# Effects of clustered nuclear geometry on the anisotropic flow in O-O
collisions at the LHC within a multiphase transport model framework

Debadatta Behera  Suraj Prasad  Neelkamal Mallick  Raghunath
Sahoo111Corresponding author<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Indore, Simrol, Indore 453552, India

###### Abstract

To understand the true origin of flowlike signatures and applicability of
hydrodynamics in small collision systems, effects of soft QCD dynamics, the
sensitivity of jetlike correlations, and nonequilibrium effects, efforts are
being made to perform p-O and O-O collisions at the LHC and RHIC energies. It
is equally interesting to look into the possible signatures of an
$\alpha$-clustered nuclear geometry in 16O-16O collisions by studying the
initial-state effects on the final-state observables. In this work, within a
multiphase transport model, we implement an $\alpha$-cluster tetrahedral
density profile in the oxygen nucleus along with the default Woods-Saxon
density profile. We study the eccentricity ($\epsilon_{2}$), triangularity
($\epsilon_{3}$), normalized symmetric cumulants [NSC(2,3)], elliptic flow
($v_{2}$), and triangular flow ($v_{3}$) in 16O-16O collisions at
$\sqrt{s_{\rm NN}}=7~{}$TeV. The constituent quark number scaling of the
elliptic flow is also reported. For the most central collisions, enhanced
effects in $\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ and
$\langle v_{3}\rangle/\langle v_{2}\rangle$ with a negative value of NSC(2,3),
and an away-side broadening in the two-particle azimuthal correlation function
[$C(\Delta\phi)$] of the identified particles are observed in the presence of
an $\alpha$-clustered geometry.

##  I Introduction

Ultrarelativistic heavy-ion collisions at the Large Hadron Collider (LHC) and
the Relativistic Heavy Ion Collider (RHIC) create high temperature and
density, which provide suitable conditions for producing a locally thermalized
and deconfined partonic medium. This hot and dense fireball is made up of QCD
matter, i.e., quarks and gluons, and thus called quark-gluon plasma (QGP).
Studies related to QGP investigate all the indirect signatures as QGP is a
highly short-lived state due to the behavior of strongly interacting matter.
QGP expands rapidly, and its evolution is well understood through relativistic
viscous hydrodynamics with dissipative effects. Thus, the initial-state
collision geometry and the fluctuations in energy and entropy density are
embedded in the final-state multiparticle correlations through this collective
expansion of the QGP Heinz:2013th ; Voloshin:2008dg ; Ollitrault:1992bk .
Usually, this is studied as the medium response to the initial eccentricity
($\epsilon_{2}$) and triangularity ($\epsilon_{3}$) by quantifying the Fourier
coefficients ($v_{2}$ and $v_{3}$) of the azimuthal momentum distribution of
the final-state hadrons voloshin-zhang . Experimental measurements of these
flow coefficients agree with the predictions from hydrodynamic calculations
suggesting that QGP behaves like a perfect fluid STAR:2005gfr . Thus, the
presence of finite-flow coefficients is considered a signature of the
hydrodynamic behavior of the QGP and hence, the thermalization in the early
stages of the collision. Recently, similar signatures have been observed in
small collision systems such as the high-multiplicity $pp$ collisions, where
hydrodynamic expansion or collectivity is usually not expected CMS:2016fnw .
These observations also raise questions on the applicability of hydrodynamics
in small collision systems formed in ultrarelativistic nuclear collisions. As
the system size of 16O-16O overlaps high-multiplicity $pp$ and peripheral Pb-
Pb collisions, it provides an opportunity to explore the origin of flowlike
signatures in small collision systems.

Another interesting direction is to explore how the final-state observables
are affected by the initial-state nuclear structure, nuclear shape
deformation, or even the presence of 4He-nuclei (known as $\alpha$-clusters)
inside the nucleus of elements having $4n$-number of nucleons, such as 8Be,
12C, and 16O, to name a few. Studies related to nuclear shape deformation have
been carried out at the RHIC Giacalone:2021udy ; Haque:2019vgi ;
PHENIX:2018lia  and at the LHC with Xe-Xe collisions at $\sqrt{s_{\rm
NN}}=5.44$ TeV ALICE:2018lao ; CMS:2019cyz ; ATLAS:2019dct . Results show
quadruple deformation in 129Xe nucleus. The presence of four 4He–clusters
inside the 16O nucleus was first proposed by Gamow back in 1930s gamow  and
then by Wheeler Wheeler:1937zza . Although there is evidence for the existence
of such a clustered structure Bijker:2014tka ; Wang:2019dpl ; He:2014iqa ;
He:2021uko , the contribution of the clustered states in the ground state of
16O was found to be less than 30% Zuker:1968zz . Recently, there are proposals
for dedicated runs for 16O-16O collisions at both RHIC and LHC Brewer:2021kiv
; Katz:2019qwv . This could clarify the origin of collectivity on small
systems and the effects of clustered nuclear geometry on the final-state
observables.

In recent years, there have been several theoretical studies reported on
oxygen collisions based on Glauber Monte Carlo Rybczynski:2019adt ;
Sievert:2019zjr ; Huang:2019tgz , different hydrodynamic models Lim:2018huo ;
Summerfield:2021oex ; Schenke:2020mbo , global observables Behera:2021zhi ,
parton energy loss Huss:2020whe , and jet-quenching effects across small to
large collision systems Zakharov:2021uza . Some observables showing evidence
of the signatures of $\alpha$-clusters are also reported in 16O-16O collisions
Li:2020vrg ; Ding:2023ibq ; Wang:2021ghq ; Rybczynski:2017nrx ; alpha-frag .
In this work, within a multiphase transport model, we implement an
$\alpha$-cluster tetrahedral density profile in the oxygen nucleus along with
the default Woods-Saxon density profile. We study the eccentricity
($\epsilon_{2}$), triangularity ($\epsilon_{3}$), normalized symmetric
cumulants [NSC(2,3)], elliptic flow ($v_{2}$), and triangular flow ($v_{3}$)
in 16O-16O collisions at $\sqrt{s_{\rm NN}}=7~{}$TeV.

In addition, the elliptic flow coefficients as a function of transverse
momentum ($v_{2}(p_{\rm T})$) for the light-flavor hadrons such as
$\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$ are studied for nuclear collisions
with default Woods-Saxon and $\alpha$-cluster tetrahedral density profiles.
The appearance of hadronic collectivity is believed to have originated from
the early deconfined partonic phase and is subsequently transferred to the
hadrons via the quark recombination mechanism of hadronization. This is also
known as the quark coalescence model Voloshin:2002wa . This behavior leads to
the observation of a higher flow of baryons than mesons in the intermediate
$p_{\rm T}$, and ideally to the number-of-constituent-quark (NCQ) scaling
Molnar:2003ff ; Sato:1981ez ; Dover:1991zn . Experimentally, at RHIC, the NCQ
scaling seems to be valid as seen in Au-Au collisions at $\sqrt{s_{\rm
NN}}=200$ GeV STAR:2003wqp ; PHENIX:2003qra . However, at the LHC energies,
the scaling is only approximate ALICE:2010suc ; ALICE:2014wao ; ALICE:2022zks
. Using AMPT in string melting mode, the NCQ scaling is observed at the top
RHIC energies in Au-Au collisions Singha:2016aim . However, NCQ scaling seems
to be violated using the same model at the LHC energies in Pb-Pb. The breaking
of NCQ scaling in AMPT string melting mode is found to be independent of the
magnitude of parton-parton cross sections and hadron cascade time
Singha:2016aim . However, the breaking of scaling is understood as the
increase in the partonic density at the LHC energy in Pb-Pb collisions.
Further, Si-Si collisions at this energy show NCQ scaling, which adds to this
understanding Singha:2016aim . This makes the case appealing to look for the
validation of NCQ scaling in 16O-16O collisions at $\sqrt{s_{\rm
NN}}=7~{}$TeV. In a recent event-shape dependent study of NCQ scaling using
transverse spherocity ($S_{0}$) in heavy-ion collisions in AMPT, it is
reported that low-$S_{0}$ (jetty-like) events show more deviation from the NCQ
scaling than the $S_{0}$-integrated (unbiased) events Mallick:2021hcs . In Pb-
Pb collisions, the deviation appears in the $S_{0}$-integrated events and gets
enhanced in low-$S_{0}$ events, whereas in Au-Au collisions, the scaling
violation appears only in the low-$S_{0}$ events and not in the
$S_{0}$-integrated events. These results show the dependence of NCQ scaling on
the event shapes, and it awaits experimental confirmation. For the time being,
we proceed to study the NCQ scaling behavior in 16O-16O collisions at
$\sqrt{s_{\rm NN}}=7~{}$TeV using the AMPT string melting model, and explore
the possible role of density profiles on the NCQ scaling in small collision
systems. Further, for the most central collisions, we observe enhanced effects
in $\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ and $\langle
v_{3}\rangle/\langle v_{2}\rangle$ with a negative value of NSC(2,3), and an
away-side broadening in the two-particle azimuthal correlation function
[$C(\Delta\phi)$] of the identified particles. Here onwards, for the sake of
simplicity, we write O-O instead of 16O-16O throughout the text.

The paper is organized as follows. It begins with a brief introduction to the
event generator(a multiphase transport model), the $\alpha$-cluster geometry
implementation, and estimation of anisotropic flow coefficients via the two-
particle correlation method in Sec. II. The paper then shows and describes the
results for eccentricity, triangularity, elliptic flow, triangular flow, and
the number-of-constituent-quarks scaling of the elliptic flow in Sec. III.
Finally, the paper concludes with the important findings summarized in Sec.
IV.

##  II Event Generation and Analysis Methodology

In this section, we briefly introduce the components of the AMPT model, the
tuning used to generate the collisions, and the implementation of the
$\alpha$-cluster geometry in the oxygen nucleus. The two-particle correlation
method used to estimate the flow coefficients is also discussed.

###  II.1 A multiphase transport model

A multiphase transport model (AMPT) is a Monte Carlo-based transport model for
heavy-ion collision. It consists of four main stages: initialization of the
collisions, parton cascade, hadronization, and hadron transport Zhang:1999mqa
; Zhang:2000bd ; Zhang:2000nc ; Lin:2001cx ; Lin:2001yd ; Pal:2001zw ;
Lin:2001zk ; Zhang:2002ug ; Pal:2002aw ; Lin:2002gc ; Lin:2003ah ; Lin:2003iq
; Wang:1991hta ; Zhang:1997ej ; He:2017tla ; Li:2001xh ; Greco:2003mm . The
initialisation of the collisions is performed by HIJING, where the
differential cross-section of produced minijets in pp collisions is converted
into AA and p-A collisions Wang:1991hta . The parton cascade or the parton
transport is performed by Zhang’s Parton Cascade (ZPC) model Zhang:1997ej . In
the string-melting version of AMPT, the coloured strings are transformed into
the low momentum partons. The transported partons are hadronized using a
spatial coalescence mechanism in the string-melting version of AMPT Lin:2001zk
; He:2017tla ; however, in the default version of AMPT, the Lund string
fragmentation mechanism is used to perform the hadronisation. A relativistic
transport model is used for the evolution of the produced hadrons Li:2001xh ;
Greco:2003mm . In the current work, we have used the string melting mode of
AMPT (version 2.26t9b) since the quark coalescence mechanism well describes
the particle flow and spectra at the mid-transverse momentum region
Fries:2003vb ; Fries:2003kq . The choice of centrality selection is taken from
the geometrical slicing based on the impact parameter distribution for both
the Woods-Saxon and $\alpha$-cluster density profile of the nuclei. The impact
parameter cuts for different centralities are the same as in Ref.
Behera:2021zhi , which are obtained via the Glauber model estimation. The AMPT
settings for the O-O system in the current work are also the same as reported
in Ref. Behera:2021zhi .

In heavy-ion collisions, the typical density profile for a nucleus is
considered to be the Woods-Saxon distribution. The Wood-Saxon charge density
is given in terms of a three-parameter fermi (3pF) distribution as,

$\rho(r)=\frac{\rho_{0}(1+w(\frac{r}{r_{0}})^{2})}{1+{\rm
exp}(\frac{r-r_{0}}{a})}.$ (1)

Here, $r$ is the radial distance from the center of the nucleus, $a$ is the
skin depth of the nucleus, $r_{0}$ is the mean radius of the nucleus, and $w$
is the deformation parameter. In the oxygen nucleus, $r_{0}$ = 2.608
$\rm{fm}$, $a$ is 0.513 $\rm{fm}$ and $w$ is -0.051 Loizides:2014vua .
$\rho_{0}$ is the nuclear density constant at $r=0$.

We also implement the $\alpha$-cluster structure inside the oxygen nucleus
using the AMPT model. The implementation is done numerically by creating a
geometric distribution of a regular tetrahedral structure having 4He nuclei
placed at the vertices. For the 4He nucleus, the distribution of nucleons
follows the Wood-Saxon density profile described in Eq. 1 with the parameters
$r_{0}$ = 0.964 $\rm{fm}$, $a$ = 0.322 $\rm{fm}$, and $w$ = 0.517. This leads
to the $\rm{rms}$ radius for 4He nucleus to be 1.676 $\rm fm$. These
$\alpha$-clustered nuclei are positioned on the vertices of a standard
tetrahedron with a side length of 3.42 $\rm{fm}$. In this configuration, the
$\rm{rms}$ radius for 16O is calculated to be 2.699 $\rm{fm}$ Behera:2021zhi ;
Li:2020vrg . The orientation of the tetrahedron is randomized for each
projectile and target on an event-by-event basis.

###  II.2 Two-particle correlation method

In noncentral heavy-ion collisions, the collision overlap region is
anisotropic in space. The pressure gradient of the thermalized partonic medium
formed in such collisions can transform the initial spatial anisotropies into
the momentum space azimuthal anisotropies. These azimuthal anisotropies of
different orders can be quantified by the coefficients of Fourier series
decomposition of the momentum distribution of final-state particles, given as:

$\displaystyle E\frac{d^{3}N}{dp^{3}}=\frac{d^{2}N}{2\pi p_{\rm T}dp_{\rm
T}dy}\bigg{(}1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\phi-\psi_{n})]\bigg{)}\,.$ (2)

Here, $\phi$ represents the azimuthal angle of the final-state particles in
the transverse plane, and $\psi_{n}$ represents the $n$th harmonic event plane
angle v2eventplane . $v_{n}$ is the $n$th-order anisotropic flow coefficient
where $n$ = 1 stands for directed flow, $n$ = 2 is the elliptic flow and $n$ =
3 quantifies the triangular flow. Anisotropic flow coefficients of different
orders can be estimated as follows:

$\displaystyle v_{n}=\langle\cos[n(\phi-\psi_{n})]\rangle$ (3)

In experiments, obtaining the event plane angle is not trivial, and Eq. 3
includes the non-flow effects, such as contributions from resonance decays and
jets. On the other hand, a two-particle correlation method to estimate the
flow coefficients can, not only efficiently reduce the non-flow contribution
by implementing a proper pseudorapidity gap but also does not require the
event-plane angle. In this study, we have ignored the pseudorapidity
dependence of $\psi_{n}$, which is observed in the experiments.

To estimate the anisotropic flow coefficients using the two-particle
correlation method, one requires the two-particle correlation function, which
can be determined using the following steps ATLAS:2012at :

  1. (i)

In each event, two sets of particles are formed based on their transverse
momenta, namely, “a” and “b”. “a” denotes the trigger particles, whereas “b”
represents the associated particle set.

  2. (ii)

Each particle from trigger group (“a”) pairs with each particle from associate
group (“b”) and the relative pseudorapidities ($\Delta\eta=\eta_{a}-\eta_{b}$)
and relative azimuthal angles ($\Delta\phi=\phi_{a}-\phi_{b}$) are determined.

  3. (iii)

Same event pairs ($S(\Delta\eta,\Delta\phi)$) and mixed event pairs are
($B(\Delta\eta,\Delta\phi)$) are determined. In the same event pair, both “a”
and “b” belong to the same event; however, in the mixed event pair, “a” and
“b” are from different events where “a” pairs with “b” from five randomly
selected events to remove physical correlations.

  4. (iv)

Two particle correlation function ($C(\Delta\eta,\Delta\phi)$) is determined
by taking the ratio of $S(\Delta\eta,\Delta\phi)$ to
$B(\Delta\eta,\Delta\phi)$.

In this analysis, we use final-state charged hadrons with kinematic cuts as
$|\eta|<$ 2.5 and $p_{\rm T}>$ 0.4 GeV/$c$ to encompass a broader spectrum of
particles. To omit the jet peak region seen in the $C(\Delta\eta,\Delta\phi)$
distribution, the $\Delta\eta$ interval is carefully chosen. The interval, in
our case, is implemented to be $1.0<|\Delta\eta|<4.8$ to obtain 1D correlation
$C(\Delta\phi)$, given as:

$\displaystyle C(\Delta\phi)=\frac{dN_{\rm
pairs}}{d\Delta\phi}=A\times\frac{\int
S(\Delta\eta,\Delta\phi)d\Delta\eta}{\int
B(\Delta\eta,\Delta\phi)d\Delta\eta}.$ (4)

Here, the normalization constant $A$ ensures that at a given $\Delta\eta$
interval, there is the same number of pairs in the same events and mixed
events.

Figure 1: (Color online) Centrality dependence of average eccentricity
($\langle\epsilon_{2}\rangle$), triangularity ($\langle\epsilon_{3}\rangle$),
and $\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ for O-O collisions
at $\sqrt{s_{\rm NN}}=7$ TeV in the AMPT string melting model for both Woods-
Saxon and $\alpha$-cluster type nuclear density profiles.

The pair distribution ($N_{\rm pairs}$) or 1D correlation function can be
expanded into a Fourier transform in $\Delta\phi$ as follows:

$\displaystyle C(\Delta\phi)=\frac{dN_{\rm
pairs}}{d\Delta\phi}\propto\bigg{[}1+2\sum_{n=1}^{\infty}v_{n,n}(p_{\rm
T}^{a},p_{\rm T}^{b})\cos(n\Delta\phi)\bigg{]}.$ (5)

Here, $v_{n,n}$ is the two-particle flow coefficient. In this definition, the
convolution of particle pairs removes the event plane angle. Now, $v_{n,n}$
can be obtained as:

$\displaystyle v_{n,n}(p_{\rm T}^{a},p_{\rm T}^{b})=\langle
cos(n\Delta\phi)\rangle$ (6)

In terms of $p_{\rm T}^{a}$ and $p_{\rm T}^{b}$, $v_{n,n}$ are symmetric
functions. The definition of harmonics in Eq. 2 enters to Eq. 5, which can be
written as:

$\displaystyle\frac{dN_{\rm
pairs}}{d\Delta\phi}\propto\bigg{[}1+2\sum_{n=1}^{\infty}v_{n}(p_{\rm
T}^{a})v_{n}(p_{\rm T}^{b})\cos(n\Delta\phi)\bigg{]}.$ (7)

If collective expansion is what causes azimuthal anisotropy, then $v_{n,n}$
can be factorized into the product of two single-particle harmonic
coefficients.

$\displaystyle v_{n,n}(p_{\rm T}^{a},p_{\rm T}^{b})=v_{n}(p_{\rm
T}^{a})v_{n}(p_{\rm T}^{b}).$ (8)

From Eq. 8, $v_{n}$ can be estimated as:

$\displaystyle v_{n}(p_{\rm T}^{a})=v_{n,n}(p_{\rm T}^{a},p_{\rm
T}^{b})/\sqrt{v_{n,n}(p_{\rm T}^{b},p_{\rm T}^{b})}$ (9)

Following the above steps, one can obtain the nth-order coefficients of the
azimuthal anisotropy of all-charged particles along with identified particles
such as $\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$ for the O-O collision system
at the LHC energies using the AMPT model.

Figure 2: (Color online) Eccentricity ($\langle\epsilon_{2}\rangle$) and
triangularity ($\langle\epsilon_{3}\rangle$) distribution for the most central
case in O-O collisions at $\sqrt{s_{\rm NN}}$ = 7 TeV in Woods-Saxon (top) and
$\alpha$-cluster (bottom) type nuclear density profiles.

##  III Results and Discussions

In this section, we start by discussing the results of the participant
eccentricity, triangularity, and the correlations among them using normalized
symmetric cumulants for both the Woods-Saxon density profile and the
$\alpha$-clustered structure. Then we discuss the evolution of elliptic and
triangular flow with centrality, their ratios, and their scalings with the
initial eccentricities of the same order. We discuss the two-particle
azimuthal correlation function for the identified hadrons. Finally, the
elliptic flow as a function of transverse momentum and their NCQ scaling with
transverse kinetic energy is discussed for different centralities and nuclear
profiles.

###  III.1 Eccentricity and triangularity

In a collision of two nuclei, the overlap region of the colliding nucleons is
not spherical and isotropic. It majorly depends upon the colliding nuclei
species, the centrality of the collision, and the distribution of the nucleons
inside the nucleus. Eccentricity represents the elliptic shape of the overlap
region of the colliding nucleons and is purely geometric; however,
triangularity represents the triangular shape of the region, and it arises due
to event-by-event density fluctuations in the collision overlap region
Prasad:2022zbr . The anisotropic flow coefficients of the final-state hadrons
have a significant contribution from the initial geometrical anisotropies.
Eccentricity greatly influences the elliptic flow; however, the influence of
triangularity on triangular flow is limited only to (65-70)% for a minimally
viscous fluid Chaudhuri:2011pa . The study of eccentricity, triangularity,
elliptic flow, and triangular flow may unveil information about the medium
response to different harmonic flow coefficients. It is not trivial to
determine the eccentricity and triangularity in experiments; however, it can
be estimated in the AMPT model using the following expression Prasad:2022zbr ;
Petersen:2010cw :

$\epsilon_{\rm n}=\frac{\sqrt{\langle{r^{\rm
n}\cos(n\phi_{\text{part}})}\rangle^{2}+\langle{r^{\rm
n}\sin(n\phi_{\text{part}})}\rangle^{2}}}{\langle{r^{\rm n}}\rangle}$ (10)

where $r$ and $\rm{\phi_{\text{part}}}$ are the polar co-ordinates of the
participants. In $\epsilon_{n}$, $\rm n$ = 2 corresponds to eccentricity
($\epsilon_{2}$) and $\rm n$ = 3 corresponds to triangularity
($\epsilon_{3}$). In Fig. 1, the event averaged eccentricity
($\langle\epsilon_{2}\rangle$) (left), triangularity
($\langle\epsilon_{3}\rangle$) (middle), their ratios (right) determined from
AMPT for the Woods-Saxon density profile and $\alpha$-clustered structure in
O-O collisions at $\sqrt{s_{\rm NN}}$ = 7 TeV are shown. As traditionally
observed in heavy-ion collisions, both initial nucleon distributions have
similar behavior of $\langle\epsilon_{2}\rangle$ with centrality. The value of
$\langle\epsilon_{2}\rangle$ is observed to be increasing towards the
peripheral collisions as the overlap region gets largely elliptic with
increasing the impact parameter of the collisions. However, for a given
centrality class, $\langle\epsilon_{2}\rangle$ is lower for $\alpha$-cluster
case compared to Woods-Saxon nuclear density profile except for the mid-
central cases where both of the profiles have similar values of
$\langle\epsilon_{2}\rangle$. This indicates that even if the number of
participants in a collision is similar, the distribution of the nucleons
inside the nucleus significantly contributes to the eccentricity, which is
expected to finally be reflected in the anisotropic flow coefficients given
the hydrodynamical behavior of the medium formed. A similar trend of
$\langle\epsilon_{3}\rangle$ is observed as a function of centrality where the
mean triangularity for both Woods-Saxon and $\alpha$-cluster density profiles
is increasing towards the peripheral collisions owing to the appearance of a
more triangular shape. This trend of $\langle\epsilon_{3}\rangle$ as a
function of centrality has a peculiar behavior for the $\alpha$-cluster case
where the value decreases from central to mid-central collisions, attains a
minimum and then starts to rise again towards the peripheral collisions.
Nevertheless, the value for the Woods-Saxon nuclear density profile dominates
over the $\alpha$-cluster structure throughout the centrality selection except
for the most central cases, i.e. (0–5)% and (5–10)%. The $\alpha$-cluster
structure thus can have more significant event-by-event fluctuations in the
participant distribution due to its larger triangularity in the most central
collisions. It is to be noted that, due to a smaller collision system, the
number of sources that contribute to $\epsilon_{n}$ decreases, which can make
$\epsilon_{n}$ more significant in O-O collisions compared to Pb-Pb or Au-Au
collision systems Bzdak:2013rya . A similar study is reported in Ref.
Li:2020vrg  for the most central ($b=0$) O-O collisions at $\sqrt{s_{\rm NN}}$
= 6.37 TeV using AMPT, where the reported values for
$\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$ follow a similar
trend for a given choice of the nuclear profile. However, the values of
$\langle\epsilon_{2}\rangle$ in Ref. Li:2020vrg  are larger as compared to the
present study, whereas $\langle\epsilon_{3}\rangle$ values are almost
comparable. This could be because of the use of the initial partons to
estimate $\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$, in
contrast to the present study, where we use the initial participant nucleons.

The right-most panel of Fig. 1 shows the ratio of mean triangularity to the
eccentricity, i.e., $\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ as
a function of centrality. Here the ratio is plotted for both Woods-Saxon
nuclear density profile and $\alpha$-clustered structure in O-O collisions at
$\sqrt{s_{\rm NN}}$ = 7 TeV using AMPT. The value of this ratio is explicitly
higher for the $\alpha$-clustered structure in the most central case and
fluctuates around the Woods-Saxon profile, consistent around unity in the mid-
central to peripheral collisions. This demonstrates a balance between the
geometry of the collisions and the fluctuations in the corresponding nucleon
distributions. The exceptionally high value of
$\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ in the most central
collisions is limited to the $\alpha$-clustered structure. Therefore, for a
hydrodynamical evolution, observation of unprecedented high value of $\langle
v_{3}\rangle/\langle v_{2}\rangle$ in the most central O-O collisions could be
a possible signature of such $\alpha$-clustered structure in the oxygen
nuclei.

The observed high value of
$\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ in the most central
O-O collisions for $\alpha$-clustered structure compared to Woods-Saxon
nuclear density profile can be understood by studying the distribution of
$\epsilon_{2}$ and $\epsilon_{3}$ for both the nuclear profiles separately.
Figure 2 shows the eccentricity and triangularity distribution of most central
(0-5)% cases in O-O collisions at $\sqrt{s_{\rm NN}}$ = 7 TeV for both Woods-
Saxon and $\alpha$-cluster density profiles estimated in AMPT. In Fig. 2, the
eccentricity distribution is represented as the solid markers, and the
triangularity distribution is represented as open markers. At the same time,
the top and bottom panels of Fig. 2 show the cases with the Woods-Saxon and
$\alpha$-cluster density profiles, respectively. In the Woods-Saxon case, one
observes that both the eccentricity and triangularity have their peaks shifted
towards the lower values, indicating relatively isotropic distributions of the
participants in the transverse plane. The eccentricity distribution for the
$\alpha$-clustered structure has comparatively less mean and standard
deviation compared to the Woods-Saxon case, showing a relatively more
isotropic distribution of participants in the $\alpha$-clustered structure
than the Woods-Saxon case. On the other hand, the distribution of
triangularity in the case of the $\alpha$-clustered structure is broader
compared to the distribution of triangularity in the Woods-Saxon profile. It
has a considerably higher mean value and standard deviation. This implies that
even if the participant distribution in the $\alpha$-clustered structure is
more isotropic in shape, it has more in-built fluctuations inside. These
features of the interplay between eccentricity and triangularity with respect
to different nucleon distribution profiles could be studied using different
correlation functions, such as the normalized symmetric cumulants discussed in
the following subsection.

Figure 3: (Color online) The normalized symmetric cumulants coefficient
NSC(2,3) as a function of centrality for both Woods-Saxon and $\alpha$-cluster
type nuclear density profiles in O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV
in AMPT string melting model.

Figure 4: (Color online) Integrated elliptic flow ($\langle v_{2}\rangle$)
(left), triangular flow ($\langle v_{3}\rangle$) (middle), and the ratio
$(\langle v_{3}\rangle/\langle v_{2}\rangle)$ (right) as a function of
centrality for both Woods-Saxon and $\alpha$-cluster type nuclear density
profiles in O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV from AMPT model.

###  III.2 Normalized symmetric cumulants NSC(n,m)

To quantify the positive or negative correlations between eccentricity and
triangularity for different nuclear density profiles with respect to collision
centrality, we define normalized symmetric cumulants coefficient NSC(n,m),
which is given by Dasgupta:2020orj :

$\displaystyle\rm
NSC(n,m)=\frac{\langle{\varepsilon_{n}}^{2}{\varepsilon_{m}}^{2}\rangle-\langle{\varepsilon_{n}}^{2}\rangle\langle{\varepsilon_{m}}^{2}\rangle}{\langle\varepsilon_{n}^{2}\rangle\langle\varepsilon_{m}^{2}\rangle}.$
(11)

Figure 3 shows the normalized symmetric cumulants coefficient as a function of
centrality for both Woods-Saxon and $\alpha-$clustered cases in O-O collisions
at $\sqrt{s_{\rm NN}}=7$ TeV in AMPT string melting model. The negative
coefficient values represent the anti-correlation between the two variables.
We see a negative correlation for the $\alpha$-clustered case up to mid-
central (20–30%). This suggests that there is an anti-correlation between
$\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$ only in the case
of $\alpha$-clustering. However, we got positive correlations for the Woods-
Saxon density profile for all centralities. It should be noted that several
recent studies investigate the properties of the initial-state eccentricities
and final hadron flow observables from the collisions of clustered carbon and
heavy ions at various beam energies in the event-by-event framework
Dasgupta:2020orj ; Broniowski:2013dia .

###  III.3 Elliptic flow and triangular flow

Figure 4 shows the $p_{\rm T}$-integrated elliptic (left) and triangular flow
(middle) and the ratio of triangular to the elliptic flow coefficient (right)
as a function of collision centrality in O-O collisions at $\sqrt{s_{\rm NN}}$
= 7 TeV for both Woods-Saxon density profile and $\alpha$-clustered structure
using AMPT. The elliptic flow for the Woods-Saxon density profile does not
have a strong centrality dependence, whereas the $\alpha$-clustered structure
is observed to have significant centrality dependence. Unlike the Woods-Saxon
profile, where the elliptic flow is finite yet almost flat with centrality, in
$\alpha$-clustered structure, the elliptic flow value increases as one moves
initially from central to midcentral collisions, and then attains a maximum
around (20-30)% centrality class. Thereafter, the value decreases towards the
peripheral collisions. On the other hand, triangular flow for both Woods-Saxon
density profile and $\alpha$-clustered structure have similar trends, i.e.,
maximum at the central collisions and decreases towards the peripheral
collisions. This structure of triangular flow is peculiar to observe,
considering the heavy-ion-like behavior where the value for the triangular
flow peaks at the mid-central collisions. The cause of the peculiar behavior
of elliptic and triangular flow may be due to the fact that the smaller system
size and shorter lifetime of the fireball do not fully help transform the
initial eccentricities to the final-state anisotropic flow coefficients. It is
to be noted that the $\alpha$-clustered structure has a more significant
triangular flow compared to the Woods-Saxon density profile throughout the
centrality classes. The observations reported in Ref Li:2020vrg  also present
higher values of triangular flow for $\alpha$-clustered structure as compared
to the Woods-Saxon density profile. However, Ref. Li:2020vrg  reports lesser
values for both elliptic and triangular flow for both the nuclear density
profiles as they employ a different set of kinematic cuts, i.e., $-0.5<y<0.5$
and $(0.2<p_{\rm T}<3)$ GeV/c. Furthermore, the studies based on the
hydrodynamic simulations in Ref. Ding:2023ibq  show a minimal dependence of
the $\alpha$-clustered structure on the elliptic and triangular flow
coefficients. In the right plot of Fig. 4, where $\langle v_{3}\rangle/\langle
v_{2}\rangle$ as a function of centrality is shown, one observes the ratio to
be below one throughout the centrality classes and for both the density
profiles and as one goes towards the peripheral collisions, the ratio seems to
decrease for both the nuclear profiles. The value of $\langle
v_{3}\rangle/\langle v_{2}\rangle$ for $\alpha$-clustered structure is larger
than the Woods-Saxon density profile towards the most central and peripheral
cases. Interestingly, one observes a sharp hike in the $\langle
v_{3}\rangle/\langle v_{2}\rangle$ value for the most central case, which is
inferred from the right panel of Fig. 1, i.e.,
$\langle\epsilon_{3}\rangle/\langle\epsilon_{2}\rangle$ vs. centrality. This
might be a possible signature of $\alpha$-clustered structure of oxygen nuclei
in O-O collisions which can be verified in future experimental studies.

Figure 5: (Color online) The ratio $\langle
v_{2}\rangle/\langle\epsilon_{2}\rangle$ (top) and $\langle
v_{3}\rangle/\langle\epsilon_{3}\rangle$ (bottom) for O-O collisions at
$\sqrt{s_{\rm NN}}=7$ TeV for both Woods-Saxon and $\alpha$-cluster type
nuclear density profiles in AMPT.  Figure 6: (Color online) Two-particle
azimuthal correlation function for $\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$ in
the most central O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using Woods-Saxon
and $\alpha$-cluster type nuclear density profiles.

Figure 7: (Color online) Transverse momentum $(p_{\rm T})$ dependence of
$v_{2}(p_{\rm T})$ for $\pi^{\pm}$, $\rm K^{\pm}$, and $\rm p+\bar{p}$ in O-O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV. Results include Woods-Saxon and
$\alpha$-cluster type nuclear charge density profiles for the oxygen nucleus.

In Fig. 5, $\langle v_{2}\rangle/\langle\epsilon_{2}\rangle$ (top) and
$\langle v_{3}\rangle/\langle\epsilon_{3}\rangle$ (bottom) as a function of
centrality for $\alpha$-clustered structure and Woods-Saxon density profile
for O-O collisions at $\sqrt{s_{\rm NN}}$ = 7 TeV are shown. In Fig. 5,
$\langle v_{2}\rangle/\langle\epsilon_{2}\rangle$ and $\langle
v_{3}\rangle/\langle\epsilon_{3}\rangle$ is observed to be decreasing towards
the peripheral collisions for both the nuclear profiles; however, both these
ratios for the $\alpha$-clustered structure are larger as compared to the
Woods-Saxon density profile. Both $\langle
v_{2}\rangle/\langle\epsilon_{2}\rangle$ and $\langle
v_{3}\rangle/\langle\epsilon_{3}\rangle$ tell about the effect of the medium
on the evolution of the flow coefficients, i.e., $\langle v_{2}\rangle$ and
$\langle v_{3}\rangle$ from initial eccentricities, i.e.,
$\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$, respectively.
As discussed by the authors in Ref. Schenke:2011bn , it is known that
anisotropic flow coefficients of different order are affected differently by
the medium formed, and as the order of the flow coefficients increase, their
sensitivity to the viscosity of the medium increase. Thus the observed
enhanced values of $\langle v_{2}\rangle/\langle\epsilon_{2}\rangle$ and
$\langle v_{3}\rangle/\langle\epsilon_{3}\rangle$ for $\alpha$-clustered
structure compared to Woods-Saxon density profile may be attributed to a
longer duration of the partonic or hadronic phase of the collision system.

Figure 8: (Color online) Transverse kinetic energy dependence of constituent-
quark scaling of $v_{2}$ for $\pi^{\pm}$, $\rm K^{\pm}$, and $\rm p+\bar{p}$
in O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV.

###  III.4 Elliptic flow of light-flavor hadrons and NCQ scaling

Figure 6 shows the two-particle azimuthal correlation function
($C(\Delta\phi)$) for $\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$ in the most
central O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV in the relative azimuthal
angle $\Delta\phi\in[-\pi/2,3\pi/2]$. The blue dots and the red triangles
represent the cases for the nucleus having the Woods-Saxon and tetrahedral
$\alpha$-cluster density profiles respectively. The correlation function is
constructed in the transverse momentum range, $0.5<p_{\rm T}^{\rm a},p_{\rm
T}^{\rm b}<5.0$ GeV/c, in the relative pseudorapidity gap
$1.0<|\Delta\eta|<4.8$. This pseudorapidity gap ensures the removal of short-
range resonance decays and mini-jets contributing to the non-flow effects. The
magnitude of the peak of the correlation function is related to the magnitude
of the anisotropic flow coefficients. Both the density profiles show similar
magnitudes of peaks at the near side ($\Delta\phi\simeq 0$). However, for the
case with $\alpha$-cluster, there is an away-side ($\Delta\phi\simeq\pi$)
broadening and suppression in the two-particle azimuthal correlation function.
This effect gets more pronounced as one moves from pion to kaon and then to
proton. This away-side valley may arise due to the more violent interactions
among the partons caused due to the more compact and denser fireball created
in nuclear collisions having $\alpha$-clusters, which also results in higher
multiplicity than the Woods-Saxon case in similar centrality bins
Behera:2021zhi . The presence of two peaks on the away side adds to this
understanding as it leads to an enhanced contribution to the triangular flow
Mallick:2021hcs ; Prasad:2022zbr . In short, by comparing the $C(\Delta\phi)$
distributions of an ordinary Woods-Saxon nucleus with the $\alpha$-clustered
nucleus, one can observe a dependence of the azimuthal correlation function on
the initial density profile of the nucleus. These results are in line with the
observations reported in Ref. Wang:2021ghq . It can be noted that there might
be residual jetlike correlations leading to the away-side signal suppression.
Proton being the massive one, shows a relatively higher suppression in the
medium than kaon and pion.

Figure 7 shows centrality dependence of single particle elliptic flow
coefficients for $\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$ in O-O collisions at
$\sqrt{s_{\rm NN}}=7$ TeV, for Woods-Saxon and $\alpha$-clustered density
profiles. Three centrality bins are chosen for this study, the most central
(0–5)%, intermediate (20–30)%, and noncentral (40–50)%. For the Woods-Saxon
case, there is a very weak dependency of $v_{2}(p_{\rm T})$ on centrality for
the three particle types. But for the $\alpha$-clustered case, in (20–30)%
centrality, there is a higher $v_{2}(p_{\rm T})$ as compared to the other
centrality bins. In the Woods-Saxon case, we argue that the smaller system
size does not allow much variation in $v_{2}$ as a function of centrality,
irrespective of an increasing $\epsilon_{2}$; however, for the
$\alpha$-cluster case, the more compact geometry tends to produce
comparatively a denser medium, and thus the variation of $v_{2}$ with respect
to centrality comes into picture. Now moving onto the particle types, at low
$p_{\rm T}$, there is a distinct mass ordering in the elliptic flow of
$\pi^{\pm}$, $K^{\pm}$, and $p+\bar{p}$. This is understood to have originated
from the competing effects of radial (symmetric) flow and anisotropic flow. In
the intermediate $p_{\rm T}$, the baryon-meson flow separation occurs, with
baryon $v_{2}$ being greater than that of the meson. This comes into existence
due to the quark coalescence mechanism of hadronization embedded in the AMPT
string melting model.

Figure 8 shows the centrality dependence of $v_{2}(p_{\rm T}^{a}$)/$n_{q}$
scaling as a function of $(m_{\rm T}-m_{0})/n_{q}$ for $\pi^{\pm}$, $K^{\pm}$
and $p+\bar{p}$ in O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV for both Woods-
Saxon and $\alpha$-cluster type nuclear density profiles. Here, $n_{q}=2$ for
mesons, $n_{q}=3$ for baryons, and the transverse kinetic energy, $KE_{\rm
T}=(m_{\rm T}-m_{0})$, where $m_{\rm T}$ is the transverse mass and $m_{0}$ is
the rest mass of the particle. These plots quantitatively show the elliptic
flow of the constituent quarks as a function of their transverse kinetic
energy. As discussed earlier, within the AMPT framework, in Pb-Pb collisions
at the LHC energies, the NCQ scaling is violated. However, at the same energy
in Si-Si collision system, the NCQ scaling is found to be valid. In O-O
collisions, which is an even smaller system, the scaling is valid for all
centrality classes irrespective of the Woods-Saxon or $\alpha$-clustered type
nucleus. Thus, the presence of $\alpha$-cluster geometry does not seem to play
a role in the NCQ-scaling behaviour. However, the away-side broadening seen in
Fig. 6, seems to have been influenced from the early stages of the collisions
due to the change in the nuclear density profiles leading to a more dense and
compact system formation in the presence of $\alpha$-cluster type nucleus.

##  IV Summary

In this paper, we have investigated the effect of Woods-Saxon and
$\alpha$-clustered nuclear geometry on the eccentricity and triangularity
along with their correlations, elliptic flow, triangularity flow, and NCQ
scaling in O-O collisions at $\sqrt{s_{\rm NN}}=7$ TeV in the framework of a
multiphase transport model. The key findings are summarized below:

  1. (i)

Eccentricity and triangularity are found to vary with a change in the density
profiles. However, the effects are more pronounced in the most central case,
where the initial state has more triangularity than eccentricity for an
$\alpha$-clustered oxygen nucleus as compared to the normal Woods-Saxon type
distribution.

  2. (ii)

Employing the normalized symmetric cumulants, we observe that the strength of
the correlation between eccentricity and triangularity for the Woods-Saxon
density profile is more than for the $\alpha$-clustered structure. Also, the
appearance of negative NSC (2,3) value for the $\alpha-$clustered nucleus in
the most central cases is observed.

  3. (iii)

In the Woods-Saxon type nucleus, the elliptic flow is found to depend weakly
on the centrality of the collision. However, in the $\alpha-$clustered
nucleus, the elliptic flow increases from central to mid-central collisions
and then decreases while moving from mid to peripheral collisions.

  4. (iv)

We report an enhancement in the $\langle v_{3}\rangle/\langle v_{2}\rangle$
towards the most central collisions for the $\alpha-$clustered nucleus than
the Woods-Saxon case.

  5. (v)

The two-particle azimuthal correlation function $[C(\Delta\phi)]$ of the
identified particles shows an away-side broadening for the $\alpha$-clustered
type nucleus. This hints towards a denser and more compact system formation in
the $\alpha$-clustered nucleus.

  6. (vi)

The NCQ scaling is valid for all centrality classes for both Woods-Saxon and
$\alpha-$clustered type of nucleus. This observation is crucial as it hints
towards the existence of a deconfined partonic medium in O-O collisions at
$\sqrt{s_{\rm NN}}=7$ TeV and the appearance of partonic collectivity.

It would be interesting to compare these findings to experimental observations
when experimental data are available in order to determine the density profile
of the oxygen nucleus that is best suited to describe ultra-relativistic
nuclear collisions. Although probing the nuclear density profile is a matter
of low-energy nuclear scattering experiments, some of the observables in TeV
nuclear collisions may be sensitive to the nuclear density profiles. In this
study, we report a few such observables in heavy-ion collisions which could be
sensitive to the nuclear density profiles and should be studied in
experimental data.

## Acknowledgements

D.B. acknowledges the financial support from CSIR, the Government of India.
S.P. acknowledges the financial support from UGC, the Government of India. R.
S. sincerely acknowledges the DAE-DST, Government of India funding under the
Mega-Science Project – “Indian participation in the ALICE experiment at CERN”
bearing Project No. SR/MF/PS-02/2021-IITI (E-37123). The authors gratefully
acknowledge the usage of resources of the LHC grid Tier-3 computing facility
at IIT Indore.

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# A characterization result of cofinite local cohomology modules

Moharram Aghapournahr  Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran<EMAIL_ADDRESS>and  Leif
Melkersson  Department of Mathematics, University of Lund
S-221 00 Lund
Sweden<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$
an arbitrary $R$-module and $N$ a finite $R$-module. We prove that [23,
Theorem 2.1] and [2, Proposition 3.3 (i)$\Leftrightarrow$(ii)] are true for
any Serre subcategory of $R$-modules. We also prove a characterization theorem
for $\operatorname{H}_{\mathfrak{a}}^{i}(M)$ and
$\operatorname{H}_{\mathfrak{a}}^{i}(N,M)$ to be $\mathfrak{a}$-cofinite for
all $i$, whenever one of the following cases holds: (a)
$\operatorname{ara}(\mathfrak{a})\leq 1$, (b) $\dim R/\mathfrak{a}\leq 1$ or
(c) $\dim R\leq 2$. In the end we study Artinianness and Artinian
$\mathfrak{a}$-cofiniteness of local cohomology modules.

###### Key words and phrases:

Local cohomology, ${\rm FD_{<n}}$ modules, cofinite modules, weakly Laskerian
modules

###### 2010 Mathematics Subject Classification:

13D45, 13E05, 14B15.

##  1\. Introduction

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

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

Hartshorne in [13] defined a module $M$ to be $\mathfrak{a}$–cofinite if
$\operatorname{Supp}_{R}(M)\subseteq\operatorname{V}(\mathfrak{a})$ and
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite for all $i\geq 0$. He
also asked the following question:

Hartshorne’s question: Let $M$ be a finite $R$-module and $\mathfrak{a}$ be an
ideal of $R$. When are $\operatorname{H}^{i}_{\mathfrak{a}}(M)$
$\mathfrak{a}$–cofinite for all $i\geq 0$?
The answer is negative in general, see [13] for a counterexample, but it is
true in the following cases:

######  Theorem 1.1.

Let $M$ be a finite $R$-module and suppose one of the following cases holds:

  * (a)

$\operatorname{cd}(\mathfrak{a},R)\leq 1$ or $\dim R\leq 2$. See [23].

  * (b)

$\dim R/{\mathfrak{a}}\leq 1$. See [5].

Then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite for
all $i\geq 0$.

Note that $\operatorname{cd}(\mathfrak{a},R)\leq 1$ (the cohomological
dimension of $\mathfrak{a}$ in $R$) is the smallest integer $n$ such that the
local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(S)$ are zero for
all $R$–modules $S$ and for all $i>n$.

The generalized local cohomology module

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

for all $R$–module $M$ and $N$ was introduced by Herzog in [16]. Clearly it is
a generalization of ordinary local cohomology module. In [25] Yassemi asked
whether the Hartshorne’s question hold for generalized local cohomology. It is
answered in the following cases:

######  Theorem 1.2.

Let $M,N$ be two finite $R$-modules and suppose one of the following cases
holds:

  * (a)

$\operatorname{cd}(\mathfrak{a},R)\leq 1$ or $\dim R\leq 2$. See [20].

  * (b)

$\dim R/{\mathfrak{a}}\leq 1$. See [10].

Then $\operatorname{H}^{i}_{\mathfrak{a}}(N,M)$ is $\mathfrak{a}$–cofinite for
all $i\geq 0$.

Recall that an arbitrary $R$-module $X$ is said to be weakly Laskerian if the
set of associated prime ideals of any quotient module of $X$ is finite (see
[11]). Recently, in [17], the author introduced the class of extension modules
of finitely generated modules by the class of all modules of finite support
and named it ${\rm FSF}$ modules. Bahmanpour in [3, Theorem 3.3] proved that
over a Noetherian ring $R$ an $R$-module $X$ is weakly Laskerian if and only
if it is ${\rm FSF}$.

Recall too that an arbitrary $R$-module $X$ is said to be an
$\operatorname{FD}_{<n}$ $R$-module if there exists a finite submodule
$X^{\prime}$ of $X$ such that $\dim_{R}(X/X^{\prime})<n$ (see [1]). Recall
also that a class of $R$-modules is said to be a Serre subcategory of the
category of $R$-modules, when it is closed under taking submodules, quotients
and extensions.

The second author of present paper in [23, Theorem 2.1] proved that for any
ideal $\mathfrak{a}$ of $R$ and any $R$-module $M$, the $R$-module
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite for all $i$ if and
only if the $R$-module $\operatorname{Tor}^{R}_{i}(R/\mathfrak{a},M)$ is
finite for all $i$ if and only if Koszul cohomology
$\operatorname{H}^{i}(x_{1}\dots,x_{s};M)$ is finite for all $i$ where
$\mathfrak{a}=(x_{1},\dots,x_{s})$. Our main result in Section 2 is in this
direction. More precisely, we prove the following result that show [23,
Theorem 2.1] and [2, Proposition 3.3 (i)$\Leftrightarrow$(ii)] are true for
any Serre subcategory of modules:

######  Theorem 1.3.

Let $\mathcal{S}$ be a Serresubcategory of the category of modules over the
Noetherian ring $R$ and let $\mathfrak{a}=(x_{1},\dots,x_{s})$ be an ideal of
$R$. Given an $R$-module $M$ the following conditions are equivalent:

  * (i)

$\operatorname{Ext}_{R}^{i}(R/{\mathfrak{a}},M)$ belongs to $\mathcal{S}$ for
all $i$.

  * (ii)

$\operatorname{Tor}_{i}^{R}(R/{\mathfrak{a}},M)$ belongs to $\mathcal{S}$ for
all $i$.

  * (iii)

$\operatorname{H}^{i}(x_{1}\dots,x_{s};M)$ belongs to $\mathcal{S}$ for all
$i$.

  * (iv)

$\operatorname{H}_{i}(x_{1},\dots,x_{s};M)$ belongs to $\mathcal{S}$ for all
$i$.

In section 3, we study relationship between cofiniteness of local cohomology
and generalized local cohomology modules and we prove that the cofiniteness of
these two kinds of local cohomolgy under the following assumptions depends to
each other. More generally we prove a characterization result (see Corollary
3.8), in the cases (a) $\operatorname{ara}(\mathfrak{a})\leq 1$, (b) $\dim
R/\mathfrak{a}\leq 1$ or (c) $\dim R\leq 2$ for
$\operatorname{H}_{\mathfrak{a}}^{i}(M)$ and
$\operatorname{H}_{\mathfrak{a}}^{i}(N,M)$ to be $\mathfrak{a}$-cofinite for
all $i$.

One of the important question in the study of local cohomology modules is
determine when the local cohomology is Artinian (see [14])? In this direction,
in Section 4, we study the Artinianness and Artinian
$\mathfrak{a}$-cofiniteness of local cohomology modules from upper bound when
$\dim R/{\mathfrak{a}}=1$. More precisely, we show that:

######  Theorem 1.4.

Let $\mathfrak{a}$ be an ideals of a Noetherian ring $R$, with $\dim
R/{\mathfrak{a}}=1$. For an $R$-module $M$ and integer an $r$ the following
conditions are equivalent;

  * (i)

$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$
for each $i>r$.

  * (ii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an Artinian $R$-module for each
$i>r$.

  * (iii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an Artinian $\mathfrak{a}$-cofinte
$R$-module for each $i>r$.

Throughout this paper, $R$ will always be a commutative Noetherian ring with
non-zero identity and $\mathfrak{a}$ will be an ideal of $R$. We denote
$\\{\mathfrak{p}\in{\rm Spec}\,R:\,\mathfrak{p}\supseteq\mathfrak{a}\\}$ by
$\operatorname{V}(\mathfrak{a})$. For any unexplained notation and terminology
we refer the reader to [6], [7] and [21].

##  2\. Equivalence of $\operatorname{Ext}$ and $\operatorname{Tor}$ in Serre
subcategories

We begin this section with a useful remark that we need in the proof of main
result of this section.

######  Remark 2.1.

Observe that if $\mathcal{S}$ is a Serre subcategory of the category of
$R$-modules then for any $R$-module $M$, such that the submodule
$0\underset{M}{:}\mathfrak{a}$ belongs to $\mathcal{S}$, then for each $l$
also $0\underset{M}{:}\mathfrak{a}^{l}$ belongs to $\mathcal{S}$ and if the
quotient module $M/{{\mathfrak{a}}M}$ belongs to $\mathcal{S}$, then
$M/{{\mathfrak{a}}^{l}M}$ belong to $\mathcal{S}$ for all $l$. This follows by
induction on $l$. Let $x_{1},\dots,x_{s}$ be generators of $\mathfrak{a}$. For
the first assertion we use the map
${0\underset{M}{:}\mathfrak{a}^{l+1}}\to{(0\underset{M}{:}\mathfrak{a}^{l})_{1}^{s}}$,
defined by $u\mapsto(x_{i}u)_{1}^{s}$. This map has the kernel
$0\underset{M}{:}{\mathfrak{a}}$. For the second assertion use the map
$(M/{\mathfrak{a}^{l}M})_{1}^{s}\to M/{\mathfrak{a}^{l+1}M}$ defined by
$({\overline{u}}_{i})_{1}^{s}\mapsto{\sum_{1}^{s}\overline{x_{i}u_{i}}}$,
whose cokernel is isomorphic to $M/{\mathfrak{a}M}$.

######  Theorem 2.2.

Let $\mathcal{S}$ be a Serresubcategory of the category of modules over the
Noetherian ring $R$ and let $\mathfrak{a}=(x_{1},\dots,x_{s})$ be an ideal of
$R$. Given an $R$-module $M$ the following conditions are equivalent:

  * (i)

$\operatorname{Ext}_{R}^{i}(R/{\mathfrak{a}},M)$ belongs to $\mathcal{S}$ for
all $i$.

  * (ii)

$\operatorname{Tor}_{i}^{R}(R/{\mathfrak{a}},M)$ belongs to $\mathcal{S}$ for
all $i$.

  * (iii)

$\operatorname{H}^{i}(x_{1}\dots,x_{s};M)$ belongs to $\mathcal{S}$ for all
$i$.

  * (iv)

$\operatorname{H}_{i}(x_{1},\dots,x_{s};M)$ belongs to $\mathcal{S}$ for all
$i$.

The equivalence between (iii) and (iv) follows from the symmetry
$\operatorname{H}^{i}(x_{1},\dots,x_{s};M)\cong\operatorname{H}_{s-i}(x_{1},\dots,x_{s};M)$
for $0\leq i\leq s$. The proof of the other equivalences is provided by a
series of lemmas.

######  Lemma 2.3.

Let $\mathfrak{a}$ be an ideal of $R$ and let $\mathcal{C}$ be a class of
$R$-modules with the follwing three properties:

  * (1)

$\mathcal{S}\subset\mathcal{C}$.

  * (2)

$0\underset{M}{:}\mathfrak{a}\in\mathcal{S}$ for each $M\in\mathcal{C}$.

  * (3)

If $0\to{M^{\prime}}\to M\to M^{\prime\prime}\to 0$ is a short exact sequence
with $M^{\prime}$ and $M$ in $\mathcal{C}$, then also $M^{\prime\prime}$ is in
$\mathcal{C}$.

Let $X^{\bullet}:0\to X^{0}\to X^{1}\to X^{2}\to\dots$ be a cochain complex
such that for every $i$, $X^{i}\in\mathcal{C}$ and
$\mathfrak{a}\subset\sqrt{\operatorname{Ann}\operatorname{H}^{i}(X^{\bullet})}$.
Then $\operatorname{H}^{i}(X^{\bullet})\in\mathcal{S}$ for all $i$.

In order to prove Lemma 2.3 we prove two lemmas. In those $\mathcal{C}$
satisfies the conditions in Lemma 2.3. These lemmas are also of some
independent interest.

######  Lemma 2.4.

Let $f:M\to N$ be $R$-linear, where both $M$ and $N$ are in $\mathcal{C}$.
Assume that $\mathfrak{a}^{l}\operatorname{Ker}f=0$ for some $l$. Then
$\operatorname{Ker}f$ belongs to $\mathcal{S}$ (hence also to $\mathcal{C}$)
and $\operatorname{Coker}f$ belongs to $\mathcal{C}$.

###### Proof.

Since $\operatorname{Ker}f\subset{0\underset{M}{:}\mathfrak{a}^{l}}$, which
belongs to $\mathcal{S}$ by remark 2.1 and the second condition in Lemma 2.3.
Using the third condition in Lemma 2.3 applied to the exact sequences
$0\to\operatorname{Ker}f\to M\to\operatorname{Im}f\to 0$ and
$0\to\operatorname{Im}f\to N\to\operatorname{Coker}f\to 0$ we first get that
$\operatorname{Im}f\in\mathcal{C}$ (Note that we assumed that
$\mathcal{S}\subset\mathcal{C}$) and then that
$\operatorname{Coker}f\in\mathcal{C}$. ∎

######  Lemma 2.5.

Let $0\to X^{0}\overset{d^{0}}{\to}X^{1}\to\dots\to
X^{n-1}\overset{d^{n-1}}{\to}X^{n}\overset{d^{n}}{\to}X^{n+1}$ be a cocomplex,
where each $X^{i}$ belongs to $\mathcal{C}$. Suppose that there is some $l$,
such that $\mathfrak{a}^{l}\operatorname{H}^{i}$ are in $\mathcal{S}$ for
$i=0,\dots,n$. Then the cohomology modules $\operatorname{H}^{i}$ are in
$\mathcal{S}$ for $i=0,\dots,n$ and $\operatorname{Coker}d^{n}\in\mathcal{C}$.

###### Proof.

We use induction on $n$. The case $n=0$ follows from Lemma 2.4. Let $n>0$ and
assume that we have proved the result for $n-1$. The map $d^{n}$ induces a map
$f:\operatorname{Coker}{d^{n-1}}\to X^{n+1}$. Here both modules belong to
$\mathcal{C}$. The first one by the induction hypothesis and the second one by
the hypothesis in the lemma. Since $\operatorname{Ker}f=\operatorname{H}^{n}$
we can apply Lemma 2.4 to prove the case $n$. ∎

Lemma 2.3 now follows from Lemma 2.5.

######  Lemma 2.6.

Let $\mathcal{C}$ be a class of $R$-modules satisfying:

  * (1)

$\mathcal{S}\subset\mathcal{C}$.

  * (2)

$M/{\mathfrak{a}M}\in\mathcal{S}$ for all $M\in\mathcal{C}$.

  * (3)

If $0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$ is a short exact with $M$
and $M^{\prime\prime}$ in $\mathcal{C}$, then also $M^{\prime}\in\mathcal{C}$.

Let $X_{\bullet}:\dots\to
X_{2}\overset{d_{1}}{\to}X_{1}\overset{d_{0}}{\to}X_{0}\to 0$ be a chain
complex such that for all $i$, $X_{i}\in\mathcal{C}$ and
$\mathfrak{a}\subset\operatorname{Ann}\sqrt{\operatorname{H}_{i}(X_{\bullet})}$.
Then the homology modules $\operatorname{H}_{i}(X_{\bullet})$ belong to
$\mathcal{S}$ for all $i$.

In the next two lemmas $\mathcal{C}$ is as in Lemma 2.6.

######  Lemma 2.7.

Let $f:M\to N$ be an $R$-linear map, where $M$ and $N$ both are in
$\mathcal{C}$. Assume that ${\mathfrak{a}}^{l}\operatorname{Coker}f=0$ for
some $l$. Then $\operatorname{Ker}f$ and $\operatorname{Coker}f$ belong to
$\mathcal{C}$.

###### Proof.

Since $\operatorname{Coker}f$ is a homomorhic image of
$N/{\mathfrak{a}}^{l}N$, it follows from (2) in Lemma 2.6 and remark 2.1 that
$\operatorname{Coker}f\in\mathcal{S}$ and hence by (1) in Lemma 2.6 that
$\operatorname{Coker}f\in\mathcal{C}$. By property (3) in Lemma 2.6 using the
exact sequences $0\to\operatorname{Im}f\to N\to\operatorname{Coker}f\to 0$ and
$0\to\operatorname{Ker}f\to M\to\operatorname{Im}f\to 0$ we first get that
$\operatorname{Im}f$ belongs to $\mathcal{C}$ and then that
$\operatorname{Ker}f$ belongs to $\mathcal{C}$. ∎

######  Lemma 2.8.

Let $X_{n+1}\overset{d_{n}}{\to}X_{n}\to\dots\to
X_{1}\overset{d_{0}}{\to}X_{0}\to 0$ be a complex. Assume that
$X_{i}\in\mathcal{C}$ for all $i=0,\dots,n$ and that there is $l$, such that
${\mathfrak{a}}^{l}$ annihilates the homology modules $\operatorname{H}_{i}$
for $i=0,\dots,n$. Then $\operatorname{Ker}d_{n}$ belongs to $\mathcal{S}$
(hence also to $\mathcal{C}$) and the homology modules $\operatorname{H}_{i}$
belong to $\mathcal{S}$ for $i=0,\dots,n$.

###### Proof.

The proof is by induction on $n$. The case case $n=0$ is just Lemma 2.7.
Assume that $n>0$ and that we know that $\operatorname{Ker}d_{n-1}$ belongs to
$\mathcal{C}$ and that the homology modules $\operatorname{H}_{i}$ belong to
$\mathcal{S}$ for $i=0,\dots,n-1$. Consider the map
$f:X_{n+1}\to\operatorname{Ker}d_{n-1}$ induced by $d_{n}$. Since
$\operatorname{Ker}f=\operatorname{Ker}d_{n}$ and
$\operatorname{Coker}f=\operatorname{H}_{n-1}$ we get by 2.7 that
$\operatorname{Ker}d_{n}\in\mathcal{C}$ and that
$\operatorname{H}_{n-1}\in\mathcal{S}$. ∎

We now prove the equivalences in Theorem 2.2.

###### Proof of Theorem 2.2.

We have already noticed that the conditions (iii) and (iv) are equivalent. Let
${\mathcal{C}}_{i}$, ${\mathcal{C}}_{ii}$, ${\mathcal{C}}_{iii}$ and
${\mathcal{C}_{iv}}$ be the classes of modules which satisfy the respective
condition in Theorem 2.2. We have already noticed that $\mathcal{C}_{iii}$ and
$\mathcal{C}_{iv}$ are equal. Let $F_{\bullet}\to R/\mathfrak{a}$ be a
resolution of $R/\mathfrak{a}$ consisting of finite free modules and let
$\operatorname{K}_{\bullet}=\operatorname{K}_{\bullet}(x_{1},\dots,x_{s})$ be
the Koszul complex of the sequence $x_{1},\dots,x_{s}$. The classes
$\mathcal{C}_{i}$ and $\mathcal{C}_{iii}$ both satisfy the three conditions in
2.3. If $M$ is in $\mathcal{C}_{i}$ then by 2.3 the cohomology modules of the
cochain complex
$X^{\bullet}=\operatorname{Hom}_{R}(\operatorname{K}_{\bullet},M)$ are in
$\mathcal{S}$ and therefore $M$ is in $\mathcal{C}_{iii}$. Conversely let $M$
be in $\mathcal{C}_{iii}$. Consider the cochain complex
$X^{\bullet}=\operatorname{Hom}_{R}(F_{\bullet},M)$. We get using Lemma 2.3,
that $M$ belongs to $\mathcal{C}_{i}$. To prove that the classes
$\mathcal{C}_{ii}$ and $\mathcal{C}_{iv}$ are the same we consider the
complexes $\operatorname{K}_{\bullet}\otimes M$ and $F_{\bullet}\otimes M$ and
use Lemma 2.6. ∎

######  Theorem 2.9.

For each $R$-module $M$ which satisfies the equivalent conditions of Theorem
2.2 and each finite $R$-module $N$ with support in
$\operatorname{V}(\mathfrak{a})$, the modules
$\operatorname{Ext}_{R}^{i}(N,M)$ and $\operatorname{Tor}_{i}^{R}(N,M)$ belong
to $\mathcal{S}$.

###### Proof.

Let $\mathcal{C}$ be the class of $R$-modules which satidfy the equivalent
conditions in Theorem 2.2. Apply Lemma 2.3 and Lemma 2.6 to the complexes
$X^{\bullet}=\operatorname{Hom}_{R}(F_{\bullet},M)$ and
$X_{\bullet}=F_{\bullet}\otimes M$. ∎

##  3\. Cofiniteness of local cohomology and generalized local cohomology

The following two result generalize and decrease the assumptions on $M$ in [1,
Theorem 3.4 (i) and Corollary 3.5 (i)].

######  Theorem 3.1.

Let $R$ be a Noetherian ring and $\mathfrak{a}$ an ideal of $R$. Let
$t\in\mathbb{N}_{0}$ be an integer and $M$ an $R$-module such that
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ are finite for all $i\leq t+1$.
Let the $R$-modules $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ are ${\rm
FD_{<2}}$ $R$-modules for all $i<t$. Then, the $R$-modules
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ are $\mathfrak{a}$-cofinite for all
$i<t$.

###### Proof.

We use induction on $t$. The exact sequence

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

induces the following exact sequence:

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

Since
${\operatorname{Hom}}_{R}(R/\mathfrak{a},M/\operatorname{\Gamma}_{\mathfrak{a}}(M))=0$
so ${\rm Hom}_{R}(R/\mathfrak{a},\operatorname{\Gamma}_{\mathfrak{a}}(M))$ and
${\rm Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{\Gamma}_{\mathfrak{a}}(M))$
are finite. Assume inductively that $t>0$ and that we have established the
result for non-negative integers smaller than $t$. Using inductive assumption,
the rest of proof is the same as [1, Theorem 3.4 (i)].∎

######  Corollary 3.2.

Let $R$ be a Noetherian ring and $\mathfrak{a}$ an ideal of $R$. Let $M$ be an
$R$-module such that the $R$-module
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite for all $i$ and the
$R$-module $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is ${\rm FD_{<2}}$ (or
weakly Laskerian) $R$-modules for all $i$. Then the $R$-modules
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ are $\mathfrak{a}$-cofinite for all
$i$.

###### Proof.

Follows by Theorem 3.1.∎

The following corollary is a characterization of cofiniteness of local
cohomology modules which are ${\rm FD_{<2}}$ $R$-module.

######  Corollary 3.3.

Let $R$ be a Noetherian ring and $\mathfrak{a}$ an ideal of $R$. Let $M$ be an
$R$-module such that the $R$-modules $\operatorname{H}^{i}_{\mathfrak{a}}(M)$
are ${\rm FD_{<2}}$ (or weakly Laskerian) $R$-modules for all $i$. Then, the
following conditions are equivalent:

(i) The $R$-module $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite
for all $i$.

(ii) The $R$-module $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is
$\mathfrak{a}$-cofinite for all $i$.

###### Proof.

(i)$\Rightarrow$(ii) Follows by Corollary 3.2.

(ii)$\Rightarrow$(i) It follows by [23, Proposition 3.9].∎

######  Theorem 3.4.

Let $M$ be an $R$-module and suppose one of the following cases holds:

  * (a)

$\operatorname{ara}(\mathfrak{a})\leq 1$;

  * (b)

$\dim R/\mathfrak{a}\leq 1$;

  * (c)

$\dim R\leq 2$.

Then, $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-cofinite for
all $i$ if and only if $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is a
finite $R$-module for all $i$.

###### Proof.

The case (c) follows by Theorem [23, Theorem 7.10].

In the cases (a) and (b), suppose $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is
$\mathfrak{a}$-cofinite for all $i$. It follows from [23, Corollary 3.10] that
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is a finite $R$-module for all
$i$.

To prove the converse, in the case (b), it follows by [22, Theorem 2.10]. In
the case (a), use [4, Theorem 3.3] and the method of proof of [22, Theorem
2.10]. ∎

######  Lemma 3.5.

Let $X$ be a finite $R$–module, $M$ be an arbitrary $R$–module. Then the
following statements hold true.

  * (a)

$\Gamma_{\mathfrak{a}}(X,M)\cong\operatorname{Hom}_{R}(X,\Gamma_{\mathfrak{a}}(M)).$

  * (b)

If
$\operatorname{Supp}_{R}(X)\cap\operatorname{Supp}_{R}(M)\subseteq\operatorname{V}(\mathfrak{a})$,
then
$\operatorname{H}^{i}_{\mathfrak{a}}(X,M)\cong\operatorname{Ext}^{i}_{R}(X,M)$
for all $i$.

###### Proof.

See [24, Lemma 2.5]. ∎

######  Theorem 3.6.

Let $M$ be an $R$-module and suppose one of the following cases holds:

  * (a)

$\operatorname{ara}(\mathfrak{a})\leq 1$;

  * (b)

$\dim R/\mathfrak{a}\leq 1$;

  * (c)

$\dim R\leq 2$.

Then, for any finite $R$-module $N$,
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M)$ is $\mathfrak{a}$-cofinite for all
$i\geq 0$ if and only if $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is
finite $R$-module for all $i\geq 0$.

###### Proof.

First suppose for any finite $R$-module $N$,
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M)$ is $\mathfrak{a}$-cofinite for all
$i\geq 0$. Let $N=R$, then it follows by Theorem 3.4, that
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite $R$-module for all
$i\geq 0$.

To prove the converse we use induction on $i$. Let $i=0$, then it follows by
Lemma 3.5 (a) that

$\operatorname{H}^{0}_{\mathfrak{a}}(N,M)=\Gamma_{\mathfrak{a}}(N,M)\cong\operatorname{Hom}_{R}(N,\Gamma_{\mathfrak{a}}(M)).$

Since $\Gamma_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-cofinite by Theorem 3.4, so
the assertion follows by [4, Theorem 4.1], [22, Corollary 2.7] and [23,
Theorem 7.4], respectively in the cases (a), (b) and (c). Now assume that
$i>0$ and that the claim holds for $i-1$. Since
$\operatorname{\Gamma}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-cofinite and
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite $R$-module for all
$i\geq 0$. Using the short exact sequence

$0\longrightarrow\operatorname{\Gamma}_{\mathfrak{a}}(M)\longrightarrow
M\longrightarrow M/\operatorname{\Gamma}_{\mathfrak{a}}(M)\longrightarrow 0,$

it is easy to see that the $R$-modules
${\operatorname{Ext}}^{i}_{R}(R/{\mathfrak{a}},M/\operatorname{\Gamma}_{\mathfrak{a}}(M))$
are finite for all $i\geq 0$. Now, by applying the derived functor
$\Gamma_{\mathfrak{a}}(N,-)$ to the same short exact sequence and using Lemma
3.5 (b), we obtain the long exact sequence

$\dots\longrightarrow\operatorname{Ext}_{R}^{i}(N,\operatorname{\Gamma}_{\mathfrak{a}}(M))\overset{f_{i}}{\longrightarrow}\operatorname{H}^{i}_{\mathfrak{a}}(N,M)\overset{g_{i}}{\longrightarrow}\operatorname{H}^{i}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))\overset{h_{i}}{\longrightarrow}\operatorname{Ext}_{R}^{i}(N,\operatorname{\Gamma}_{\mathfrak{a}}(M))\overset{f_{i+1}}{\longrightarrow}\operatorname{H}^{i+1}_{\mathfrak{a}}(N,M)\longrightarrow\dots$

which yields short exact sequences

$0\longrightarrow\operatorname{Ker}{f_{i}}\longrightarrow\operatorname{Ext}_{R}^{i}(N,\operatorname{\Gamma}_{\mathfrak{a}}(M))\longrightarrow\operatorname{Im}{f_{i}}\longrightarrow
0,$

$0\longrightarrow\operatorname{Im}{f_{i}}\longrightarrow\operatorname{H}^{i}_{\mathfrak{a}}(N,M)\longrightarrow\operatorname{Im}{g_{i}}\longrightarrow
0$

and

$0\longrightarrow\operatorname{Im}{g_{i}}\longrightarrow\operatorname{H}^{i}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))\longrightarrow\operatorname{Ker}{f_{i+1}}\longrightarrow
0.$

Since $\operatorname{Ext}_{R}^{i}(N,\operatorname{\Gamma}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$-cofinite $R$-module for all $i\geq 0$ by [4, Theorem 4.1],
[22, Corollary 2.7] and [23, Theorem 7.4], respectively in the cases (a), (b)
and (c), it follows by definition that,
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M)$ is $\mathfrak{a}$-cofinite
$R$-module if and only if
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$-cofinite $R$-module for all $i\geq 0$. Therefore it suffices
to show that
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$-cofinite $R$-module for all $i\geq 0$. To this end consider
the exact sequence

$0\longrightarrow M/\operatorname{\Gamma}_{\mathfrak{a}}(M)\longrightarrow
E\longrightarrow L\longrightarrow 0,\,\,\,\,\,\,\,\,(\dagger)$

in which $E$ is an injective $\mathfrak{a}$-torsion free module. Since
$\operatorname{\Gamma}_{\mathfrak{a}}(M/\operatorname{\Gamma}_{\mathfrak{a}}(M))=0=\operatorname{\Gamma}_{\mathfrak{a}}(E)$,
thus $\operatorname{Hom}_{R}(R/\mathfrak{a},E)=0$ and
$\operatorname{\Gamma}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))=0=\operatorname{\Gamma}_{\mathfrak{a}}(N,E)$
by Lemma 3.5 (a). Applying the derived functors of
$\operatorname{Hom}_{R}(R/\mathfrak{a},-)$ and $\Gamma_{\mathfrak{a}}(N,-)$ to
the short exact sequence $(\dagger)$ we obtain, for all $i>0$, the
isomorphisms

$\operatorname{H}^{i-1}_{\mathfrak{a}}(N,L)\cong\operatorname{H}^{i}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))$.

and

${\operatorname{Ext}}^{i-1}_{R}(R/{\mathfrak{a}},L)\cong{\operatorname{Ext}}^{i}_{R}(R/{\mathfrak{a}},M/\operatorname{\Gamma}_{\mathfrak{a}}(M))$.

From what has already been proved, we conclude that
${\operatorname{Ext}}^{i-1}_{R}(R/{\mathfrak{a}},L)$ is finite for all $i>0$.
Hence $\operatorname{H}^{i-1}_{\mathfrak{a}}(N,L)$ is $\mathfrak{a}$-cofinite
by induction hypothesis for all $i>0$, which yields that
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M/\operatorname{\Gamma}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$-cofinite for all $i\geq 0$, this completes the inductive
step. ∎

######  Lemma 3.7.

Let $N$ be a finitely generated $R$-module and $M$ be an arbitrary $R$-module.
Let $t$ be a non-negative integer such that
${\operatorname{Ext}}_{R}^{i}(N,M)$ is finite for all $0\leq i\leq t$. Then
for any finitely generated $R$-module $L$ with
$\operatorname{Supp}_{R}(L)\subseteq\operatorname{Supp}_{R}(N)$,
${\operatorname{Ext}}_{R}^{i}(L,M)$ is finite for all integer $0\leq i\leq t$.

###### Proof.

See [8, Proposition 1]. ∎

The following corollary is our main result of this section which is a
characterization of cofinite local cohomology and generalized local cohomology
modules under the assumptions (a) $\operatorname{ara}(\mathfrak{a})\leq 1$,
(b) $\dim R/\mathfrak{a}\leq 1$ and (c) $\dim R\leq 2$. This corollary shows
that with the above assumptions, cofiniteness of local cohomology and
generalized local cohomology modules are equivalent.

######  Corollary 3.8.

Let $M$ be an $R$-module and suppose one of the following cases holds:

  * (a)

$\operatorname{ara}(\mathfrak{a})\leq 1$;

  * (b)

$\dim R/\mathfrak{a}\leq 1$;

  * (c)

$\dim R\leq 2$.

Then the following conditions are equivalent:

  * (i)

$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite for all $i$.

  * (ii)

$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite for $i=0,1$ in the
cases (a) and (b) (resp. for $i=0,1,2$ in the case (c));

  * (iii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-cofinite for all
$i$;

  * (iv)

$\operatorname{H}^{i}_{\mathfrak{a}}(N,M)$ is $\mathfrak{a}$-cofinite for all
$i$ and for any finite $R$-module $N$;

  * (v)

$\operatorname{Ext}^{i}_{R}(N,M)$ is finite for all $i$ and for any finite
$R$-module $N$ with
$\operatorname{Supp}_{R}(N)\subseteq\operatorname{V}(\mathfrak{a})$;

  * (vi)

$\operatorname{Ext}^{i}_{R}(N,M)$ is finite for all $i$ and for some finite
$R$-module $N$ with
$\operatorname{Supp}_{R}(N)=\operatorname{V}(\mathfrak{a})$;

  * (vii)

$\operatorname{Ext}^{i}_{R}(N,M)$ is finite for $i=0,1$ in the cases (a) and
(b) (resp. for $i=0,1,2$ in the case (c)) and for any finite $R$-module $N$
with $\operatorname{Supp}_{R}(N)\subseteq\operatorname{V}(\mathfrak{a})$;

  * (viii)

$\operatorname{Ext}^{i}_{R}(N,M)$ is finite for $i=0,1$ in the cases (a) and
(b) (resp. for $i=0,1,2$ in the case (c)) and for some finite $R$-module $N$
with $\operatorname{Supp}_{R}(N)=\operatorname{V}(\mathfrak{a})$.

###### Proof.

In order to prove (i)$\Leftrightarrow$(ii), use [4, Theorem 3.3], [22, Theorem
2.3] and [23, Theorem 7.10], respectively in the cases (a), (b) and (c).

(i)$\Leftrightarrow$(iii) follows by Theorem 3.4.

(i)$\Leftrightarrow$(iv) follows by Theorem 3.6.

In order to prove (i)$\Leftrightarrow$(v) and (ii)$\Leftrightarrow$(vii) use
[19, Lemma 1].

(v)$\Rightarrow$(vi) and (vii)$\Rightarrow$(viii) are trivial.

In order to prove (vi)$\Rightarrow$(v) and (viii)$\Rightarrow$(vii), let $L$
be a finitely generated $R$-module with
$\operatorname{Supp}_{R}(L)\subseteq\operatorname{V}(\mathfrak{a})$. Then
$\operatorname{Supp}_{R}(L)\subseteq\operatorname{Supp}_{R}(N)$. Now the
assertion follows by Lemma 3.7. ∎

It is easy to see that
$\operatorname{cd}(\mathfrak{a},R)\leq\operatorname{ara}\mathfrak{a}$ but the
equality is not true in general (see [9, Example 2.3]). We close this section
by offering a question and problem for further research.

Question: Let $R$ be a commutative Noetherian ring with non-zero identity and
$\mathfrak{a}$ an ideal of $R$. Is the charaterization in Corollary 3.8 true,
when we change $\operatorname{ara}\mathfrak{a}\leq 1$ with
$\operatorname{cd}(\mathfrak{a},R)\leq 1$ in the case (a)?

##  4\. Artinianness of local cohomology modules

In general it is not true that when $M$ is a finite $R$-module over the
Noetherian ring $R$, such that the local cohomology modules
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ have support in $\operatorname{Max}R$
for all $i>r$, then they are Artinian $R$-module for all $i>r$. For this see
the example by Hartshorne in [13, page 151]. However we prove that this is
true in the case when $\dim R/{\mathfrak{a}}=1$. To do this we need the
following lemma.

######  Lemma 4.1.

Let $\mathfrak{a}$ be an ideals of a Noetherian ring $R$, such that $\dim
R/{\mathfrak{a}}=1$. Let $M$ be a finite $R$-module and $r$ an integer. Then
$\dim M_{\mathfrak{p}}\leq r$ for each prime ideal $\mathfrak{p}$ minimal over
$\mathfrak{a}$ if and only if
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$
for all $i>r$.

###### Proof.

This is the direct consequence of Grothendieck’s vanishing theorem and the
isomorphism
$\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{p}}\cong\operatorname{H}^{i}_{{\mathfrak{a}}R_{\mathfrak{p}}}(M_{\mathfrak{p}})$
for each prime ideal $\mathfrak{p}$ minimal over $\mathfrak{a}$. ∎

######  Theorem 4.2.

Let $\mathfrak{a}$ be an ideals of a Noetherian ring $R$, with $\dim
R/{\mathfrak{a}}=1$. For an $R$-module $M$ and integer an $r$ the following
conditions are equivalent;

  * (i)

$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$
for each $i>r$.

  * (ii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an Artinian $R$-module for each
$i>r$.

  * (iii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an Artinian $\mathfrak{a}$-cofinte
$R$-module for each $i>r$.

###### Proof.

The implications (iii)$\Rightarrow$(ii)$\Rightarrow$(i) always hold, so let us
assume (i) and prove that (iii) hold. We do induction on $r$. First we note
that if $N$ is a finite $R$-module, such that
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(N))\subseteq\operatorname{Max}(R)$,
then $\operatorname{H}^{i}_{\mathfrak{a}}(N)$ is $\mathfrak{a}$-cofinite
Artinian for all $i$. Let $r=0$ and put
$N=M/\operatorname{\Gamma}_{\mathfrak{a}}(M)$. Then
$\operatorname{H}^{0}_{\mathfrak{a}}(N)=0$ and
$\operatorname{H}^{i}_{\mathfrak{a}}(N)\cong\operatorname{H}^{i}_{\mathfrak{a}}(M)$
for all $i>0$. Hence by the above remark
$\operatorname{H}^{i}_{\mathfrak{a}}(N)$ is $\mathfrak{a}$-cofinite Artinian
for all $i$. It follows that $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is
Artinian and $\mathfrak{a}$-cofinite for all $i>0$.

Next let $r>0$ and put $N=M/\operatorname{\Gamma}_{\mathfrak{a}}(M)$. Then
$\operatorname{H}^{0}_{\mathfrak{a}}(N)=0$ and we can take $x\in\mathfrak{a}$
such that $x$ is $N$-regular.

Then by Lemma 4.1, $\dim(N/xN)_{\mathfrak{p}}\leq r-1$ for each prime ideal
$\mathfrak{p}$ minimal over $\mathfrak{a}$. Applying the induction hypothesis
to $N/xN$, we have that $\operatorname{H}^{i}_{\mathfrak{a}}(N/xN)$ is
$\mathfrak{a}$-cofinite Artinian for all $i>r-1$. Next applying local
cohomology functor to the exact sequence $0\longrightarrow N\longrightarrow
N\longrightarrow N/xN\longrightarrow 0,$ we can conclude that
$(0:_{\operatorname{H}^{i}_{\mathfrak{a}}(N)}x)$ is $\mathfrak{a}$-cofinite
Artinian for all $i>r$. It follows that
$\operatorname{H}^{i}_{\mathfrak{a}}(N)$ is Artinian and
$\mathfrak{a}$-cofinite for all $i>r$. Therefore also
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ , for all $i>r$.

∎

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