is introductory, Section $2$ introduces
the metric and field equations. Solutions and discussion of results are given
in Section $3$ and finally conclusions are given in Section $4$.

##  2 Metric and Field Equations

To investigate the physical behaviour of the universe, we consider the action
principle as

$I=\int{\left[\frac{1}{2}f(R)+L_{m}\right]}\sqrt{-g}dx^{4}$ (1)

where $f(R)$ is a function of Ricci scalar curvature $R$ and its higher
powers, $g$ is the determinant of the metric tensor $g_{ij}$. $L_{m}$ is
matter Lagrangian, in cosmic unit taking $8\pi G=c=1$. By the variation of
action $(1)$ with respect to metric field $g_{ij}$ and scalar field $\phi$
respectively, we get

$f^{\prime}(R)R_{ij}-\frac{1}{2}f(R)g_{ij}+f^{\prime\prime}(R)(\nabla_{i}\nabla_{j}R-\Box
Rg_{ij})+f^{\prime\prime\prime}(R)(\nabla_{i}R\nabla_{j}R-\nabla^{k}R\nabla_{k}Rg_{ij})=-T_{ij}$
(2)

and

$\sqrt{-g}\phi_{,i}^{,i}=(\sqrt{-g}g^{ij}\phi_{,i})_{,j}=0$ (3)

where $\Box=g_{ij}\nabla^{i}\nabla^{j}$ and $\nabla_{i}$ is the covariant
differential operator, and prime represents the derivative with respect to
$R$. The Stress-energy momentum tensor for the fluid coupled with zero-mass
scalar-field is given as

$T_{ij}=T^{F}_{ij}+T^{\phi}_{ij}$ (4)

where

$T^{F}_{ij}=(\rho+p)u_{i}u_{j}+pg_{ij}$ (5)

and

$T^{\phi}_{ij}=\phi_{,i}\phi_{,j}-\frac{1}{2}g_{ij}\phi_{,k}\phi^{,k}$ (6)

where $\rho$ is the energy density and $p$ is the isotropic pressure of the
fluid. In a co-moving coordinate system, the four-velocity vector $u_{i}$ are
satisfies the condition $u_{i}u^{i}=-1$. $\phi$ is the zero-mass scalar-field.

Now, we have considered the flat Robertson-Walker spacetime metric, given by

$ds^{2}=-dt^{2}+a^{2}(t)[dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\varphi^{2})]$ (7)

where $a(t)$ is the scale factor of the universe. The scalar curvature $R$ and
trace $T$ of the energy momentum tensor $T_{ij}$ of the universe are given as
respectively

$R=-6(\dot{H}+2H^{2})$ (8)

and

$T=-(\rho-3p)-\dot{\phi}^{2}$ (9)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter and an over dot represents
a derivative with respect to time. The trace and $(0,0)$ components of $(2)$
are respectively, given by

$f^{\prime}(R)R-2f(R)-3f^{\prime\prime}(R)(\ddot{R}+3H\dot{R})-3f^{\prime\prime\prime}(R)\dot{R}^{2}+T=0$
(10)

and

$f^{\prime}(R)R^{0}_{0}-\frac{1}{2}f(R)-3f^{\prime\prime}(R)H\dot{R}+T^{0}_{0}=0$
(11)

Now, since, we have considered a higher derivative theory of gravity defined
by $f(R)=R+\alpha R^{2}$. Using Eq. $(8)$ in $(10)$ and $(11)$, we obtain

$\dot{H}+2H^{2}+6\alpha[\dddot{H}+7H\ddot{H}+4\dot{H}^{2}+12H^{2}\dot{H}]=-\frac{1}{6}T$
(12)

and

$3H^{2}+18\alpha[2H\ddot{H}+\dot{H}^{2}+10H^{2}\dot{H}]=\rho+\frac{\dot{\phi}^{2}}{2}$
(13)

From Eq. $(3)$, we get

$\ddot{\phi}+3H\dot{\phi}=0$ (14)

Taking covariant derivative of the field equation $(2)$ and applying energy
conservation laws, we obtain

$36\alpha(\ddot{H}+4H\dot{H})(2\dot{H}+H^{2})=0$ (15)

and

$\dot{\rho}+3H(\rho+p)=0$ (16)

where over dot denotes ordinary derivatives with respect to cosmic time
$`t^{\prime}$.

##  3 Solutions of the field equations and discussion of results

From redshift relation $\frac{a(t_{0})}{a(t)}=1+z$ with $a(t_{0})=1$, we have
found

$\dot{H}=-(1+z)HH^{\prime}\hskip 441.01772pt$ (17)
$\ddot{H}=(1+z)H^{2}H^{\prime}+(1+z)HH^{\prime
2}+(1+z)^{2}H^{2}H^{\prime\prime}\hskip 284.52756pt$ (18)
$\dddot{H}=-(1+z)H^{3}H^{\prime}-4(1+z)^{2}H^{2}H^{\prime
2}-3(1+z)^{2}H^{3}H^{\prime\prime}-(1+z)^{3}H^{2}H^{\prime
3}-4(1+z)^{3}H^{2}H^{\prime}H^{\prime\prime}-(1+z)^{3}H^{3}H^{\prime\prime\prime}$
(19)

where $H^{\prime}=\frac{dH}{dz}$.
Now, integrating Eq. $(14)$, we have obtained the expression for zero-mass
scalar field as

$\dot{\phi}=\frac{n}{a^{3}}=n(1+z)^{3}$ (20)

or

$\phi=-n\int\frac{(1+z)^{2}}{H}dz$ (21)

where $n$ is an integrating constant.
Therefore, using these results in Eqs. $(12)$ and $(13)$ we obtain

$T=6(1+z)HH^{\prime}-12H^{2}+36\alpha(1+z)H^{2}[6HH^{\prime}-4(1+z)HH^{\prime\prime}+(1+z)^{2}H^{\prime
3}+4(1+z)^{2}H^{\prime}H^{\prime\prime}+(1+z)^{2}HH^{\prime\prime\prime}-7H^{\prime
2}]$ (22)

and

$\rho=3H^{2}+18\alpha(1+z)H^{2}[(3+z)H^{\prime
2}+2(1+z)HH^{\prime\prime}-8HH^{\prime}]-\frac{n^{2}}{2}(1+z)^{6}$ (23)

If the equation of state (EoS) for the considered fluid is read as
$p=\rho\omega$, then EoS parameter $\omega$ is calculated from Eq. $(16)$ as

$\omega=-1-\frac{\dot{\rho}}{3H\rho}$ (24)

or

$2(1+z)HH^{\prime}-n^{2}(1+z)^{6}$
$\omega=-1+\frac{-12\alpha(1+z)H[2(1+z)(2HH^{\prime 2}-H^{\prime
3}+H^{2}H^{\prime\prime})-(1+z)^{2}H(H^{\prime}H^{\prime\prime}+H^{\prime
3}+HH^{\prime\prime\prime})+4H^{2}H^{\prime}+5HH^{\prime
2}]}{[3H^{2}+18\alpha(1+z)H^{2}[(3+z)H^{\prime
2}+2(1+z)HH^{\prime\prime}-8HH^{\prime}]-\frac{n^{2}}{2}(1+z)^{6}]}$ (25)

Now, from Eq. $(15)$, we have

$\ddot{H}+4H\dot{H}=0~{}~{}~{}~{}~{}\text{or}~{}~{}~{}~{}2\dot{H}+H^{2}=0$
(26) $2\dot{H}+H^{2}=0~{}\implies~{}H=\frac{k}{\sqrt{a}}=k(1+z)^{\frac{1}{2}}$
(27)

and

$\ddot{H}+4H\dot{H}=0~{}\implies~{}\dot{H}=\frac{c_{1}}{a^{4}}=c_{1}(1+z)^{4}$
(28)

where $k$ and $c_{1}$ are arbitrary integrating constant.
Thus, there are two cases to obtain the Hubble parameter and scale-factor
which gives two different types of universe model.

###  3.1 Accelerating Model: Case-I: when $H=\frac{k}{\sqrt{a}}$

$H=\frac{k}{\sqrt{a}}~{}\implies~{}a(t)=\left(\frac{kt+l}{2}\right)^{2}$ (29)

which shows power-law cosmology and also, in terms of redshift, the Hubble
parameter $(H)$ obtained as

$H=k(1+z)^{\frac{1}{2}}$ (30)

It is known that at present $z=0$ putting in Eq. $(30)$, we get the present
value of Hubble parameter $H_{0}=k$. Recently, Maurya and Zia [29] estimated
the present value of the Hubble parameter $H_{0}=71.27$. Figure $1$ represents
the variation of Hubble parameter $H(z)$ over redshift $z$. We see that $H(z)$
is an increasing function of redshift $z$ which reveals the expanding nature
of the universe.

The deceleration parameter for the model is obtained as a constant negative
value $q=-0.5$ that shows that the accelerating scenario of the universe.

Now, Eq. $(21)$ becomes

$\phi=\phi_{0}+\frac{2n}{5k}[1-(1+z)^{\frac{5}{2}}]$ (31)

where $\phi_{0}$ is the present value of scalar field $\phi$. Figures 2(a) and
2(b) represent the variation of scalar field $\phi(z)$ versus redshift $z$.

Figure 1: The plot of Hubble parameter $H(z)$ versus redshift $z$ for
$H_{0}=71.27$.

(a) (b)

Figure 2: The plots of scalar field $\phi(z)$ versus redshift $z$ for
$H_{0}=71.27$.

Using Eqs. $(30)$ and $(31)$ in Eqs. $(22)-(25)$, we obtain the expressions
for the trace $T$, energy density $\rho$ of the fluid, and EoS parameter
$\omega$ as

$T=-9k^{2}(1+z)+\frac{9\alpha
k^{4}}{2}(1+z)^{2}\left[31+(1+z)^{\frac{1}{2}}-\frac{14k^{2}}{1+z}\right]$
(32) $\rho=3k^{2}(1+z)-\frac{9\alpha
k^{4}}{2}(1+z)(22+23z)-\frac{n^{2}}{2}(1+z)^{6}$ (33)
$\omega=-1+\frac{1-\frac{n^{2}}{k^{2}}(1+z)^{5}-\frac{3}{2}\alpha
k^{2}(1+z)\left[2(1+2z)+\frac{8}{1+z}-\frac{k}{(1+z)^{\frac{1}{2}}}+16\right]}{3-\frac{9\alpha
k^{2}}{2}(22+23z)-\frac{n^{2}}{2k^{2}}(1+z)^{5}}$ (34)

Eq. $(32)$ represents the expression for trace of the total energy momentum
tensor $T_{ij}$ and Figure $3$ represents the plot of the trace $T$ over the
variation of redshift $z$ for the different values of $\alpha$ and constrained
by the Hubble constant $H_{0}$. At present $z=0$, $T_{0}=-9k^{2}+144\alpha
k^{4}-63\alpha k^{6}$ which depends on the value of $k$ and $\alpha$.

Eq. $(33)$ represents the expression for energy density $\rho$ in terms of
redshift $z$ and Figures 4(a), 4(b), and 4(c) show the variation of energy
density $\rho(z)$ over the variation of redshift $z$ constrained by Hubble
constant $H_{0}$ for different choices of the value of $\alpha$ and $n$. At
present $z=0$, $\rho_{0}=3k^{2}(1-33\alpha k^{2})-\frac{n^{2}}{2}$ and in past
as $z$ increases, $\rho$ increases to a large value. Here, $\rho_{0}>0$ for
$3k^{2}(1-33\alpha k^{2})>\frac{n^{2}}{2}$.

Eq. $(34)$ denotes the expression for EoS parameter $\omega(z)$ and Figure
5(a), 5(b), and 5(c) represent the evolution of EoS parameter $\omega$ over
the variation of redshift $z$ constrained by $H_{0}$. One can see that at
present $z=0$,

$\omega_{0}=-1+\frac{2k^{2}-78\alpha k^{4}+3\alpha
k^{5}-2n^{2}}{6k^{2}-198\alpha k^{4}-n^{2}}$

and $\lim\limits_{z\to\infty}\omega(z)=1$ which are consistent with recent
observations [1]-[7].

Figure 3: The plots of trace $T(z)$ versus redshift $z$ for $H_{0}=71.27$.

(a) (b) (c)

Figure 4: The plots of energy density $\rho(z)$ versus redshift $z$ for
$H_{0}=71.27$.

(a) (b) (c)

Figure 5: The plots of EoS parameter $\omega(z)$ versus redshift $z$ for
$H_{0}=71.27$.

### Age of the Universe

The age of the cosmic universe is calculated as

$t_{0}-t=\int_{0}^{z}\frac{dz}{(1+z)H(z)}$ (35)
$H_{0}(t_{0}-t)=2\left[1-\frac{1}{\sqrt{1+z}}\right]$ (36)

Figure 6: The plot of cosmic time $t$ versus redshift $z$

From Figure $6$, the constant graph of time shows the age of the universe i.e.
$\lim\limits_{z\to\infty}H_{0}(t_{0}-t)=2\implies H_{0}t_{0}=2$ and hence, the
present age of the universe $t_{0}=27.44Gyrs$.

###  3.2 Transit Model: Case-II: when $\dot{H}=\frac{c_{1}}{a^{4}}$

From equations
$\dot{H}=\frac{c_{1}}{a^{4}}~{}~{}\text{and}~{}~{}\dot{H}=-(1+z)H\frac{dH}{dz}$
we have obtained the Hubble parameter $H$ as

$H=[2c_{2}-\frac{c_{1}}{2}(1+z)^{4}]^{\frac{1}{2}}$ (37)

where $c_{1}$ and $c_{2}$ are arbitrary constants. At present $z=0$, we found
the $H_{0}=[2c_{2}-\frac{c_{1}}{2}]^{\frac{1}{2}}$. Figure $7$ represents its
variation over redshift $z$.

Now, we have

$\frac{\dot{a}}{a}=\sqrt{2c_{2}-\frac{c_{1}}{2a^{4}}}$ (38)

Let us choose the arbitrary constants $c_{1},c_{2}$ as $c_{2}=-c_{1}=m_{1}$
and $c_{2}=c_{1}=m_{1}$, in these cases, integrating Eq. $(38)$, we get the
scale factor as

$a(t)=\frac{1}{\sqrt{2}}[sinh(2\sqrt{2}m_{1}t+4m_{2})]^{\frac{1}{2}}$ (39)

and

$a(t)=\frac{1}{\sqrt{2}}[cosh(2\sqrt{2}m_{1}t+4m_{2})]^{\frac{1}{2}}$ (40)

respectively. Where $m_{1},m_{2}$ are integrating constants. Here, we have
considered first case.
Now, we have calculated the deceleration parameter $q(z)$ as

$q=\frac{d}{dt}\left(\frac{1}{H}\right)-1~{}~{}\implies~{}~{}q(z)=-\frac{4c_{2}+c_{1}(1+z)^{4}}{4c_{2}-c_{1}(1+z)^{4}}$
(41)

At present $z=0$, $q_{0}=-\frac{4c_{2}+c_{1}}{4c_{2}-c_{1}}$ and in the past
as $z\to\infty$, we have $\lim\limits_{z\to\infty}q(z)=1$. Also, Figure 7(b)
represents the variation of deceleration parameter $q(z)$ over the variation
of redshift $z$. One can see that $q(z)$ is an increasing function of redshift
$z$ and at present $z=0$, $q_{0}=-0.6$ and at $z_{t}=0.414$ expansion transits
its phase from deceleration to acceleration. Thus, our universe model is a
transit phase model.

(a) (b)

Figure 7: (a)The plot of Hubble parameter $H$ versus redshift $z$ and (b) The
plot of deceleration parameter $q(z)$ versus redshift $z$ for
$c_{1}=-\frac{2}{5}H_{0}^{2},~{}c_{2}=\frac{2}{5}H_{0}^{2}$.

Now, using Eq. $(37)$ in Eqs. $(22)-(25)$, we obtain trace $T$, energy density
$\rho$ and EoS parameter $\omega$ in terms of redshift $z$ as

$T=-24c_{2}-\frac{36\alpha
c_{1}^{2}(1+z)^{8}}{[2c_{2}-\frac{c_{1}}{2}(1+z)^{4}]}[3-2c_{2}-3c_{1}(1+z)^{4}+c_{1}(1+z)^{4}[2c_{2}-\frac{c_{1}}{2}(1+z)^{4}]^{\frac{1}{2}}]$
(42) $\rho=6c_{2}-\frac{3c_{1}}{2}(1+z)^{4}+36\alpha
c_{1}(1+z)^{4}[2c_{2}+c_{1}(1+z)^{3}-2c_{1}(1+z)^{4}]-\frac{n^{2}}{2}(1+z)^{6}$
(43) $\omega=-1+\frac{L(z)}{\rho(z)}$ (44)

where

$L(z)=-2c_{1}(1+z)^{4}+\frac{12\alpha
c_{1}(1+z)^{3}}{[2c_{2}-\frac{c_{1}}{2}(1+z)^{4}]}\times$
$[4c_{2}(5-6c_{2})(1+z)+24c_{1}c_{2}(1+z)^{5}-2c_{1}^{2}(1+z)^{8}+\frac{3c_{1}^{2}}{2}(1+z)^{9}+[30c_{2}+\frac{5}{2}c_{1}(1+z)^{4}-c_{1}^{2}(1+z)^{9}][2c_{2}-\frac{c_{1}}{2}(1+z)^{4}]^{\frac{1}{2}}]-n^{2}(1+z)^{6}$
(45)

Eq. $(42)$ represents the expression for trace $T(z)$ of the total energy
momentum tensor $T_{ij}$ and Figure $8$ represents the plot of the trace $T$
over the variation of redshift $z$ for the different values of $\alpha$ and
constrained by the Hubble constant $H_{0}$. At present $z=0$,
$T_{0}=-24c_{2}-\frac{72\alpha
c_{1}^{2}}{4c_{2}-c_{1}}[3(1-c_{1})-2c_{2}+c_{1}(2c_{2}-\frac{c_{1}}{2})^{\frac{1}{2}}]$
which depends on the value of $c_{1},c_{2}$ and $\alpha$.

Eq. $(43)$ represents the expression for energy density $\rho(z)$ in terms of
redshift $z$ and Figures 9(a), 9(b), and 9(c) show the variation of energy
density $\rho(z)$ over the variation of redshift $z$ constrained by Hubble
constant $H_{0}$ for different choices of the value of $\alpha$ and $n$. At
present $z=0$, $\rho_{0}=6c_{2}-\frac{3}{2}c_{1}+36\alpha
c_{1}(2c_{2}-c_{1})-\frac{n^{2}}{2}$ and in past as $z$ increases, $\rho$
increases to a large value.

Eq. $(44)$ denotes the expression for EoS parameter $\omega(z)$ and Figures
10(a), 10(b), and 10(c) represent the evolution of EoS parameter $\omega$ over
the variation of redshift $z$ constrained by $H_{0}$. One can see that at
present $z=0$,

$\omega_{0}=-1+\frac{-2c_{1}+\frac{24\alpha
c_{1}}{4c_{2}-c_{1}}[4c_{2}(5-6c_{2})+24c_{1}c_{2}-\frac{1}{2}c_{1}^{2}+(30c_{2}+\frac{5}{2}c_{1}-c_{1}^{2})(2c_{2}-\frac{1}{2}c_{1})^{\frac{1}{2}}]-n^{2}}{6c_{2}-\frac{3}{2}c_{1}+36\alpha
c_{1}(2c_{2}-c_{1})-\frac{n^{2}}{2}}$

and $\lim\limits_{z\to\infty}\omega(z)=\infty$ which are consistent with
recent observations [1]-[7].

Figure 8: The plots of trace $T(z)$ versus redshift $z$ for $H_{0}=71.27$.

(a) (b) (c)

Figure 9: The plots of energy density $\rho(z)$ versus redshift $z$ for
$H_{0}=71.27$.

(a) (b) (c)

Figure 10: The plots of EoS parameter $\omega(z)$ versus redshift $z$ for
$H_{0}=71.27$. Figure 11: The plot of cosmic time $t$ versus redshift $z$

### Age of the Universe

The age of the cosmic universe is calculated as

$t_{0}-t=\int_{0}^{z}\frac{dz}{(1+z)H(z)}$ (46)
$(t_{0}-t)=\frac{1}{4}\sqrt{\frac{2}{c_{2}}}\left[tanh^{-1}\left(\frac{4c_{2}-c_{1}}{4c_{2}}\right)^{\frac{1}{2}}-tanh^{-1}\left(\frac{4c_{2}-c_{1}(1+z)^{4}}{4c_{2}}\right)^{\frac{1}{2}}\right]$
(47)

From Figure $11$, the constant graph of time shows the age of the universe
i.e. $\lim\limits_{z\to\infty}H_{0}(t_{0}-t)=0.8233\implies H_{0}t_{0}=0.8233$
and hence, the present age of the universe $t_{0}=11.3Gyrs$.

##  4 Concluding Remarks

In this paper, we have investigated a flat FRW cosmological model filled with
perfect fluid coupled with the zero-mass scalar field in the higher derivative
theory of gravity. We have obtained two types universe models, first one is
accelerating universe (power-law cosmology) and second one is transit phase
model (hyperbolic expansion-law). The main features of the models are as:

  * •

The scale-factor $a(t)$ is obtained from the field equations without
assumptions.

  * •

These models evolves with high redshift values $0\leq z<\infty$.

  * •

For the transit phase model the deceleration parameter shows signature-
flipping with transition value $z_{t}=0.414$ which is consistent with recent
observations. The present value deceleration parameter is obtained as
$q_{0}=-0.6$ for transit model and $q_{0}=-0.5$ for accelerating universe.

  * •

The EoS parameter $\omega$ varies from positive to negative values with the
evolution the universe, which reveals the formation of different structures
during the evolution of the universe.

  * •

At $t=0$, the scale factor $a(t)$ has a finite value.

  * •

The present age of the universe for transit phase model is obtained as
$11.3Gyrs$.

## Acknowledgment

One of the authors (DM) is thankful to IASE (Deemed to be University),
Sardarshahar, Rajsthan, India for providing facilities and support where part
of this work is carried out.

## References

  * [1] P. M. Garnavich et al., Constraints on cosmological models from Hubble space telescope observations of high Z supernovae, Astrophys. J. 493 (1998) L53-L57.
  * [2] P. M. Garnavich et al., Supernova limits on the cosmic equation of state, Astrophys. J. 509 (1998) 74-79.
  * [3] S. Perlmutter et al., Measurements of the cosmological parameters Omega and Lambda from the first 7 supernovae at $z\geq 0.35$, Astrophys. J. 483 (1997) 565-581.
  * [4] S. Perlmutter et al., Discovery of a supernova explosion at half the age of the universe and its cosmological implications, Nature 391 (1998) 51-54.
  * [5] S. Perlmutter et al., Measurements of omega and lambda from 42 high-redshift supernovae, Astrophys. J. 517 (1999) 565-586.
  * [6] A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009-1038.
  * [7] B. P. Schmidt et al., The high-Z supernova search: Measuring cosmic deceleration and global curvature of the universe using type IA supernovae, Astrophys. J. 507 (1998) 46-63.
  * [8] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B, 91 (1980) 99-102.
  * [9] A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23 (1981) 347-356.
  * [10] K. S. Stelle, Renormalization of higher-derivative quantum gravity, Phys. Rev. D 16 (1977) 953-969.
  * [11] A. D. Linde, Particle Physics and Cosmology. Gordon and Breach, New York (1990).
  * [12] E. Kolb and M. S. Turner, The early Universe. Addison-Wesley, New York (1990).
  * [13] L. P. Chimento and A. S. Jakubi, Exact Solutions and Scalar Fields in Gravity, Int. J. Mod. Phys. D 5 (1996) 71.
  * [14] I. Muller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys. 198 (1967) 329-344.
  * [15] W. A. Hiscock and J. Salmonson, Dissipative Boltzmann-Robertson-Walker cosmologies, Phys. Rev. D 43 (1991) 3249-3258.
  * [16] W. Israel and J. M. Stewart, Nonstationary irreversible thermodynamics: A causal relativistic theory, Ann. Phys. (N.Y.) 100 (1976) 310-331.
  * [17] D. Pavon, D. Jou and J. Casas-Vazquez, On a covariant formulation of dissipative phenomena, Ann. Inst. Henri Poincare, A. Phys. Theor. 36 (1982) 79-88.
  * [18] D. Pavon, Nonequilibrium fluctuations in cosmic vacuum decay, Phys. Rev. D 43 (1991) 375-378.
  * [19] B. Ratra and P. J. E. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D 37 (1988) 3406-3427.
  * [20] D. Pavon and I. W. Zimdahl, Dark matter and dissipation, Phys. Lett. A. 179 (1993) 261-265.
  * [21] I. W. Zimdahl et al., Cosmic antifriction and accelerated expansion, Phys. Rev. D 64 (2001) 063501.
  * [22] B. Ram, Infinitely many solutions of Einstein cosmology in “slow-roll”, Phys. Lett. A 172 (1993) 404-406.
  * [23] G. F. R. Ellis and M. S. Madsen, Exact scalar field cosmologies, Class. Quantum Gravity 8 (1991) 667-676.
  * [24] J. D. Barrow and P. Saich, Scalar-field cosmologies, Class. Quantum Gravity 10 (1993) 279-283.
  * [25] J. D. Barrow and J. P. Mimoso, Perfect fluid scalar-tensor cosmologies, Phys. Rev. D 50 (1994) 3746-3754.
  * [26] J. P. Mimoso and D. Wands, Massless fields in scalar-tensor cosmologies, Phys. Rev. D 51 (1995) 477-489.
  * [27] R. Casana, C. A. M. de Melo and B. M. Pimentel, Massless DKP field in a Lyra manifold, Class. Quantum Gravity 24 (2007) 723-736.
  * [28] N. I. Singh, S. S. Singh and S. R. Devi, Casual viscous universe coupled with zero-mass scalar field in higher derivative theory, Astrophys. Space Sci. 334 (2011) 187-191.
  * [29] D. C. Maurya and R. Zia, Brans-Dicke Scalar Field Cosmological Model in Lyra’s Geometry, Phys. Rev. D 100 (2019) 023503.
  * [30] A. R. Roy and M. N. Verma, R-W cosmological models with zero-mass scalar fields for different equation of state, Astrophys. Space Sci. 138 (1987) 217-219.
  * [31] D. R. K. Reddy, R. Venkateswarlu, Bianchi type-I universe in the presence of zero-mass scalar fields, Astrophys. Space Sci. 136 (1987) 17-20.
  * [32] A. Pradhan, K. L. Tiwari and A. Beesham, LRS Bianchi I cosmological models in the presence of zero-mass scalar field, Indian J. pure appl. Math. 32 (2001) 789-795.
  * [33] R. Venkateswarlu and K. P. Kumar, Plane symmetric string cosmological models in zero-mass scalar field, Int. J. Theor. Phys. 49 (2010) 1894-1898.


# Annihilators of local cohomology modules via a classification theorem of the
dominant resolving subcategories

Takeshi Yoshizawa  National Institute of Technology, Toyota College, 2-1
Eiseicho, Toyota, Aichi, Japan, 471-8525<EMAIL_ADDRESS>
###### Abstract.

This paper investigates when local cohomology modules have an annihilator that
does not depend on the choice of an ideal. Takahashi classified the dominant
resolving subcategories of the category of finitely generated modules over a
commutative Noetherian ring. We show that his classification theorem describes
annihilation results of local cohomology modules over a finite-dimensional
ring with certain assumptions or a Cohen-Macaulay ring.

###### Key words and phrases:

Local cohomology module, resolving subcategory

###### 2010 Mathematics Subject Classification:

13C60, 13D45

##  1\. Introduction

Throughout this paper, let $R$ be a commutative Noetherian ring.

The vanishing of local cohomology modules has been widely studied in local
cohomology theory. Grothendieck’s vanishing theorem [1, Theorem 6.1.2] and the
Lichtenbaum-Hartshorne vanishing theorem [1, Theorem 8.2.1] are perfect
examples of indispensable vanishing results.

Vanishing phenomena of local cohomology modules can also be seen with the help
of other factors. In one example, Faltings’ annihilator theorem [5] states
that some power of the second ideal annihilates local cohomology modules over
a homomorphic image of a regular ring; see also [1, Theorem 9.5.1]. In other
examples, Raghavan established the uniform annihilation theorem over a
homomorphic image $R$ of a biequidimensional regular ring with a finite
dimension; see [7, Theorem 3.1]. His theorem guarantees that there exists an
integer $n$ (depending only on a finitely generated $R$-module $M$) such that
$\mathfrak{b}^{n}H^{i}_{\mathfrak{a}}(M)=0$ for all ideals $\mathfrak{a}$,
$\mathfrak{b}$ of $R$ and all integers
$i<\lambda^{\mathfrak{b}}_{\mathfrak{a}}(M)=\inf\left\\{0pt\,M_{\mathfrak{p}}+0pt(\mathfrak{a}+\mathfrak{p})/\mathfrak{p}\mid\mathfrak{p}\in\mathrm{Spec}(R)\setminus
V(\mathfrak{b})\right\\}$. In particular, the ideal $\mathfrak{b}^{n}$ has
some annihilator $s$ such that $sH_{\mathfrak{a}}^{i}(M)=0$ for all ideals
$\mathfrak{a}$ of $R$ and all integers
$i<\lambda^{\mathfrak{b}}_{\mathfrak{a}}(M)$.

In 2006, Zhou showed that, if a locally equidimensional ring $R$ of finite
positive dimension is a homomorphic image of a Cohen-Macaulay ring of finite
dimension or an excellent local ring, then $R$ has a uniform local
cohomological annihilator; see [10, Corollary 3.3]. A uniform local
cohomological annihilator is an element $s\in
R\setminus(\cup_{\mathfrak{p}\in\mathrm{Min}(R)}\,\mathfrak{p})$ such that,
for each maximal ideal $\mathfrak{m}$ of $R$, one has
$sH^{i}_{\mathfrak{m}}(R)=0$ for all integers $i<0pt\,\mathfrak{m}$. It should
be noted that these annihilators do not depend on the choice of the maximal
ideal.

The purpose of this paper is to study vanishing phenomena of local cohomology
modules. We start by focusing on [1, Lemma 9.4.3], which relates the difficult
part of a proof of Faltings’ annihilator theorem; see [1, Chapter 9.4]. This
lemma is concerned with the fact that, if $M_{\mathfrak{p}}$ is a non-zero
free $R_{\mathfrak{p}}$-module for a finitely generated $R$-module $M$ and
$\mathfrak{p}\in\mathrm{Spec}(R)$, then there exists an element $s\in
R\setminus\mathfrak{p}$ such that $sH^{i}_{\mathfrak{a}}(M)=0$ for all ideals
$\mathfrak{a}$ of $R$ and all integers $i<\mathrm{grade}(\mathfrak{a},R)$.

In this regard, the paper deals with the following question as to when a
similar vanishing phenomenon occurs. It goes without saying that the
interesting case is when a prime ideal $\mathfrak{p}$ belongs to
$\mathrm{Supp}_{R}(M)$; see also [1, Lemma 9.4.1].

######  Question.

Let $M$ be a finitely generated $R$-module and let
$\mathfrak{p}\in\mathrm{Spec}(R)$. When does there exist an element $s\in
R\setminus\mathfrak{p}$ such that $sH^{i}_{\mathfrak{a}}(M)=0$ for all ideals
$\mathfrak{a}$ of $R$ and all integers $i<\mathrm{grade}(\mathfrak{a},R)$?

Our strategy stems from Dao and Takahashi’s classification theorem of the
dominant resolving subcategories of the category of finitely generated modules
over a Cohen-Macaulay ring; see [4, Theorem 1.4]. Furthermore, it should be
noted that Takahashi recently classified such subcategories without the Cohen-
Macaulayness of rings; see [8, Theorem 5.4]. These theorems suggest that
dominant resolving subcategories have a close relationship with the existence
of annihilators of local cohomology modules. Indeed, using [4, Theorem 1.4],
we provided an answer to the above question in a finite-dimensional Cohen-
Macaulay ring; see [9, Theorem 4.3].

One of our aims is to establish the following theorem, which shows that [8,
Theorem 5.4] removes the assumption of finite dimension from [9, Theorem 4.3].

######  Theorem 1.1.

Suppose that $R$ is a Cohen-Macaulay ring, and let
$\mathfrak{p}\in\mathrm{Spec}(R)$. Then, for a finitely generated $R$-module
$M$, the following conditions are equivalent.

  1. (1)

There exists an element $s\in R\setminus\mathfrak{p}$ such that
$sH_{\mathfrak{a}}^{i}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i<\mathrm{grade}(\mathfrak{a},R)$.

  2. (2)

The $R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is a maximal Cohen-Macaulay
$R_{\mathfrak{p}}$-module.

Theorem 1.1 leads to the following corollary: the Cohen-Macaulay dimension of
a finitely generated module influences whether local cohomology modules have
an annihilator that does not depend on the choice of the ideal.

######  Corollary 1.2.

Suppose that $R$ is a Cohen-Macaulay ring, and let
$\mathfrak{p}\in\mathrm{Spec}(R)$. Then, for a finitely generated $R$-module
$M$, there exists an element $s\in R\setminus\mathfrak{p}$ such that

$sH^{i}_{\mathfrak{a}}(M)=0$

for all ideals $\mathfrak{a}$ of $R$ and all integers
$i<\mathrm{grade}(\mathfrak{a},R)-\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}$.

Another purpose is to investigate the above question under (not necessarily
Cohen-Macaulay) finite-dimensional rings. The following theorem is one of the
main results, which provides an answer to our question under some assumptions.

######  Theorem 1.3.

Suppose that $R$ is a finite-dimensional ring, and let $\mathfrak{p}$ be a
prime ideal of $R$ with $0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$
for all $\mathfrak{q}\in
U(\mathfrak{p})=\\{\mathfrak{q}\in\mathrm{Spec}(R)\mid\mathfrak{q}\subseteqq\mathfrak{p}\\}$.
Then, for a finitely generated $R$-module $M$, the following conditions are
equivalent.

  1. (1)

There exists an element $s\in R\setminus\mathfrak{p}$ such that
$sH_{\mathfrak{a}}^{i}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i<\mathrm{grade}(\mathfrak{a},R)$.

  2. (2)

One has $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\leqq 0$.

The above assumption of prime ideals is not an unacceptable condition. Indeed,
our assumption is very similar to the Cohen-Macaulayness of rings: [3, Lemma
3.1] states that one has
$0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$ for all
$\mathfrak{q}\in\mathrm{Spec}(R)$ if and only if $R$ is an almost Cohen-
Macaulay ring; for details, see also Remark 4.2 (2).

Finally, we can investigate all finitely generated modules by combining
Theorem 1.3 with the following result.

######  Corollary 1.4.

Let $R$ and $\mathfrak{p}$ be as in Theorem 1.3, and let $M$ be a finitely
generated $R$-module with $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\geqq 0$.
Then there exists an element $s\in R\setminus\mathfrak{p}$ such that

$sH^{i}_{\mathfrak{a}}(M)=0$

for all ideals $\mathfrak{a}$ of $R$ and all integers
$i<\mathrm{grade}(\mathfrak{a},R)-\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}$.

The organization of this paper is as follows. We dedicate section 2 to the
preparation: notations, definitions, and fundamental facts. Section 3 provides
the essential result in this paper; see Theorem 3.4. This result describes
that Takahashi’s classification theorem [8, Theorem 5.4] provides a necessary
and sufficient condition that local cohomology modules have an annihilator
that does not depend on the choice of the ideal. After showing Theorem 1.3 and
Corollary 1.4 in section 4, we establish Theorem 1.1 and Corollary 1.2 in
section 5.

##  2\. Preliminaries

Throughout this paper, let $R$ be a commutative Noetherian ring, and all
modules are assumed to be unitary. We denote by $R\text{-}\mathrm{mod}$ the
category of finitely generated $R$-modules. We suppose that all subcategories
of $R\text{-}\mathrm{mod}$ are full and closed under taking isomorphisms. The
symbol $\mathbb{N}_{0}$ denotes the set of non-negative integers. We adopt the
convention that the grade, the depth, and the dimension of the zero module are
$\infty$.

First of all, we will recall a set of $\mathbb{N}_{0}$-valued functions on
$\mathrm{Spec}(R)$ that was introduced by Takahashi in [8, Definition 5.1
(1)]. We note that the symbol $\mathbb{N}$ in the paper [8] stands for the set
of non-negative integers.

######  Definition 2.1.

Let $\mathcal{C}$ be a subcategory of $R\text{-}\mathrm{mod}$. We denote by
$\mathbb{F}(\mathcal{C})$ the set of maps
$f\colon\mathrm{Spec}(R)\to\mathbb{N}_{0}$ such that, for every
$\mathfrak{p}\in\mathrm{Spec}(R)$, there exists $E\in\mathcal{C}$ satisfying
the following two conditions:

  1. (1)

One has $0pt\,R_{\mathfrak{p}}-0pt\,E_{\mathfrak{p}}=f(\mathfrak{p})$;

  2. (2)

One has $0pt\,R_{\mathfrak{q}}-0pt\,E_{\mathfrak{q}}\leqq f(\mathfrak{q})$ for
all $\mathfrak{q}\in\mathrm{Spec}(R)$.

Next, we need to recall the notions of a dominant subcategory and a resolving
subcategory of $R\text{-}\mathrm{mod}$. We denote by $\Omega_{R}^{n}M$ the
$n$th syzygy module of an $R$-module $M$. In particular, we will write
$\Omega_{R}^{1}M=\Omega_{R}M$, which is the kernel of some epimorphism from a
projective module to $M$. Note that $\Omega_{R}^{n}M$ is uniquely determined
up to projective summands.

######  Definition 2.2.

Let $\mathcal{X}$ be a subcategory of $R\text{-}\mathrm{mod}$.

  1. (1)

We denote by $\mathrm{add}\,\mathcal{X}$ the subcategory of
$R\text{-}\mathrm{mod}$ consisting of the modules isomorphic to direct
summands of finite direct sums of copies of modules in $\mathcal{X}$.

  2. (2)

We say that $\mathcal{X}$ is dominant if, for each
$\mathfrak{p}\in\mathrm{Spec}(R)$, there exists a non-negative integer $n$
such that
$\Omega_{R_{\mathfrak{p}}}^{n}\kappa(\mathfrak{p})\in\mathrm{add}\,\mathcal{X}_{\mathfrak{p}}$,
where $\kappa(\mathfrak{p})=R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}$ and
$\mathcal{X}_{\mathfrak{p}}=\\{X_{\mathfrak{p}}\in
R_{\mathfrak{p}}\text{-}\mathrm{mod}\mid X\in\mathcal{X}\\}$.

  3. (3)

We say that $\mathcal{X}$ is a resolving subcategory if $\mathcal{X}$ contains
all finitely generated projective modules and is closed under taking direct
summands, extensions, and syzygies.

Takahashi showed that the set $\mathbb{F}(R\text{-}\mathrm{mod})$ classifies
the dominant resolving subcategories of $R\text{-}\mathrm{mod}$; see [8,
Definition 5.1 and Theorem 5.4].

######  Theorem 2.3.

(Takahashi). There exist mutually inverse order-preserving bijections

$\left\\{\begin{matrix}\text{ dominant resolving subcategories of
}R\text{-}\mathrm{mod}\
\end{matrix}\right\\}\begin{matrix}\overset{\phi}{\longrightarrow}\cr\underset{\psi}{\longleftarrow}\cr\end{matrix}\
\mathbb{F}(R\text{-}\mathrm{mod})$

where the maps $\phi$ and $\psi$ are given by

$\displaystyle\phi(\mathcal{X})(\mathfrak{p})=0pt\,R_{\mathfrak{p}}-\inf_{X\in\mathcal{X}}\\{0pt\,X_{\mathfrak{p}}\\}\
\text{ for }\mathfrak{p}\in\mathrm{Spec}(R),\text{ and }$
$\displaystyle\psi(f)=\left\\{X\in R\text{-}\mathrm{mod}\mid
0pt\,R_{\mathfrak{p}}-0pt\,X_{\mathfrak{p}}\leqq f(\mathfrak{p})\text{ for all
}\mathfrak{p}\in\mathrm{Spec}(R)\right\\}.$

Finally, we will give the notion of a subcategory associated with local
cohomology modules and summarize some properties. We adopt the convention that
the local cohomology functor $H^{i}_{\mathfrak{a}}(-)$ is the zero functor for
all ideals $\mathfrak{a}$ of $R$ and all negative integers $i$.

######  Definition 2.4.

  1. (1)

We define the subcategory $\mathcal{R}(\mathfrak{p})$ of
$R\text{-}\mathrm{mod}$ relative to a prime ideal $\mathfrak{p}$ of $R$ by

$\mathcal{R}(\mathfrak{p})=\left\\{M\in
R\text{-}\mathrm{mod}\mathrel{}\middle|\mathrel{}\begin{matrix}\text{There
exists }s\in R\setminus\mathfrak{p}\text{ such that
}sH^{i}_{\mathfrak{a}}(M)=0\cr\text{ for all ideals }\mathfrak{a}\text{ of
}R\text{ and all integers
}i<\mathrm{grade}(\mathfrak{a},R)\end{matrix}\right\\}.$

  2. (2)

We denote by $U(\mathfrak{p})$ the generalization closed subset
$\\{\mathfrak{q}\in\mathrm{Spec}(R)\mid\mathfrak{q}\subseteqq\mathfrak{p}\\}$
of $\mathrm{Spec}(R)$ for $\mathfrak{p}\in\mathrm{Spec}(R)$.

A finitely generated $R$-module $M$ over an arbitrary ring $R$ is called a
maximal Cohen-Macaulay module if $0pt\,M_{\mathfrak{p}}\geqq\dim
R_{\mathfrak{p}}$ for all $\mathfrak{p}\in\mathrm{Spec}(R)$. We note that the
depth of the zero module is $\infty$, and thus we consider the zero module to
be maximal Cohen-Macaulay.

######  Proposition 2.5.

Let $\mathfrak{p}$ be a prime ideal of $R$.

  1. (1)

For each $\mathfrak{q}\in U(\mathfrak{p})$, one has
$\mathcal{R}(\mathfrak{p})\subseteqq\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,M_{\mathfrak{q}}\geqq\mathrm{grade}(\mathfrak{q},R)\\}$.

  2. (2)

For each $\mathfrak{q}\in\mathrm{Spec}(R)\setminus U(\mathfrak{p})$, the
$R$-module $R/\mathfrak{q}$ is in $\mathcal{R}(\mathfrak{p})$.

  3. (3)

The subcategory $\mathcal{R}(\mathfrak{p})$ is a resolving subcategory of
$R\text{-}\mathrm{mod}$.

  4. (4)

We suppose that $R$ is a finite-dimensional ring. Then the subcategory
$\mathcal{R}(\mathfrak{p})$ is dominant.

  5. (5)

We suppose that $R$ is a Cohen-Macaulay ring. Then the subcategory
$\mathcal{R}(\mathfrak{p})$ contains all maximal Cohen-Macaulay $R$-modules.
In particular, the subcategory $\mathcal{R}(\mathfrak{p})$ is dominant.

###### Proof.

(1) Let $\mathfrak{q}\in U(\mathfrak{p})$. We suppose that a finitely
generated $R$-module $M$ is in the subcategory $\mathcal{R}(\mathfrak{p})$.
The definition of $\mathcal{R}(\mathfrak{p})$ gives an element $s\in
R\setminus\mathfrak{p}$ such that $sH^{i}_{\mathfrak{q}}(M)=0$ for all
integers $i<\mathrm{grade}(\mathfrak{q},R)$. The flat base change theorem [1,
Theorem 4.3.2] yields
$(s/1)H^{i}_{\mathfrak{q}R_{\mathfrak{q}}}(M_{\mathfrak{q}})=0$ where $s/1$ is
an element of $R_{\mathfrak{q}}$. Since the assumption of $\mathfrak{q}\in
U(\mathfrak{p})$ implies that $s\in R\setminus\mathfrak{q}$, the element $s/1$
is a unit of $R_{\mathfrak{q}}$. Therefore, we obtain
$H^{i}_{\mathfrak{q}R_{\mathfrak{q}}}(M_{\mathfrak{q}})=0$ for all integers
$i<\mathrm{grade}(\mathfrak{q},R)$.

When we suppose that
$\mathfrak{q}R_{\mathfrak{q}}M_{\mathfrak{q}}=M_{\mathfrak{q}}$, Nakayama’s
lemma says that $M_{\mathfrak{q}}=0$. Thus, we have
$0pt\,M_{\mathfrak{q}}=\infty>\mathrm{grade}(\mathfrak{q},R)$. On the other
hand, if we suppose that $\mathfrak{q}R_{\mathfrak{q}}M_{\mathfrak{q}}\neq
M_{\mathfrak{q}}$, then [1, Theorem 6.2.7] deduces
$0pt\,M_{\mathfrak{q}}\geqq\mathrm{grade}(\mathfrak{q},R)$.

(2) Our statement is proved by the same argument in the proof of [9,
Proposition 4.2] without the assumption of the Cohen-Macaulayness for the ring
$R$.

(3) We have already shown in [9, Proposition 3.6 (1)].

(4) Our assertion follows from [9, Proposition 3.6 (2)].

(5) The former assertion follows from [9, Remark 4.4 (2)]. Furthermore, since
$R$ is a Cohen-Macaulay ring, we deduce from [8, Corollary 4.7] that the
resolving subcategory $\mathcal{R}(\mathfrak{p})$ is dominant. (We note that
the assertion of [8, Corollary 4.7] has been proved without assuming that the
base ring has a finite dimension.) ∎

##  3\. Conditions for the existence of an annihilator of local cohomology
modules

[8, Theorem 5.4] implies that, if the resolving subcategory
$\mathcal{R}(\mathfrak{p})$ for $\mathfrak{p}\in\mathrm{Spec}(R)$ is dominant,
then there exists the map
$f_{\mathcal{R}(\mathfrak{p})}\in\mathbb{F}(R\text{-}\mathrm{mod})$
corresponding to $\mathcal{R}(\mathfrak{p})$. In this section, we will
investigate the relationship between values of the map
$f_{\mathcal{R}(\mathfrak{p})}$ and the existence of annihilators of local
cohomology modules.

First of all, in the case when the resolving subcategory
$\mathcal{R}(\mathfrak{p})$ is dominant, the properties in Proposition 2.5 (1)
and (2) determine a range of value of the map $f_{\mathcal{R}(\mathfrak{p})}$.

######  Lemma 3.1.

Let $\mathfrak{q}$ be a prime ideal of $R$. Suppose that a dominant resolving
subcategory $\mathcal{X}$ of $R\text{-}\mathrm{mod}$ corresponds to a map
$f\in\mathbb{F}(R\text{-}\mathrm{mod})$ that is given by Theorem 2.3. Then the
following assertions hold.

  1. (1)

Let $n(\mathfrak{q})$ be an integer with $0\leqq n(\mathfrak{q})\leqq
0pt\,R_{\mathfrak{q}}$. If $\mathcal{X}$ is contained in $\\{M\in
R\text{-}\mathrm{mod}\mid 0pt\,M_{\mathfrak{q}}\geqq n(\mathfrak{q})\\}$, then
one has the inequalities $0\leqq f(\mathfrak{q})\leqq
0pt\,R_{\mathfrak{q}}-n(\mathfrak{q})$.

  2. (2)

If the $R$-module $R/\mathfrak{q}$ is in $\mathcal{X}$, then one has the
equality $f(\mathfrak{q})=0pt\,R_{\mathfrak{q}}$.

###### Proof.

[8, Theorem 5.4] states that the subcategory $\mathcal{X}$ can be described as
follows:

$None$ $\mathcal{X}=\left\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,R_{\mathfrak{p}}-0pt\,M_{\mathfrak{p}}\leqq f(\mathfrak{p})\text{ for all
}\mathfrak{p}\in\mathrm{Spec}(R)\right\\}.$

(1) We note that the map $f\in\mathbb{F}(R\text{-}\mathrm{mod})$ is an
$\mathbb{N}_{0}$-valued map. Therefore, we need to establish the inequality
$f(\mathfrak{q})\leqq 0pt\,R_{\mathfrak{q}}-n(\mathfrak{q})$.

By the definition of $\mathbb{F}(R\text{-}\mathrm{mod})$, for the prime ideal
$\mathfrak{q}$, there exists a finitely generated $R$-module $E(\mathfrak{q})$
satisfying

$0pt\,R_{\mathfrak{q}}-0pt\,E(\mathfrak{q})_{\mathfrak{q}}=f(\mathfrak{q}),\text{
and }$ $0pt\,R_{\mathfrak{p}}-0pt\,E(\mathfrak{q})_{\mathfrak{p}}\leqq
f(\mathfrak{p})\text{ for all }\mathfrak{p}\in\mathrm{Spec}(R).$

The above equality $(*)$ implies that the module $E(\mathfrak{q})$ is in the
subcategory $\mathcal{X}$. Therefore, our assumption yields the inequality
$0pt\,E(\mathfrak{q})_{\mathfrak{q}}\geqq n(\mathfrak{q})$. Consequently, we
achieve the equality and the inequality

$f(\mathfrak{q})=0pt\,R_{\mathfrak{q}}-0pt\,E(\mathfrak{q})_{\mathfrak{q}}\leqq
0pt\,R_{\mathfrak{q}}-n(\mathfrak{q}).$

(2) Since our assumption says that the module $R/\mathfrak{q}$ is in the
subcategory $\mathcal{X}$, the above equality $(*)$ yields
$0pt\,R_{\mathfrak{q}}-0pt\,(R/\mathfrak{q})_{\mathfrak{q}}\leqq
f(\mathfrak{q})$. On the other hand, the definition of
$\mathbb{F}(R\text{-}\mathrm{mod})$ gives the equality
$0pt\,R_{\mathfrak{q}}-0pt\,E(\mathfrak{q})_{\mathfrak{q}}=f(\mathfrak{q})$
for some finitely generated $R$-module $E(\mathfrak{q})$. Consequently, we
have

$\displaystyle
0pt\,R_{\mathfrak{q}}=0pt\,R_{\mathfrak{q}}-0pt\,\kappa\left(\mathfrak{q}\right)\leqq
f(\mathfrak{q})\leqq 0pt\,R_{\mathfrak{q}},$

that is the equality $f(\mathfrak{q})=0pt\,R_{\mathfrak{q}}$. ∎

######  Remark 3.2.

Let $\mathfrak{q}$ be a prime ideal of $R$.

(1) We note that each map $f\in\mathbb{F}(R\text{-}\mathrm{mod})$ satisfies
inequalities $0\leqq f(\mathfrak{p})\leqq 0pt\,R_{\mathfrak{p}}$ for all
$\mathfrak{p}\in\mathrm{Spec}(R)$. Considering this fact, the assumption of
inequalities $0\leqq n(\mathfrak{q})\leqq 0pt\,R_{\mathfrak{q}}$ is
appropriate to investigate the assertion (1) in Lemma 3.1.

(2) [2, Proposition 1.2.10 (a)] states that one has

$0\leqq\mathrm{grade}(\mathfrak{q},R)=\inf\\{0pt\,R_{\mathfrak{p}}\mid\mathfrak{p}\in
V(\mathfrak{q})\\}\leqq 0pt\,R_{\mathfrak{q}}.$

Next, Lemma 3.1 (2) makes the equality (*) in the proof for Lemma 3.1 a
slightly simpler form.

######  Lemma 3.3.

We suppose that a resolving subcategory $\mathcal{X}$ of
$R\text{-}\mathrm{mod}$ is dominant. By Theorem 2.3, the subcategory
$\mathcal{X}$ corresponds to a map $f\in\mathbb{F}(R\text{-}\mathrm{mod})$.
Let $S$ be a subset of the set $\\{\mathfrak{q}\in\mathrm{Spec}(R)\mid
R/\mathfrak{q}\in\mathcal{X}\\}$. Then one has the following equality of
subcategories of $R\text{-}\mathrm{mod}$:

$\mathcal{X}=\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq f(\mathfrak{q})\text{ for all
}\mathfrak{q}\in\mathrm{Spec}(R)\setminus S\\}.$

###### Proof.

Let $\mathfrak{q}$ be a prime ideal in the set $S$, and let $M$ be a finitely
generated $R$-module. Since the $R$-module $R/\mathfrak{q}$ is in the
subcategory $\mathcal{X}$, Lemma 3.1 (2) implies that

$0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq
0pt\,R_{\mathfrak{q}}=f(\mathfrak{q}).$

Consequently, we can establish the following equalities of subcategories of
$R\text{-}\mathrm{mod}$

$\displaystyle\mathcal{X}$ $\displaystyle=\left\\{M\in
R\text{-}\mathrm{mod}\mid 0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq
f(\mathfrak{q})\text{ for all }\mathfrak{q}\in\mathrm{Spec}(R)\right\\}$
$\displaystyle=\left\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq f(\mathfrak{q})\text{ for all
}\mathfrak{q}\in\mathrm{Spec}(R)\setminus S\right\\}$

by [8, Theorem 5.4]. ∎

Let $\mathfrak{p}$ be a prime ideal of $R$. According to Proposition 2.5 (2),
the subcategory $\mathcal{R}(\mathfrak{p})$ has the relation
$\mathrm{Spec}(R)\setminus
U(\mathfrak{p})\subseteqq\\{\mathfrak{q}\in\mathrm{Spec}(R)\mid
R/\mathfrak{q}\in\mathcal{R}(\mathfrak{p})\\}$. Consequently, as an immediate
consequence of Lemma 3.3, we now present the essential result in this paper.
The following theorem provides a necessary and sufficient condition that local
cohomology modules have a dominant annihilator, in the sense that this
annihilator does not depend on the choice of the ideal for local cohomology
modules.

######  Theorem 3.4.

Let $\mathfrak{p}$ be a prime ideal of $R$. Suppose that the resolving
subcategory $\mathcal{R}(\mathfrak{p})$ is dominant, and also that
$\mathcal{R}(\mathfrak{p})$ corresponds to the map
$f_{\mathcal{R}(\mathfrak{p})}\in\mathbb{F}(R\text{-}\mathrm{mod})$ that is
given by Theorem 2.3. Then, for a finitely generated $R$-module $M$, the
following conditions are equivalent.

  1. (1)

There exists an element $s\in R\setminus\mathfrak{p}$ such that
$sH_{\mathfrak{a}}^{i}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i<\mathrm{grade}(\mathfrak{a},R)$.

  2. (2)

One has $0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq
f_{\mathcal{R}(\mathfrak{p})}(\mathfrak{q})$ for all $\mathfrak{q}\in
U(\mathfrak{p})$.

Theorem 3.4 reveals that Takahashi’s classification theorem [8, Theorem 5.4]
suggests the following conclusion: the questions below have a closed
relationship with the existence of a dominant annihilator of local cohomology
modules.

######  Question.

Let $\mathfrak{p}$ be a prime ideal of $R$.

  1. (1)

When is the resolving subcategory $\mathcal{R}(\mathfrak{p})$ dominant?

  2. (2)

What value does $f_{\mathcal{R}(\mathfrak{p})}(\mathfrak{q})$ take for each
$\mathfrak{q}\in U(\mathfrak{p})$?

Proposition 2.5 (4) (respectively, (5)) states that the above question (1) has
an affirmative answer when $R$ is a finite-dimensional ring (respectively, a
Cohen-Macaulay ring). Regarding the above question (2), the next section will
give an answer when $R$ has a finite dimension under certain assumptions.
Furthermore, the last section will provide a complete answer when $R$ is a
Cohen-Macaulay ring.

##  4\. The case of finite-dimensional rings

The purpose of this section is to establish annihilation results for local
cohomology modules over a finite-dimensional ring with certain assumptions.

In the case when a prime ideal $\mathfrak{p}$ of a finite-dimensional ring $R$
has the equality $0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$ for
all $\mathfrak{q}\in U(\mathfrak{p})$, we can describe a condition for the
existence of an annihilator of local cohomology modules without using the
notion of maps in $\mathbb{F}(R\text{-}\mathrm{mod})$.

######  Theorem 4.1.

Let $\mathfrak{p}$ be a prime ideal of $R$ with
$0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$ for all
$\mathfrak{q}\in U(\mathfrak{p})$. Suppose that the resolving subcategory
$\mathcal{R}(\mathfrak{p})$ is dominant (e.g., the ring $R$ has a finite
dimension). Then, for a finitely generated $R$-module $M$, the following
conditions are equivalent.

  1. (1)

There exists an element $s\in R\setminus\mathfrak{p}$ such that
$sH_{\mathfrak{a}}^{i}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i<\mathrm{grade}(\mathfrak{a},R)$.

  2. (2)

One has $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\leqq 0$.

###### Proof.

Note that finite-dimensional rings have the dominant resolving subcategory
$\mathcal{R}(\mathfrak{p})$ by Proposition 2.5 (4).

Let $R$ be a (not necessarily finite-dimensional) ring such that the resolving
subcategory $\mathcal{R}(\mathfrak{p})$ is dominant. Then [8, Theorem 5.4]
gives the map $f\in\mathbb{F}(R\text{-}\mathrm{mod})$ corresponding to the
subcategory $\mathcal{R}(\mathfrak{p})$.

We will show that the equality $f(\mathfrak{q})=0$ holds for each
$\mathfrak{q}\in U(\mathfrak{p})$. Indeed, Proposition 2.5 (1) and our
assumption yield the inclusion relation

$\mathcal{R}(\mathfrak{p})\subseteqq\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,M_{\mathfrak{q}}\geqq 0pt\,R_{\mathfrak{q}}\\}.$

Thus, we can deduce from Lemma 3.1 (1) that $0\leqq
f(\mathfrak{q})=0pt\,R_{\mathfrak{q}}-0pt\,R_{\mathfrak{q}}=0$.

Theorem 3.4 establishes the equalities

$\displaystyle\mathcal{R}(\mathfrak{p})$ $\displaystyle=\\{M\in
R\text{-}\mathrm{mod}\mid 0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq
f(\mathfrak{q})\text{ for all }\mathfrak{q}\in U(\mathfrak{p})\\}$
$\displaystyle=\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq 0\text{ for all
}\mathfrak{q}\in U(\mathfrak{p})\\}.$

These equalities mean that an $R$-module $M$ is in the subcategory
$\mathcal{R}(\mathfrak{p})$ if and only if one has $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\leqq 0$. ∎

######  Remark 4.2.

Let $\mathfrak{p}$ be a prime ideal of $R$.

  1. (1)

In the preceding section, we talked about the question for the value of
$f_{\mathcal{R}(\mathfrak{p})}(\mathfrak{q})$ for each $\mathfrak{q}\in
U(\mathfrak{p})$. As in the proof for Theorem 4.1, if $R$ is a finite-
dimensional ring and $\mathfrak{p}$ satisfies the equality
$0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$ for all
$\mathfrak{q}\in U(\mathfrak{p})$, then one has the value
$f_{\mathcal{R}(\mathfrak{p})}(\mathfrak{q})=0$ for each $\mathfrak{q}\in
U(\mathfrak{p})$.

  2. (2)

A ring $R$ is said to be almost Cohen-Macaulay if it has the Cohen-Macaulay
defect $\mathrm{cmd}\,R=\sup_{\mathfrak{q}\in\mathrm{Spec}(R)}\\{\dim
R_{\mathfrak{q}}-0pt\,R_{\mathfrak{q}}\\}\leqq 1$. The characterization [3,
Lemma 3.1] states that a ring $R$ is almost Cohen-Macaulay if and only if one
has $0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$ for all
$\mathfrak{q}\in\mathrm{Spec}(R)$.

  3. (3)

An ideal $I$ of $R$ is called perfect if the projective dimension of the
$R$-module $R/I$ is equal to $\mathrm{grade}(I,R)$; see [2, Definition
1.4.15]. In the case when a prime ideal $\mathfrak{q}$ of $R$ is perfect, one
has $0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$; see [2,
Proposition 1.4.16].

  4. (4)

For a finitely generated $R$-module $M$, the large restricted flat dimension
$\mathrm{Rfd}_{R}\,M$ is defined by
$\sup_{\mathfrak{q}\in\mathrm{Spec}(R)}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}$.
The following inequalities are an immediate consequence of [2, Proposition
1.2.12]:

$\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\leqq\min\\{\dim
R_{\mathfrak{p}},\mathrm{Rfd}_{R}\,M\\}\leqq\dim R.$

See also [3, Proposition 2.2 and Theorem 2.4].

Theorem 4.1 concludes that the discussion now turns to the existence of
annihilator of local cohomology modules for a finitely generated $R$-module
$M$ with $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}>0$. Theorem
4.1 describes the following corollary as an answer to the discussion about
such modules.

######  Corollary 4.3.

Let $\mathfrak{p}$ be a prime ideal of $R$ with
$0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$ for all
$\mathfrak{q}\in U(\mathfrak{p})$. Suppose that the resolving subcategory
$\mathcal{R}(\mathfrak{p})$ is dominant (e.g., the ring $R$ has a finite
dimension). Let $M$ be a finitely generated $R$-module with
$\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\geqq 0$.
Then there exists an element $s\in R\setminus\mathfrak{p}$ such that

$sH^{i}_{\mathfrak{a}}(M)=0$

for all ideals $\mathfrak{a}$ of $R$ and all integers
$i<\mathrm{grade}(\mathfrak{a},R)-\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}$.

###### Proof.

We have the inequalities $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}\leqq\dim
R_{\mathfrak{p}}<\infty$ by Remark 4.2 (4). Let $n=\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}$. Our
assumption guarantees that $n$ is a non-negative integer.

In the case when $n=0$, our assertion is immediately from Theorem 4.1.

Now suppose that $n\geqq 1$. Since each prime ideal $\mathfrak{q}$ in
$U(\mathfrak{p})$ has $0pt\,R_{\mathfrak{q}}\leqq 0pt\,M_{\mathfrak{q}}+n$,
the depth lemma [2, Proposition 1.2.9] yields the inequalities and the
equality

$\displaystyle 0pt\left((\Omega^{n}_{R}M)_{\mathfrak{q}}\right)$
$\displaystyle\geqq\min\\{0pt\,R_{\mathfrak{q}},0pt\left(\Omega_{R}^{n-1}M\right)_{\mathfrak{q}}+1\\}\geqq\cdots$
$\displaystyle\geqq\min\\{0pt\,R_{\mathfrak{q}},0pt\,M_{\mathfrak{q}}+n\\}=0pt\,R_{\mathfrak{q}}.$

Therefore, one has the inequality $\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\left((\Omega^{n}_{R}M)_{\mathfrak{q}}\right)\\}\leqq
0$. We now apply Theorem 4.1 to the module $\Omega^{n}_{R}M$. Then there
exists an element $s\in R\setminus\mathfrak{p}$ such that
$sH^{i}_{\mathfrak{a}}(\Omega_{R}^{n}M)=0$ for all ideals $\mathfrak{a}$ of
$R$ and all integers $i<\mathrm{grade}(\mathfrak{a},R)$.

Our aim is to show that the above element $s$ yields
$sH^{i}_{\mathfrak{a}}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i<\mathrm{grade}(\mathfrak{a},R)-n$. To prove our assertion, we now
fix an ideal $\mathfrak{a}$ of $R$.

We suppose that one has $\mathrm{grade}(\mathfrak{a},R)-n\leqq 0$. Since the
local cohomology functor $H^{i}_{\mathfrak{a}}(-)$ is the zero functor for all
integers $i<0$, we can achieve the equality $sH^{i}_{\mathfrak{a}}(M)=0$ for
all integers $i<\mathrm{grade}(\mathfrak{a},R)-n$.

Next suppose that we have $\mathrm{grade}(\mathfrak{a},R)-n\geqq 1$. It should
be noted that one has the inequality $\mathrm{grade}(\mathfrak{a},R)\geqq 2$.
A projective resolution

$\cdots\to P_{k}\to\cdots\to P_{1}\to P_{0}\to M\to 0$

of the $R$-module $M$ provides short exact sequences

$H^{i-1}_{\mathfrak{a}}(P_{k-1})\to
H^{i-1}_{\mathfrak{a}}(\Omega_{R}^{k-1}M)\to
H^{i}_{\mathfrak{a}}(\Omega_{R}^{k}M)\to H^{i}_{\mathfrak{a}}(P_{k-1})$

for all positive integers $i$ and $k$.

Since the $R$-module $P_{k-1}$ is projective, or a direct summand of a free
module, [1, Theorem 6.2.7] yields $H^{i}_{\mathfrak{a}}(P_{k-1})=0$ for all
integers $i<\mathrm{grade}(\mathfrak{a},R)$. Consequently, the above element
$s$ provides that

$sH^{i}_{\mathfrak{a}}(M)\cong sH^{i+1}_{\mathfrak{a}}(\Omega_{R}M)\cong
sH^{i+n}_{\mathfrak{a}}(\Omega^{n}_{R}M)=0$

for all integers $i$ with $i+n<\mathrm{grade}(\mathfrak{a},R)$. The proof of
our corollary is completed. ∎

##  5\. The case of Cohen-Macaulay rings

This section will investigate the existence of annihilators of local
cohomology modules over a Cohen-Macaulay ring. In particular, our purpose is
to verify that Cohen-Macaulay rings establish results similar to Theorem 4.1
and Corollary 4.3.

We begin with the following easy lemma.

######  Lemma 5.1.

We suppose that $R$ is a Cohen-Macaulay ring. Let
$\mathfrak{p}\in\mathrm{Spec}(R)$, and let $M$ be a finitely generated
$R$-module with $0pt\,M_{\mathfrak{p}}\geqq 0pt\,R_{\mathfrak{p}}$. Then one
has $0pt\,M_{\mathfrak{q}}\geqq 0pt\,R_{\mathfrak{q}}$ for all
$\mathfrak{q}\in U(\mathfrak{p})$.

###### Proof.

Let $\mathfrak{q}$ be a prime ideal of $R$ with
$\mathfrak{q}\subseteqq\mathfrak{p}$. Since $R$ is a Cohen-Macaulay ring, the
module $M$ satisfies the inequalities and the equalities

$\displaystyle 0pt\,M_{\mathfrak{q}}$ $\displaystyle\geqq
0pt\,M_{\mathfrak{p}}-0pt\,\mathfrak{p}/\mathfrak{q}\geqq
0pt\,R_{\mathfrak{p}}-0pt\,\mathfrak{p}/\mathfrak{q}$ $\displaystyle=\dim
R_{\mathfrak{p}}-(0pt\mathfrak{p}-0pt\mathfrak{q})=\dim
R_{\mathfrak{q}}=0pt\,R_{\mathfrak{q}}$

by [1, Lemma 9.3.2] or [8, Lemma 6.2]. ∎

We now achieve the following conclusion about when local cohomology modules
over a Cohen-Macaulay ring have an annihilator that does not depend on the
choice of the ideal. The result below removes from [9, Theorem 4.3] the
assumption that the base ring has a finite dimension.

######  Theorem 5.2.

Suppose that $R$ is a Cohen-Macaulay ring, and let
$\mathfrak{p}\in\mathrm{Spec}(R)$. Then, for a finitely generated $R$-module
$M$, the following conditions are equivalent.

  1. (1)

There exists an element $s\in R\setminus\mathfrak{p}$ such that
$sH_{\mathfrak{a}}^{i}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i<\mathrm{grade}(\mathfrak{a},R)$.

  2. (2)

The $R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is a maximal Cohen-Macaulay
$R_{\mathfrak{p}}$-module.

###### Proof.

Note that the Cohen-Macaulay ring $R$ has the dominant resolving subcategory
$\mathcal{R}(\mathfrak{p})$ by Proposition 2.5 (5). Hence, [8, Theorem 5.4]
gives the map $f\in\mathbb{F}(R\text{-}\mathrm{mod})$ corresponding to the
subcategory $\mathcal{R}(\mathfrak{p})$.

We claim that the equality $f(\mathfrak{q})=0$ holds for each $\mathfrak{q}\in
U(\mathfrak{p})$. Indeed, since $R$ is a Cohen-Macaulay ring, [2, Theorem
2.1.3] yields the equality
$0pt\,R_{\mathfrak{q}}=\mathrm{grade}(\mathfrak{q},R)$. It therefore follows
from Proposition 2.5 (1) that

$\mathcal{R}(\mathfrak{p})\subseteqq\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,M_{\mathfrak{q}}\geqq 0pt\,R_{\mathfrak{q}}\\}.$

Applying Lemma 3.1, we obtain $0\leqq
f(\mathfrak{q})=0pt\,R_{\mathfrak{q}}-0pt\,R_{\mathfrak{q}}=0$.

We now establish the equalities

$\displaystyle\mathcal{R}(\mathfrak{p})$ $\displaystyle=\\{M\in
R\text{-}\mathrm{mod}\mid 0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq
f(\mathfrak{q})\text{ for all }\mathfrak{q}\in U(\mathfrak{p})\\}$
$\displaystyle=\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\leqq 0\text{ for all
}\mathfrak{q}\in U(\mathfrak{p})\\}$ $\displaystyle=\\{M\in
R\text{-}\mathrm{mod}\mid 0pt\,M_{\mathfrak{p}}\geqq 0pt\,R_{\mathfrak{p}}\\}$
$\displaystyle=\\{M\in R\text{-}\mathrm{mod}\mid
0pt\,M_{\mathfrak{p}}\geqq\dim R_{\mathfrak{p}}\\}$

using Theorem 3.4, the above claim, Lemma 5.1, and the Cohen-Macaulayness for
the ring $R$, respectively. Consequently, we can conclude that an $R$-module
$M$ is in the subcategory $\mathcal{R}(\mathfrak{p})$ if and only if an
$R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is a maximal Cohen-Macaulay
$R_{\mathfrak{p}}$-module. ∎

######  Remark 5.3.

Let $\mathfrak{p}$ be a prime ideal of $R$.

  1. (1)

When $R$ is a Cohen-Macaulay ring, the proof for Theorem 5.2 presents the
complete answer to the question (2) in section 3. Indeed, with the notation of
Theorem 3.4, we have already established the equality
$f_{\mathcal{R}(\mathfrak{p})}(\mathfrak{q})=0$ for each $\mathfrak{q}\in
U(\mathfrak{p})$.

  2. (2)

We can regard Theorem 5.2 as an immediate consequence of Theorem 4.1. To prove
this fact, we now suppose that $R$ is a Cohen-Macaulay ring. By [6, Theorems
3.8, 3.9, and Proposition 3.10], a finitely generated $R$-module $M$ has the
equalities

$\displaystyle\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{0pt\,R_{\mathfrak{q}}-0pt\,M_{\mathfrak{q}}\\}$
$\displaystyle=\sup_{\mathfrak{q}\in
U(\mathfrak{p})}\\{\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{q}}}\,M_{\mathfrak{q}}\\}$
$\displaystyle=\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}$
$\displaystyle=0pt\,R_{\mathfrak{p}}-0pt\,M_{\mathfrak{p}}$
$\displaystyle=\dim R_{\mathfrak{p}}-0pt\,M_{\mathfrak{p}}.$

Note that, for each $\mathfrak{q}\in\mathrm{Spec}(R)$, one has
$0pt\,M_{\mathfrak{q}}=\infty$ if and only if $M_{\mathfrak{q}}=0$ if and only
if
$\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{q}}}\,M_{\mathfrak{q}}=-\infty$.

The corollary below is a generalization of [9, Corollary 4.6], which
guarantees the existence of annihilators of local cohomology modules for
finitely generated modules over a finite-dimensional Cohen-Macaulay ring.
Using Theorem 5.2, the same proof for [9, Corollary 4.6] works without the
assumption that the base ring has a finite dimension.

######  Corollary 5.4.

Suppose that $R$ is a Cohen-Macaulay ring, and let
$\mathfrak{p}\in\mathrm{Spec}(R)$. Then, for a finitely generated $R$-module
$M$, there exists an element $s\in R\setminus\mathfrak{p}$ such that

$sH^{i}_{\mathfrak{a}}(M)=0$

for all ideals $\mathfrak{a}$ of $R$ and all integers
$i<\mathrm{grade}(\mathfrak{a},R)-\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}$.

######  Remark 5.5.

Let $R$ be a Cohen-Macaulay ring, let $\mathfrak{p}$ be a prime ideal of $R$,
and let $M$ be a finitely generated $R$-module.

  1. (1)

We can regard Corollary 5.4 as an immediate consequence of Corollary 4.3 by
the equalities in Remark 5.3 (2).

Note that, in the case when
$\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}<0$, we
have the equality
$\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}=-\infty$.
Namely, one has $M_{\mathfrak{p}}=0$. It is easy to see that there exists an
element $s\in R\setminus\mathfrak{p}$ such that $sM=0$. Since local cohomology
functors are $R$-linear by [1, Properties 1.2.2], we achieve the equalities
$sH^{i}_{\mathfrak{a}}(M)=0$ for all ideals $\mathfrak{a}$ of $R$ and all
integers $i$; see also [1, Lemma 9.4.1].

  2. (2)

Let $t$ be a non-negative integer. We suppose that $M$ has an element $s\in
R\setminus\mathfrak{p}$ such that $sH^{i}_{\mathfrak{a}}(M)=0$ for all ideals
$\mathfrak{a}$ of $R$ and all integers $i<\mathrm{grade}(\mathfrak{a},R)-t$.
Then the following investigation will lead to the inequality
$t\geqq\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}$.

When the equality $M_{\mathfrak{p}}=0$ holds, one has
$\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}=-\infty$.
Therefore, there is nothing to prove.

Next suppose that $M_{\mathfrak{p}}\neq 0$. We now take the prime ideal
$\mathfrak{p}$ as the above ideal $\mathfrak{a}$. The flat base change theorem
[1, Theorem 4.3.2] yields
$H^{i}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ for all integers
$i<\mathrm{grade}(\mathfrak{p},R)-t$. Since Nakayama’s lemma guarantees that
$\mathfrak{p}R_{\mathfrak{p}}M_{\mathfrak{p}}\neq M_{\mathfrak{p}}$, we deduce
from [1, Theorem 6.2.7] and [2, Theorem 2.1.3] that

$0pt\,M_{\mathfrak{p}}\geqq\mathrm{grade}(\mathfrak{p},R)-t=0pt\,R_{\mathfrak{p}}-t.$

Consequently, we can conclude $t\geqq
0pt\,R_{\mathfrak{p}}-0pt\,M_{\mathfrak{p}}=\mathrm{CM}\text{-}\mathrm{dim}_{R_{\mathfrak{p}}}\,M_{\mathfrak{p}}$
by [6, Theorems 3.8 and 3.9].

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Number JP20K03549.

## References

  * [1] M. P. Brodmann; R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cambridge, 1998.
  * [2] W. Bruns; J. Herzog, Cohen-Macaulay rings, revised edition. Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998.
  * [3] L. W. Christensen; H.-B. Foxby; A. Frankild, Restricted homological dimensions and Cohen-Macaulayness. J. Algebra 251 (2002), no. 1, 479–502.
  * [4] H. Dao; R. Takahashi, Classification of resolving subcategories and grade consistent functions. Int. Math. Res. Not. IMRN 2015 (2015), no. 1, 119–149.
  * [5] G. Faltings, $\ddot{\text{U}}$ber die Annulatoren lokaler Kohomologiegruppen. Arch. Math. (Basel) 30 (1978), no. 5, 473–476.
  * [6] A. A. Gerko, On homological dimensions. Mat. Sb. 192 (2001), no. 8, 79–94; translation in Sb. Math. 192 (2001), no. 7–8, 1165–1179.
  * [7] K. Raghavan, Uniform annihilation of local cohomology and of Koszul homology. Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 3, 487–494.
  * [8] R. Takahashi, Classification of dominant resolving subcategories by moderate functions. To appear in Illinois J. Math., https://www.math.nagoya-u.ac.jp/$\sim$takahashi/papers.html (accessed 2021-4-30).
  * [9] T. Yoshizawa, Annihilators of local cohomology modules over a Cohen-Macaulay ring. Submitted, 2020.
  * [10] C. Zhou, Uniform annihilators of local cohomology. J. Algebra 305 (2006), no. 1, 585–602.