††thanks: Corresponding author

# Supernova Neutrinos as a Precise Probe of Nuclear Neutron Skin

Xu-Run Huang  School of Physics and Astronomy, Shanghai Key Laboratory for
Particle Physics and Cosmology, and Key Laboratory for Particle Astrophysics
and Cosmology (MOE), Shanghai Jiao Tong University, Shanghai 200240, China
Lie-Wen Chen<EMAIL_ADDRESS>School of Physics and Astronomy, Shanghai Key
Laboratory for Particle Physics and Cosmology, and Key Laboratory for Particle
Astrophysics and Cosmology (MOE), Shanghai Jiao Tong University, Shanghai
200240, China

###### Abstract

A precise and model-independent determination of the neutron distribution
radius $R_{\rm n}$ and thus the neutron skin thickness $R_{\rm skin}$ of
atomic nuclei is of fundamental importance in nuclear physics, particle
physics and astrophysics but remains a big challenge in terrestrial labs. We
argue that the nearby core-collapse supernova (CCSN) in our Galaxy may render
a neutrino flux with unprecedentedly high luminosity, offering perfect
opportunity to determine the $R_{\rm n}$ and $R_{\rm skin}$ through the
coherent elastic neutrino-nucleus scattering (CE$\nu$NS). We evaluate the
potential of determining the $R_{\rm n}$ of lead (Pb) via CE$\nu$NS with the
nearby CCSN neutrinos in the RES-NOVA project which is designed to hunt CCSN
neutrinos using an array of archaeological Pb based cryogenic detectors. We
find that an ultimate precision of $\sim 0.1\%$ for the $R_{\rm n}$ ($\sim
0.006$ fm for the $R_{\rm skin}$) of Pb can be achieved via RES-NOVA if the
CCSN explosion were to occur at a distance of $\sim 1{\ \rm{kpc}}$ from the
Earth.

_Introduction._ — Neutrons are expected to be distributed more extensively
than protons in heavy neutron-rich nuclei, forming a neutron skin which is
featured quantitatively by the skin thickness $R_{\rm skin}=R_{n}-R_{p}$ where
$R_{n}$ and $R_{p}$ are the (point) neutron and proton rms radii of the
nucleus, respectively. Theoretically, it has been established that the $R_{\rm
skin}$ provides an ideal probe for the density dependence of the symmetry
energy $E_{\rm sym}(\rho)$ Brown (2000); Typel and Brown (2001); Furnstahl
(2002); Yoshida and Sagawa (2004); Chen  _et al._ (2005); Centelles  _et al._
(2009); Chen  _et al._ (2010); Reinhard and Nazarewicz (2010); Roca-Maza  _et
al._ (2011); Agrawal  _et al._ (2012); Zhang and Chen (2013); Mondal  _et al._
(2016); Raduta and Gulminelli (2018); Newton and Crocombe (2021); Lynch and
Tsang (2022), which quantifies the isospin dependent part of the equation of
state (EOS) for isospin asymmetric nuclear matter and plays a critical role in
many issues of nuclear physics and astrophysics Danielewicz  _et al._ (2002);
Lattimer and Prakash (2004); Steiner  _et al._ (2005); Baran  _et al._ (2005);
Li  _et al._ (2008); Horowitz  _et al._ (2014); Gandolfi  _et al._ (2015);
Özel and Freire (2016); Baldo and Burgio (2016); Roca-Maza and Paar (2018);
Drischler  _et al._ (2021); Li  _et al._ (2021).

Experimentally, while the $R_{p}$ can be precisely inferred from its
corresponding charge rms radius $R_{\rm ch}$ which has been measured precisely
via electromagnetic processes Fricke  _et al._ (1995); Angeli and Marinova
(2013), the $R_{n}$ remains elusive since it is usually determined from strong
processes, generally involving in model dependence (see, e.g., Ref. Thiel  _et
al._ (2019)). A clean approach to determine the $R_{n}$ is to measure the
parity-violating asymmetry $A_{\rm PV}$ in the elastic scattering of polarized
electrons from the nucleus since the $A_{\rm PV}$ is particularly sensitive to
the neutron distribution due to its large weak charge compared to the tiny one
of the proton Donnelly  _et al._ (1989); Horowitz  _et al._ (2001). Following
this strategy, the 208Pb radius experiment (PREX-2) Adhikari  _et al._ (2021)
and 48Ca radius experiment (CREX) Adhikari  _et al._ (2022) recently reported
the determination of the $R_{n}$ with a precision of $\sim 1\%$, i.e.,
$R^{208}_{\rm skin}=0.283\pm 0.071~{}\rm{fm}$ for 208Pb Adhikari  _et al._
(2021) and $R^{48}_{\rm skin}=0.121\pm 0.026(\rm exp)\pm 0.024(\rm model)$ fm
for 48Ca ($1\sigma$ uncertainty). Very remarkably, analyses within modern
energy density functionals Reinhard  _et al._ (2022); Yüksel and Paar (2022);
Zhang and Chen (2022) conclude a tension between the CREX and PREX-2 results,
with the former favoring a very soft $E_{\rm sym}(\rho)$ while the latter a
very stiff one, calling for further critical theoretical and experimental
investigations. Especially, the Bayesian analysis Zhang and Chen (2022)
suggests that a higher precision for the $R_{n}$ of 208Pb is of particular
importance to address this issue. The Mainz Radius Experiment (MREX) Becker
_et al._ (2018) is expected to shrink the uncertainty by a factor of two with
a precision of $0.5\%$ (or $\pm 0.03{\ \rm{fm}}$) for the $R_{n}$ of 208Pb,
but the experiment’s start time is still largely uncertain Middleton (2021).

Another clean and model-independent way to extract the $R_{\rm skin}$ is
through the coherent elastic neutrino-nucleus scattering (CE$\nu$NS) Freedman
(1974); Freedman  _et al._ (1977), which was firstly observed by the COHERENT
Collaboration via a CsI detector with the neutrino beam from the Spallation
Neutron Source at Oak Ridge National Laboratory Akimov  _et al._ (2017). Based
on the COHERENT data, the $R_{\rm skin}$ of CsI has been extracted Cadeddu
_et al._ (2018); Huang and Chen (2019) but the uncertainty is too large to
claim a determination, due to the low statistics of CE$\nu$NS events. In
nature, the nearby core-collapse supernova (CCSN) may render a neutrino flux
with unprecedentedly high luminosity, which provides an excellent chance to
explore CE$\nu$NS. Indeed, detecting the next galactic SN neutrinos has
received much attention both from large neutrino observatories and modern dark
matter experiments Scholberg (2012); Aalbers  _et al._ (2016); An  _et al._
(2016); Abi  _et al._ (2021); Agnes  _et al._ (2021); Abe  _et al._ (2021);
Pattavina  _et al._ (2020, 2021). One of the most powerful projects is the
RES-NOVA experiment which will hunt CCSN neutrinos via CE$\nu$NS by adopting
an archaeological Pb based cryogenic detector Pattavina  _et al._ (2020,
2021). One merit of RES-NOVA is that using CE$\nu$NS as its detection channel
allows a flavor-blind neutrino measurement and thus avoids the uncertainties
from the neutrino oscillation. The other merit is that archaeological Pb
ensures the large CE$\nu$NS cross section and the ultra-low levels of
background, literally guaranteeing a high statistics.

In this work, we demonstrate that the very configuration of the RES-NOVA
experiment provides an ideal site to determine the $R_{\rm n}$ of Pb, and an
ultimate precision of $\sim 0.1\%$ for the $R_{\rm n}$ ($\sim 0.006$ fm for
the $R_{\rm skin}$) of Pb can be achieved if the galactic CCSN would explode
at a distance of $\sim 1{\ \rm{kpc}}$ from the Earth.

_Supernova neutrinos._ — The detailed knowledge of a SN neutrino flux is still
missing in experiments since we have only observed two dozen neutrino events
from the SN 1987A Bionta  _et al._ (1987); Hirata  _et al._ (1987). However,
after three decades, current neutrino experiments have stepped into an era
with unprecedented accuracy. The robust reconstruction of SN neutrino spectra
with multiple detectors has been investigated Minakata  _et al._ (2008);
Dasgupta and Beacom (2011); Lu  _et al._ (2016); Nikrant  _et al._ (2018);
Gallo Rosso  _et al._ (2018); Li  _et al._ (2018, 2019); Gallo Rosso (2021);
Nagakura (2020) and an accurate measurement is promising for a nearby SN
(e.g., $<5{\ \rm{kpc}}$). Furthermore, modern SN simulations have achieved a
tremendous progress in unveiling the mysteries of SN phenomena Janka  _et al._
(2016); Müller (2016); Just  _et al._ (2018); O’Connor  _et al._ (2018);
Nagakura  _et al._ (2020). Based on current understanding, the spectral shape
of CCSN neutrino fluxes for each flavor can be well approximated by a pinched
thermal distribution Keil  _et al._ (2003); Tamborra  _et al._ (2012)

$f_{\nu}(E_{\nu})=A\left(\dfrac{E_{\nu}}{\left<E_{\nu}\right>}\right)^{\alpha}\rm{exp}\left[-(\alpha+1)\dfrac{E_{\nu}}{\left<E_{\nu}\right>}\right].$
(1)

Here, $E_{\nu}$ and $\left<E_{\nu}\right>$ are the neutrino energy and the
averaged energy, $\alpha$ describes the amount of spectral pinching, and
$A=\dfrac{(\alpha+1)^{\alpha+1}}{\left<E_{\nu}\right>\Gamma(\alpha+1)}$ is the
normalization constant, where $\Gamma$ is the gamma function. So the neutrino
fluence per flavor on the Earth from a CCSN at a distance $d$ can be obtained
as

$\Phi(E_{\nu})=\dfrac{1}{4\pi d^{2}}\dfrac{E_{\nu}^{\rm
tot}}{\left<E_{\nu}\right>}f_{\nu}(E_{\nu}),$ (2)

where $E_{\nu}^{\rm tot}$ denotes the total emitted energy per flavor. In a
real CCSN explosion, both the amounts and spectra of the emitted neutrinos
change with time as the star evolves into different stages. However, to our
goal, we only need information of total neutrino emission. Therefore, we adopt
here the time-integrated neutrino emission parameters from a typical long-term
axisymmetric CCSN simulation, which can be found in Table I of Ref. Nikrant
_et al._ (2018). Note that although the neutrino emission of a CCSN also
depends on the details of the transient, e.g., the progenitor mass,
compactness, explosion dynamics, etc., it has a rough profile of
$\left<E_{\nu}\right>\sim 10{\ \rm{MeV}}$, $2<\alpha<4$ and $E^{\rm tot}\sim
10^{53}{\ \rm{erg}}$. Nevertheless, the accurate information can be extracted
from various detection data once a nearby CCSN explosion occurs.

_Detection prospects in RES-NOVA._ — To explore the potential of determining
the $R_{\rm n}$ of Pb with RES-NOVA, we consider the RN-3 configuration in
Table I of Ref. Pattavina  _et al._ (2020) which has a detector mass of 465 t
and an energy threshold of $1{\ \rm{keV}}$. The absorber with pure Pb is also
adopted. For the detection channel, the differential cross section in the
standard model has the form:

$\displaystyle\dfrac{{\rm{d}}\sigma}{{\rm{d}}T}(E_{\nu},T)=\dfrac{G_{F}^{2}M}{4\pi}Q_{W}^{2}F_{W}^{2}(q)\Big{[}1-\dfrac{T}{E_{\nu}}-\dfrac{MT}{2E_{\nu}^{2}}\Big{]},$
(3)

where $G_{F}$ is the Fermi coupling constant; $M$ denotes the mass of the
target nucleus with $N(Z)$ neutrons (protons); $Q_{W}$ is the weak charge and
$F_{W}(q)$ is the weak form factor; $E_{\nu}$ and $T$ represent the neutrino
energy and the kinetic recoil energy of the nucleus, respectively; and the
momentum transfer $q$ is given by $q^{2}\simeq 2MT$. Note Eq. (3) is for a
nucleus with spin-0 and the result for a spin-1/2 target (i.e., 207Pb in our
case) will gain a tiny correction Lindner  _et al._ (2017) which is neglected
in this work.

The weak charge $Q_{W}$ can be obtained as

$Q_{W}=\int\mathrm{d}^{3}r\rho_{W}(r)=Nq_{n}+Zq_{p},$ (4)

where $\rho_{W}(r)$ is the weak charge density. At tree level, the nucleon
weak charges are $q_{n}=q_{n}^{0}=2g_{V}^{n}$ and
$q_{p}=q_{p}^{0}=2g_{V}^{p}$, where the neutron (proton) vector coupling is
defined as $g_{V}^{n}=-\frac{1}{2}$
($g_{V}^{p}=\frac{1}{2}-2\sin^{2}\theta_{W}$) with the low-energy weak mixing
angle $\sin^{2}\theta_{W}=0.23857(5)$ Kumar  _et al._ (2013); Tanabashi  _et
al._ (2018). In the present work, we adopt the values $q_{n}=-0.9878$ and
$q_{p}=0.0721$ to include radiative corrections Horowitz  _et al._ (2012). The
weak form factor $F_{W}(q)$ is expressed as

$F_{W}(q)=\dfrac{1}{Q_{W}}\int\mathrm{d}^{3}r\dfrac{\sin qr}{qr}\rho_{W}(r).$
(5)

Here we use the Helm parametrization for the $F_{W}(q)$ Helm (1956);
Piekarewicz  _et al._ (2016), which has been proven to be very successful for
analyzing electron scattering form factors Rosen  _et al._ (1967); Raphael and
Rosen (1970). The $F_{W}(q)$ is then expressed as

$F_{W}(q)=3\dfrac{j_{1}(qR_{0})}{qR_{0}}{\rm{e}}^{-q^{2}s^{2}/2},$ (6)

where $j_{1}(x)=\sin(x)/x^{2}-\cos(x)/x$ is the spherical Bessel function of
order one, $R_{0}$ is the diffraction radius and $s$ quantifies the surface
thickness. The rms radius $R_{W}$ of weak charge density can then be obtained
as

$R_{W}^{2}=\int\mathrm{d}^{3}r\dfrac{r^{2}\rho_{W}(r)}{Q_{W}}=\dfrac{3}{5}R_{0}^{2}+3s^{2}.$
(7)

We use $s=1.02{\ \rm{fm}}$ following the discussion in Ref. Horowitz  _et al._
(2012). The $R_{n}$ and $R_{p}$ are related to $R_{W}$ and $R_{\rm ch}$ with
the following relations Ong  _et al._ (2010); Horowitz  _et al._ (2012),

$R_{p}^{2}=R^{2}_{\rm ch}-\langle r_{p}^{2}\rangle-\dfrac{N}{Z}\langle
r_{n}^{2}\rangle$ (8)

and

$R_{n}^{2}=\dfrac{Q_{W}}{q_{n}N}R_{W}^{2}-\dfrac{q_{p}Z}{q_{n}N}R_{\rm
ch}^{2}-\langle r_{p}^{2}\rangle-\dfrac{Z}{N}\langle
r_{n}^{2}\rangle+\dfrac{Z+N}{q_{n}N}\langle r_{s}^{2}\rangle.$ (9)

Here $\langle r_{p}^{2}\rangle^{1/2}=0.8414(19){\ \rm{fm}}$ Hammer and Meißner
(2020) is the charge radius of a proton and $\langle
r_{n}^{2}\rangle=-0.1161(22){\ \rm{fm}}^{2}$ Tanabashi  _et al._ (2018) is
that of a neutron, the squared strangeness radius of nucleon is taken to be
$\langle r_{s}^{2}\rangle=-0.0054(16){\ \rm{fm}}^{2}$ according to Lattice QCD
calculations Horowitz (2019); Green  _et al._ (2015). Note that the
contributions of the Darwin-Foldy term and the spin-orbit current are
neglected here since both of them are quite small and will not affect our
conclusions on the relative precision evaluation of the $R_{n}$ determination
for Pb.

Table 1: Charge radii $R_{\rm ch}$ Angeli and Marinova (2013), binding energies per nucleon $E_{B}/A$ Wang  _et al._ (2021) and abundance $Y_{A}$ Meija  _et al._ (2016) of Pb isotopes. Isotopes |  $R_{\rm ch}$ (fm) |  $E_{B}/A$ ($\rm{MeV}$) | $Y_{A}$
---|---|---|---
204 | 5.4803(14) | 7.87993 | 0.014(6)
206 | 5.4902(14) | 7.87536 | 0.241(30)
207 | 5.4943(14) | 7.86987 | 0.221(50)
208 | 5.5012(13) | 7.86745 | 0.524(70)

The archaeological Pb crystal in RES-NOVA is mainly composed of four isotopes,
i.e., 204,206,207,208Pb. The charge rms radius $R_{\rm ch}$, the binding
energies per nucleon $E_{B}/A$ and the natural abundance $Y_{A}$ of these
isotopes can be found in Table 1. Since it is impossible to distinguish the
CE$\nu$NS events from different isotopes in RES-NOVA, what we can extract from
such detection is the averaged $R_{\rm skin}$ of Pb and that is what we really
mean for the $R_{\rm skin}$ of Pb in this work. The mass of a nucleus is
defined as $M=N\times m_{n}+Z\times m_{p}-E_{B}$ where $E_{B}$ is the binding
energy and $m_{n(p)}$ is the rest mass of neutrons(protons).

We first assume a constant $100\%$ acceptance efficiency in the detector, and
the expected event counts can then be obtained as

$\dfrac{{\rm{d}}N}{{\rm{d}}T}=\sum Y_{A}N_{t}\int_{E_{\rm
min}}{\rm{d}}E_{\nu}\ \Phi(E_{\nu})\
\dfrac{{\rm{d}}\sigma}{{\rm{d}}T}(E_{\nu},T).$ (10)

Here, $N_{t}=N_{A}m_{\rm det}/M_{\rm{Pb}}$ is the number of nuclei in the
crystals with $N_{A}$ being the Avogadro constant, $m_{\rm det}$ the detector
mass and $M_{\rm{Pb}}=0.2072\ \rm{kg/mol}$ the molar mass of Pb. Strictly
speaking, the value of $E_{\rm min}$ depends on $T$ due to the relation:
$T_{\rm max}=2E^{2}_{\nu}/(M+2E_{\nu})$. We adopt $T_{\rm max}\simeq
2E^{2}_{\nu}/M$ since $M\gg E_{\nu}$ in this scattering. The final count will
sum over both the isotopes and neutrino flavors, and integrate over the
corresponding energy bin.

The final result will vitally depend on the distance of the SN. On the one
hand, the target SN cannot be too far since the event rate has an inverse
quadratic dependence on the distance, as shown in Eq. (2) and Eq. (10).
Especially, the relative high energy part ($T>10{\ \rm{keV}}$) has rather low
event rate and even gets hidden under the background for $d=10{\ \rm{kpc}}$
Pattavina  _et al._ (2020). However, recent study shows there exists shock
acceleration in SN to create more high energy heavy flavor neutrinos Nagakura
and Hotokezaka (2021). Note that the nucleon distribution radii exhibit
stronger sensitivity to the form factor at higher momentum transfer in this
low energy range. We thus choose $d=5{\ \rm{kpc}}$ for a typically far
distance SN target. On the other hand, a very nearby SN will lead to a
neutrino flux intense enough to cause signal pile-up in the detector. This
phenomenon has been studied recently by the RES-NOVA Collaboration, and the
results show that the pile-up probability will decrease to almost zero for
$d\gtrsim 1{\ \rm{kpc}}$ Pattavina  _et al._ (2021). Therefore, $d=1{\
\rm{kpc}}$ turns out to be an optimal choice.

Figure 1: The predicted event counts per 1 keV versus nuclear kinetic recoil
energy $T$ at the RES-NOVA detector for a SN at $1{\ \rm{kpc}}$ ($N_{1}$, blue
solid line) and $5{\ \rm{kpc}}$ ($N_{5}$, red solid line), using the PREX-2
result as the averaged $R_{\rm skin}$ of Pb. The dashed lines ($\delta N_{1}$
and $\delta N_{5}$) show the change amplitude of counts with a $0.06{\
\rm{fm}}$ variation of the neutron skin thickness. The relative variation
$\delta N/N$ of the counts is given for both SN distances by the black line
(right axis).

The expected results are shown in Fig. 1. We adopt an energy bin of $1{\
\rm{keV}}$, which is allowable since the energy resolution of RES-NOVA is
expected to be $0.2{\ \rm{keV}}$. The statistics is promising for both cases
of $d=1{\ \rm{kpc}}$ and $d=5{\ \rm{kpc}}$. In particular, the count per bin
ranges from $10^{2}$ to $10^{5}$ in the recoil energy range ($3-19{\
\rm{keV}}$) for a SN at $1{\ \rm{kpc}}$, while it becomes approximately one
order of magnitude smaller for $d=5{\ \rm{kpc}}$. We also estimate the
sensitivity to the variation of the $R_{\rm skin}$. As an example, we plot the
count change for a $R_{\rm skin}$ varaition of $0.06{\ \rm{fm}}$ ($\sim 1\%$).
The modifications on counts for $d=1(5){\ \rm{kpc}}$ stay within about
$10^{1}(1)-10^{3}(10^{2})$ depending on the recoil energy $T$. To elucidate
this effect more clearly, the relative difference $\delta N/N$ is also shown
by the black line. As anticipated, the $\delta N/N$ only depends on the recoil
energy for a certain variation of the $R_{\rm skin}$ and increases with the
recoil energy. This is because it directly quantifies the modifications on the
form factor. On the other hand, one sees that both the expected counts and its
variation $\delta N$ decrease exponentially with the recoil energy. This is
due to the roughly exponential decay of SN neutrino flux as a function of
neutrino energy at higher energies as shown in Eq. (1). As a result,
considering both the sensitivity and event statistics, the bins around the
center ($T\sim 15$ keV) are more suitable to measure the $R_{\rm skin}$.

_Neutron skin thickness sensitivity._ — In order to carry out a statistical
evaluation on the precision of the $R_{n}$ determination, we follow Ref. Cañas
_et al._ (2018) and adopt the following chi-squares function:

$\chi^{2}=\sum\dfrac{(N_{\rm exp}-N_{\rm th})^{2}}{\sigma_{\rm
stat}^{2}+\sigma^{2}_{\rm syst}}.$ (11)

Here, the “experimental” data $N_{\rm exp}$ is taken as the expected counts
with $R_{\rm skin}=0.283{\ \rm{fm}}$ and the theoretical counts $N_{\rm th}$
will vary according to the $R_{\rm skin}$ value. Except for the statistical
uncertainty $\sigma_{\rm stat}=\sqrt{N_{\rm exp}}$, we also introduce an
effective systematic error $\sigma_{\rm syst}=pN_{\rm th}/100$ to quantify the
possible uncertainties originating from the future RES-NOVA detector, the
extracted SN neutrino spectra from other detection, and so on, with $p$
representing the percentage of the future systematic error. We only select the
energy bins around the center part ($T\sim 15$ keV) for the sum in Eq. (11).
For bins with lower energy, they have higher statistics but much less
sensitivity to the neutron form factor due to their almost full coherence.
Moreover, the previous assumption of an energy-independent $100\%$ acceptance
efficiency is more likely to become unacceptable in the energy bins near the
threshold of detector. We thus drop the first 3 bins above the energy
threshold and choose the minimum recoil energy to be $T_{\rm min}=4.5{\
\rm{keV}}$, which corresponds to a minimum neutrino energy of $E_{\rm
min}\simeq 20.9{\ \rm{MeV}}$. That means that only the high energy SN
neutrinos contributes to this analysis. In contrast, bins in higher energy
region show better sensitivity but poor statistics. We adopt $T_{\rm
max}=29.5(20.5){\ \rm{keV}}$ [corresponding to a minimum neutrino energy of
$E_{\rm min}\simeq 53.4(44.6){\ \rm{MeV}}$] for $d=1(5){\ \rm{kpc}}$ just to
ensure that every bins have a reasonable event number (i.e. $N>10$).

Figure 2: The expected sensitivity to the variation of neutron skin thickness
$\delta R_{\rm{skin}}$ for a SN at $1{\ \rm{kpc}}$ (blue lines) and $5{\
\rm{kpc}}$ (red lines) with $1\%$ (solid linces) and $5\%$ (dashed lines)
systematic error.

The resulting $\chi^{2}$ as a function of the variation $\delta R_{\rm{skin}}$
for the neutron skin thickness is shown in Fig. 2 where the future systematic
error is presumed to be $1\%$ or $5\%$. In particular, the $1\sigma$
uncertainty for a $1(5)\%$ systematic error is $\pm 0.008(0.016){\ \rm{fm}}$
if the SN is located at $d=1{\ \rm{kpc}}$, and $\pm 0.030(0.044){\ \rm{fm}}$
at $d=5{\ \rm{kpc}}$. It is remarkable that our approach can achieve an
ultimate precision of $\sim 0.1\%$ in the optimal case of $d=1{\ \rm{kpc}}$,
even much higher than the expected precision of the future MREX Becker  _et
al._ (2018).

Figure 3: The expected precision of the averaged neutron radius $R_{n}$ (in
percent) as a function of the systematic uncertainty for a $1{\ \rm{kpc}}$
(blue solid line) and $5{\ \rm{kpc}}$ (red dashed line) SN. The results from
PREX-2 Adhikari  _et al._ (2021) and the future MREX Becker  _et al._ (2018)
are also included for comparison.

The systematic error $\sigma_{\rm syst}$ is of great importance in future
experiments and observations, and our current knowledge on $\sigma_{\rm syst}$
is rather insufficient. To see more clearly how the precision depends on the
$\sigma_{\rm syst}$, we plot in Fig. 3 the expected $1\sigma$ precision (in
percentage) for the $R_{n}$ determination as a function of the $\sigma_{\rm
syst}$. It is seen that the $\sigma_{\rm syst}$ dependence is nearly
unaffected by the SN distance and is almost linear in the range of $1<p<10$.
Generally, the closer the SN is located, the better its neutrino flux can be
measured, and thus the smaller $\sigma_{\rm syst}$ from the spectra will be
achieved. Therefore, it is quite promising to achieve a precision better than
MREX for a nearby SN at $d\simeq 1{\ \rm{kpc}}$ as shown in Fig. 3. Even under
a worse condition at $d\simeq 5{\ \rm{kpc}}$, one can still anticipate a
precision better than that of PREX-2. At this point, we would like to
emphasize that the nearby presupernova stars are not too rare in our Galaxy.
For example, a list of 31 candidates within 1 kpc, including the famous
Betelgeuse, can be found in Ref. Mukhopadhyay  _et al._ (2020), and therefore
much more candidates are expected to exist in $1\sim 5$ kpc.

Finally, we would like to mention that the advanced ab initio approaches using
nuclear forces from chiral effective field theory can now describe the
properties of heavy nuclei such as 208Pb Hu  _et al._ (2022). In particular,
the ab initio calculations predict $R^{48}_{\rm skin}=0.141-0.187$ fm and
$R^{208}_{\rm skin}=0.139-0.200$ fm Hu  _et al._ (2022), consistent with the
CREX result of $R^{48}_{\rm skin}=0.121\pm 0.026(\rm exp)\pm 0.024(\rm model)$
fm Adhikari  _et al._ (2022) but exhibiting a mild tension with the PREX-2
result of $R^{208}_{\rm skin}=0.283\pm 0.071~{}\rm{fm}$ Adhikari  _et al._
(2021). The approach proposed in the present work with the expected high
precision for the $R^{208}_{\rm skin}$ determination can thus crosscheck the
PREX-2 result and test the ab initio prediction on $R^{208}_{\rm skin}$. In
addition, our approach in principle can be also applied to determine the
$R_{\rm skin}$ of other nuclei which are adopted as large-scale detector
medium to hunt dark matter and neutrinos, e.g., Xe isotopes in next-generation
xenon-based detector Aalbers  _et al._ (2022).

_Conclusions._ — We have demonstrated that the neutrinos from a nearby CCSN in
our Galaxy can be used to precisely determine the $R_{n}$ and $R_{\rm skin}$
of Pb via CE$\nu$NS in RES-NOVA. In particular, an ultimate precision of $\sim
0.1\%$ ($\sim 0.006$ fm) for the $R_{n}$ ($R_{\rm skin}$) of Pb is expected to
be achieved if the CCSN explosion were to occur at a distance of $\sim 1{\
\rm{kpc}}$ from the Earth. Such a precision of the $R_{\rm skin}$ is
significantly higher than that of the existing and planned experiments in
terrestrial labs, and will eventually pin down the density dependence of the
symmetry energy and clarify the issue of the tension between CREX and PREX-2
experiments.

_Acknowledgements._ — The authors thank Ning Zhou for useful discussions. This
work was supported by the National Natural Science Foundation of China under
Grant Nos. 12235010 and 11625521, and National SKA Program of China No.
2020SKA0120300.

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# Modules of finite injective dimension

Mohsen Asgharzadeh  M. Asgharzadeh<EMAIL_ADDRESS>
###### Abstract.

We study some aspects of distinguished finitely generated modules of finite
injective dimension. Following Bass’ conjecture, this implies that the ring is
Cohen-Macaulay. We derive some properties of rings from them. Our list of
properties contains the reduced, domain, normality, and regularity (complete-
intersection, Gorensteiness, etcetera) of rings.

###### Key words and phrases:

Cohen-Macualay rings; injective dimension; integral domains; regular
(Gorenstein) rings; Auslander’s zero-divisor; regular sequences.

###### 2020 Mathematics Subject Classification:

13C14; 13H10; 13D22

###### Contents

  1. 1 Introduction
  2. 2 A family of ideals of small height
  3. 3 Modules with few generators
  4. 4 Auslander’s zero-divisor conjecture
  5. 5 Almost complete-intersection
  6. 6 Matlis and canonical module

##  1\. Introduction

Bass [9] conjectured that the existence of a finitely generated module of
finite injective dimension imposes some restrictions on the ring, namely the
Cohen-Macaulay property. Peskine-Szpiro [23] proved Bass’ conjecture in many
cases, and the conjecture has been settled by Roberts [24] in the full
setting. Despite this, it is referred to as Bass’ conjecture. The work of
Peskine-Szpiro contains a lot of related beautiful results. In particular,
they searched for more restrictions on the ring if one focuses on very special
modules of finite injective dimension. We also note that Matlis was interested
in the ideals of injective dimension one. These are our motivations to do this
note.

Let $\mathfrak{p}$ be in $\operatorname{Spec}(R)$, and suppose
$\operatorname{pd}_{R}(\mathfrak{p})<\infty$, equivalently
$\operatorname{pd}_{R}(R/\mathfrak{p})<\infty$. Peskine-Szpiro [23, Corollary
II.3.3] proved that $R$ is an integral domain. It is easy to see the ring is
domain if $\operatorname{id}_{R}(R/\mathfrak{p})<\infty$. A natural question
arises:

######  Question 1.1.

What can say about $R$ if $\operatorname{id}_{R}(\mathfrak{p})<\infty$?

Some of the results from Section 2 are summarized in the following item:

  1. 1)

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for all
${\mathfrak{p}}\in\operatorname{Assh}(R)$. Then $R$ is reduced.

  2. 2)

Suppose $\min(R)$ is singleton. If
$\operatorname{id}_{R}(\mathfrak{p})<\infty$ for some
$\mathfrak{p}\in\operatorname{Spec}(R)$, then $R$ is domain.

  3. 3)

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for all
${\mathfrak{p}}\in\operatorname{Spec}(R)$ of height at most one. Then $R$ is
normal.

  4. 4)

Let $R$ be excellent $\operatorname{UFD}$ containing $k$ and of dimension
$>3$. Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for all
${\mathfrak{p}}\in\operatorname{Spec}(R)$ of height two. Then $R$ is regular.

We also present a quasi-version of the reduced and normal items. For example,
we show:

######  Corollary 1.2.

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for some
${\mathfrak{p}}\in\operatorname{Spec}(R)$ of positive height. Then $R$ is
quasi-normal.

Suppose we allow more ideals are of finite injective dimension. For instance,
suppose that $\operatorname{id}_{R}(\mathfrak{a})<\infty$ for all ideals of
big-height at most two. Then, we apply our displaced results plus a famous
result of Auslander [6] to show $R$ is regular.

In Section 3 we focus on modules of finite injective dimensions with few
generators. In fact, Vaconcelos asked in [27, Page 51] “how do Gorenstein
rings arise?”, and he studied the question by looking at certain modules with
few number of generators, see [27, 2.40]. Our first result in §3 is as
follows:

######  Proposition 1.3.

Let $(R,\mathfrak{m})$ be quasi-normal, and let $M$ be such that

  1. i)

$\operatorname{id}_{R}(M)<\infty$,

  2. ii)

$M$ is reflexive,

  3. iii)

$\mu(M)\leq 2$.

Then $R$ is Gorenstein.

As an application, we extend Proposition 1.3 to the case of four generated
modules over rings of type at most two. Also, we present some few comments to
§2. Then, we slightly extend [27, 2.40] to more general setting.

Suppose $M$ is a nonzero tor-rigid module, Auslander [7, 4.3] proved each
$M$-regular sequence is an $R$-regular sequence. As modules of finite
projective dimension are not in general tor-rigid, he conjectured (proved by
Roberts [25, 6.2.3]) that the above property is true for modules of finite
projective dimension. This is referred to as Auslander’s zero-divisor
conjecture. In Section 4 we present the injective analogue of Auslander’s
zero-divisor conjecture. We do this via two different methods. We use these to
reconstruct Auslander’s zero-divisor conjecture over Cohen-Macaulay rings.
Also, this enables us to reprove some things from §2.

In Section 5 we deal with

######  Question 1.4.

(See [15]) Suppose $\operatorname{id}_{R}(I/I^{2})$ is finite. Is $I$
generated by a regular sequence?

We present two cases that the answer is positive. Namely, in the almost
complete-intersection case. This immediately recovers the main result of [17]
by Kunz.

Recall that [27, 2.38] suggests canonical module is an important source to
produce, in a constructive way, modules of finite injective dimension. In
fact, Matlis applied this to study the following over 1-dimensional integral
domains:

  1. i)

When is $Q/R\twoheadrightarrow E_{R}(k)$?

  2. ii)

When is $Q/I=E_{R}(k)$?

In order to see these are related to modules of finite injective dimension,
recall that ii) is equivalent to ideals of finite injective dimension. In
Section 6 we extend them to the higher dimensional situation. It may be nice
to recall that $R$ is called quasi-Gorenstein, if
$H^{\dim(R)}_{\mathfrak{m}}(R)\cong E_{R}(k)$, so Matlis was interested on
this and its semi-form as well. Finally, we extend an observation of Auslander
[5] about cohomology of $Q/R$ and present its converse part, see Observation
6.6.

For all unexplained notation and definitions see the book [10].

##  2\. A family of ideals of small height

In this note $(R,\mathfrak{m},k)$ is a commutative noetherian local ring, and
modules are finitely generated, otherwise specialized. The notation
$\operatorname{pd}_{R}(-)$ (resp. $\operatorname{id}_{R}(-)$) stands for the
projective (resp. injective) dimension of $(-)$.

######  Notation 2.1.

We denote the set of all associated prime ideals by
$\operatorname{Ass}_{R}(-)$.

######  Proposition 2.2.

Suppose $\min(R)$ is singleton. If for some
$\mathfrak{p}\in\operatorname{Spec}(R)$ one has
$\operatorname{id}_{R}(\mathfrak{p})<\infty$, then $R$ is an integral domain.

###### Proof.

Following Bass’ conjecture (see [10, Section 9.6]) $R$ is Cohen-Macaulay. In
particular, $\operatorname{Ass}(R)=\min(R).$ Suppose $\min(R)=\\{P\\}$. By
definition, there is an $x\in R$ such that $(0:_{R}x)=P$. Suppose
$x\in\mathfrak{m}$. Also, recall that

$x\in\operatorname{Zd}(R)=\bigcup_{\mathfrak{q}\in\operatorname{Ass}(R)}\mathfrak{q}=\bigcup_{\mathfrak{q}\in\min(R)}\mathfrak{q}=P=(0:x).$

In other words, $x^{2}=0$. Since $\min(R)$ is singleton,
$P\subseteq\mathfrak{p}$. Since $\operatorname{id}_{R}(\mathfrak{p})<\infty$,
and due to [10, Proposition 3.1.9], we know that
$\operatorname{id}_{R_{\mathfrak{p}}}(\mathfrak{p}R_{\mathfrak{p}})<\infty$.
According to Auslander-Buchsbaum-Serre, $R_{\mathfrak{p}}$ is regular (see
[10, Theorem 2.2.7]). Since $R_{\mathfrak{p}}$ is regular its localizations
are as well. In particular, $R_{P}=(R_{\mathfrak{p}})_{PR_{\mathfrak{p}}}$ is
regular. Recall that $x\in P$. Since $x/1\in PR_{P}$, it should be zero. There
is an $y\in R\setminus P$ such that $xy=0$. Since $xy=0$ we have
$y\in(0:x)=P$. This contradiction says that $x\notin\mathfrak{m}$. Since the
ring is local, $x$ is unit. From this $P=(0:x)=0$. Consequently, $R$ is an
integral domain. ∎

We denote the set of all associated prime ideal $\mathfrak{p}$ with the
property that $\dim R/\mathfrak{p}=\dim R$, by $\operatorname{Assh}_{R}(-)$.

######  Proposition 2.3.

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for all
${\mathfrak{p}}\in\operatorname{Assh}(R)$. Then $R$ is reduced.

###### Proof.

Recall that $R$ is Cohen-Macaulay. In particular, $R$ is equidimensional. Let

$\operatorname{Assh}(R)=\operatorname{Ass}(R)=\min(R)=\\{\mathfrak{p}_{1},\ldots,\mathfrak{p}_{t}\\}.$

In view of Proposition 2.2 we may assume $t>1$. Let
$P\in\min(\bigcap_{i=1}^{t}\mathfrak{p}_{i})$. There is an $i$ such that
$\mathfrak{p}_{i}\subseteq P$. It follows that $P$ is minimal over
$\mathfrak{p}_{i}$. Thus, $P=\mathfrak{p}_{i}$. Since
$\operatorname{id}_{R}(\mathfrak{p}_{i})<\infty$,
$\operatorname{id}_{R_{\mathfrak{p}_{i}}}(\mathfrak{p}_{i}R_{\mathfrak{p}_{i}})<\infty$.
According to Auslander-Buchsbaum-Serre, $R_{\mathfrak{p}_{i}}$ is regular.
Then

$(\bigcap_{i=1}^{t}\mathfrak{p}_{i})R_{P}\subseteq PR_{P}=0.$

Since $\min(-)$ is the minimal part of $\operatorname{Supp}(-)$, we proved
that

$\operatorname{Supp}(\bigcap_{i=1}^{t}\mathfrak{p}_{i})=\emptyset,$

and so $\bigcap_{i=1}^{t}\mathfrak{p}_{i}=0$. Therefore, $R$ is reduced. ∎

######  Notation 2.4.

By $(G_{i})$ (resp. $(R_{i})$) we mean $R_{\mathfrak{p}}$ is Gorenstein (resp.
regular) for all $\mathfrak{p}\in\operatorname{Spec}(R)$ of height at most
$i$.

Recall that a module $M$ satisfies $(S_{i})$ if
$0pt(M_{\mathfrak{p}})\geq\min\\{i,\dim(M_{\mathfrak{p}})\\}$ for all
$\mathfrak{p}\in\operatorname{Spec}(R)$.

######  Proposition 2.5.

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for some
${\mathfrak{p}}\in\operatorname{Assh}(R)$. Then $R$ is quasi-reduced.

###### Proof.

Following Bass’ conjecture, $R$ is Cohen-Macaulay. In particular, $R$
satisfies Serre’s condition $(S_{1})$. Let
${\mathfrak{q}}\in\operatorname{Ass}(R)=\min(R)$. We have two possibilities i)
$\mathfrak{q}=\mathfrak{p}$ or ii) $\mathfrak{q}\neq\mathfrak{p}$.

  1. i)

In the first case, $R_{\mathfrak{q}}$ is a field. To see this, recall that
$\operatorname{id}_{R_{\mathfrak{q}}}(\mathfrak{q}R_{\mathfrak{q}})=\operatorname{id}_{R_{\mathfrak{p}}}(\mathfrak{p}R_{\mathfrak{p}})$
is finite, because $\operatorname{id}_{R}(\mathfrak{p})<\infty$. This in turn
implies that $R_{\mathfrak{q}}$ is zero-dimensional local and regular. From
this, $R_{\mathfrak{q}}$ is a field.

  2. ii)

In the second case, $R_{\mathfrak{q}}$ is Gorenstein. Indeed, we have
$R_{\mathfrak{q}}=\mathfrak{p}R_{\mathfrak{q}}$. Consequently,
$\operatorname{id}_{R_{\mathfrak{q}}}(R_{\mathfrak{q}})=\operatorname{id}_{R_{\mathfrak{q}}}(\mathfrak{p}R_{\mathfrak{q}})$,
and recall that the later is finite, because $\mathfrak{p}$ is of finite
injective dimension. By this, $R_{\mathfrak{q}}$ is Gorenstein.

In both cases, $R_{\mathfrak{q}}$ is Gorenstein. It remains to note that
$(S_{1})$ and $(G_{0})$ imply the quasi-reduced condition. ∎

######  Proposition 2.6.

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for all
${\mathfrak{p}}\in\operatorname{Spec}(R)$ of height at most one. Then $R$ is
normal.

###### Proof.

Recall from the assumptions that $R$ satisfies Serre’s condition $(S_{2})$.
Let ${\mathfrak{p}}\in\operatorname{Spec}(R)$ be of height at most one. Since
$\operatorname{id}_{R}(\mathfrak{p})<\infty$,
$\operatorname{id}_{R_{\mathfrak{p}}}(\mathfrak{p}R_{\mathfrak{p}})<\infty$.
According to Auslander-Buchsbaum-Serre, $R_{\mathfrak{p}}$ is regular. So, $R$
satisfies $(R_{1})$. It remains to note that if a ring satisfies Serre’s
condition $(S_{2})$ and $(R_{1})$ is normal. ∎

We cite [26] for basics of quasi-normal rings.

######  Proposition 2.7.

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for some
${\mathfrak{p}}\in\operatorname{Spec}(R)$ of positive height. Then $R$ is
quasi-normal.

###### Proof.

As Proposition 2.6, the ring $R$ satisfies $(S_{2})$. Let
${\mathfrak{q}}\in\operatorname{Spec}(R)$ be of height at most one. We may
assume that $\mathfrak{p}$ in not maximal, otherwise the ring becomes regular.
We have two possibilities i) $\mathfrak{q}\subseteq\mathfrak{p}$ or ii)
$\mathfrak{q}\nsubseteq\mathfrak{p}$.

  1. i)

We localize the ring at $\mathfrak{p}$ and by using the assumption
$\operatorname{id}_{R}(\mathfrak{p})<\infty$, we know that $R_{\mathfrak{p}}$
is regular. Since $\mathfrak{q}\subseteq\mathfrak{p}$, we have
$R_{\mathfrak{q}}\cong(R_{\mathfrak{p}})_{\mathfrak{q}R_{\mathfrak{p}}}$.
Since localization of a regular ring is regular we deduce that
$R_{\mathfrak{q}}$ is regular. So, $R$ satisfies $(R_{1})$.

  2. ii)

Suppose $\mathfrak{q}\nsubseteq\mathfrak{p}$. We claim that
$\mathfrak{p}\nsubseteq\mathfrak{q}$. Indeed, suppose not. Then
$\mathfrak{p}\subseteq\mathfrak{q}$. Due to the assumption
$\operatorname{ht}(\mathfrak{p})\geq 1$, and the fact that
$\operatorname{ht}(\mathfrak{q})\leq 1$, we deduce that
$\mathfrak{p}=\mathfrak{q}$. This contradicts the assumption from ii). So,
$\mathfrak{p}\nsubseteq\mathfrak{q}$. This gives us the natural isomorphism
$\mathfrak{p}R_{\mathfrak{q}}\cong R_{\mathfrak{q}}$. Since
$\operatorname{id}_{R}(\mathfrak{p})<\infty$, its localizations are as well.
In particular,
$\operatorname{id}_{R_{\mathfrak{q}}}(R_{\mathfrak{q}})=\operatorname{id}_{R_{\mathfrak{q}}}(\mathfrak{p}R_{\mathfrak{q}})<\infty$.
By definition ${R_{\mathfrak{q}}}$ is Gorenstein. So, $R$ satisfies $(G_{1})$.

In both cases $R$ is $(G_{1})$. It remains to note that if a ring satisfies
$(S_{2})$ and $(G_{1})$, then it is quasi-normal. ∎

######  Corollary 2.8.

Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$ for some
${\mathfrak{p}}\in\operatorname{Spec}(R)$. Then $R$ is quasi-reduced.

###### Proof.

If $\mathfrak{p}$ is in $\operatorname{Assh}(R)$, then the desired claim is
subject of Proposition 2.5. So, without loss of generality, we may assume that
$\dim(R)>0$ and that
$\mathfrak{p}\in\operatorname{Spec}(R)\setminus\operatorname{Assh}(R)$. By
Bass’ conjecture, $R$ is Cohen-Macaulay. Since the ring is equi-dimensional we
have $\operatorname{ht}(\mathfrak{p})>0$. It remains to apply Proposition 2.7.
∎

######  Proposition 2.9.

Let $(R,\mathfrak{m},k)$ be excellent $\operatorname{UFD}$ containing $k$ and
of dimension $>3$. Suppose that $\operatorname{id}_{R}(\mathfrak{p})<\infty$
for all ${\mathfrak{p}}\in\operatorname{Spec}(R)$ of height two. Then $R$ is
regular.

###### Proof.

This is straightforward to see $R$ satisfies Serre’s condition $(S_{3})$ and
$(R_{2})$. Let $M:=\operatorname{Syz}_{2}(k)$. It is reflexive and locally
free over the punctured spectrum. According to the Miller’s paper [21], there
is a free module $F$ and a prime ideal $\mathfrak{p}$ of height at most two
that fits in the following Bourbaki sequence

$0\longrightarrow F\longrightarrow M\longrightarrow\mathfrak{p}\longrightarrow
0\quad(\ast)$

Recall that $R$ is Cohen-Macaulay. Thanks to a result of Murthy [10, 3.3.19]
$R$ is Gorenstein. This shows that $\operatorname{id}_{R}(F)<\infty$. We apply
this in $(\ast)$ to see $\operatorname{id}_{R}(M)$ is finite. Thus,
$\operatorname{id}_{R}(k)<\infty$. Thanks to Auslander-Buchsbaum-Serre, $R$ is
regular. ∎

What about rings of mixed characteristic? More formally, we ask:

######  Problem 2.10.

Find a mixed characteristic analogue of Bourbaki sequence and prime
ideal?***We are interested on it, even if the ring is regular.

If we allow more ideals are of finite injective dimension, then we present the
following 3 positive answers:

######  Proposition 2.11.

Let $(R,\mathfrak{m})$ be local. Suppose that
$\operatorname{id}_{R}(\mathfrak{a})<\infty$ for all ideals of big-height at
most two. Then $R$ is regular.

###### Proof.

By Proposition 2.6 $R$ is normal. We assume that $\dim R>0$. Thanks to Bass,
$R$ is Cohen-Macaulay. For any nonzero $x\in\mathfrak{m}$, $(xR)$ is height-
unmixed and of height one. By assumption $\operatorname{id}_{R}(xR)$ is
finite. Since $R$ is domain, $xR\cong R$. Thus,
$\operatorname{id}_{R}(R)<\infty$. In other words, $R$ is Gorenstein. This
allows us to assume that $\operatorname{pd}_{R}(\mathfrak{a})<\infty$ for all
ideal $\mathfrak{a}$ of big-height at most two. This plus the normality and
Cohen-Macaulay enable us to apply a result of Auslander [6, Corollary 5], and
deduce that $R$ is regular. ∎

######  Proposition 2.12.

Let $(R,\mathfrak{m},k)$ be $\operatorname{UFD}$. Suppose that
$\operatorname{id}_{R}(\mathfrak{a})<\infty$ for all unmixed ideal of height
two. Then $R$ is regular.

###### Proof.

We may assume that $\dim(R)>1$. Let $x,y$ be a parameter sequence. It is
height unmixed. By assumption, $\operatorname{id}_{R}(x,y)<\infty$. Again,
Bass’ conjecture says that the ring is Cohen-Macaulay. By the mentioned result
of Murthy, $R$ is Gorenstein. So, $\operatorname{pd}_{R}(\mathfrak{a})<\infty$
for all unmixed ideal of height two. It remains to apply the discussion below
[6, Corollary 5]. ∎

Here, is another variation of Proposition 2.11:

######  Observation 2.13.

Suppose any two generated ideal of $R$ is of finite injective dimension. Then
$R$ is a Gorenstein $\operatorname{UFD}$.

###### Proof.

The ring $R:=(1)$ is of finite injective dimension. So, $R$ is Gorenstein.
***Alternatively, use [23, II.5.5]. Over Gorenstein rings, finite injective
dimension implies finite projective dimension (see [10, 3.1.25]). Now, result
is clear by [11, 5.3]. ∎

##  3\. Modules with few generators

We start by the following easy exercise:

######  Fact 3.1.

Let $R$ be a local domain. Suppose there is a cyclically presented module $M$
of finite injective dimension. Then $R$ is Gorenstein.

###### Proof.

Let $M=R/xR$ for some $x$. Set $i:=\operatorname{id}_{R}(M)<\infty$ and look
at $0\to R\stackrel{{\scriptstyle x}}{{\longrightarrow}}R\to M\to 0$. This
gives $\operatorname{Ext}^{i+1}_{R}(k,R)\stackrel{{\scriptstyle
x}}{{\longrightarrow}}\operatorname{Ext}^{i+1}_{R}(k,R)\to\operatorname{Ext}^{i+1}_{R}(k,M)=0$.
By Nakayama, $\operatorname{Ext}^{i+1}_{R}(k,R)=0$, and so
$\operatorname{id}_{R}(R)<\infty$. By definition, $R$ is Gorenstein. ∎

The following is nontrivial:

######  Fact 3.2.

(Peskine-Szpiro) Suppose there is a cyclic module $M$ of finite injective
dimension. Then $R$ is Gorenstein.

######  Corollary 3.3.

Suppose $R$ is Gorenstein. If for some $\mathfrak{p}\in\operatorname{Spec}(R)$
one has $\operatorname{id}_{R}(\mathfrak{p})<\infty$, then $R$ is an integral
domain.

###### Proof.

We have $\operatorname{pd}_{R}(\mathfrak{p})<\infty$. Then $R$ is an integral
domain. ∎

######  Corollary 3.4.

Suppose for some principal ideal $\mathfrak{p}\in\operatorname{Spec}(R)$ one
has $\operatorname{id}_{R}(\mathfrak{p})<\infty$. Then $R$ is an integral
domain.

###### Proof.

Thanks to Fact 3.2, $R$ is Gorenstein. By Corollary 3.3, $R$ is an integral
domain. ∎

Here, we present a new proof of [27, Theorem 5.2] and [22, Theorem 2.11(iv)].

######  Corollary 3.5.

Let $I\lhd R$ be nonzero. If $\operatorname{id}_{R}(R/I)<\infty$ then $I$ has
an $R$-regular element.

###### Proof.

Thanks to Fact 3.2, $R$ is Gorenstein. By [10, 3.1.25],
$\operatorname{pd}_{R}(R/I)<\infty$. Equivalently, $I$ has a finite free
resolution. By a famous result of Burch [10, Corollary 1.4.7] $I$ contains a
regular element. ∎

######  Notation 3.6.

By $\mu(-)$ we mean the minimal number of elements that needs to generates a
finitely generated module $(-)$.

######  Proposition 3.7.

Let $(R,\mathfrak{m})$ be quasi-normal, and let $M$ be such that

  1. i)

$\operatorname{id}_{R}(M)<\infty$,

  2. ii)

$M$ is reflexive,

  3. iii)

$\mu(M)\leq 2$.

Then $R$ is Gorenstein.

###### Proof.

By Fact 3.2 we may assume that $M$ is not cyclic. In the light of Bass’
conjecture, $R$ is Cohen-Macaulay. Let $d:=\dim(R)$, and
$\underline{x}:=x_{1},\ldots,x_{d}$ be a system of parameter. In the light of
[23, II.5.4] we have

$\operatorname{Ext}^{d}_{R}(\operatorname{Hom}_{R}(k,R/\underline{x}R),M)\cong
k\otimes_{R}\frac{M}{\underline{x}M}\cong\frac{M}{\mathfrak{m}M}.$

The notation $r(R)$ stands for type of $R$. We deduce by this that

$\oplus_{r(R)}\operatorname{Ext}^{d}_{R}(k,M)\cong k\oplus k.$

Let us denote $\mu_{i}(M)$ for the $i^{th}$ Bass number of $M$, i.e., the
number of copies of $E_{R}(k)$ in the minimal injective resolution of $M$.
Following this terminology, we deduce $\mu_{d}(M)r(R)=2$. In the case
$r(R)=1$, there is nothing to proof, as the rings of type one are Gorenstein.
So, without loss of generality, $r(R)=2$. This implies that $\mu_{d}(M)=1$.
Since $M$ is torsion-free, it is of maximal dimension. Then, in view of [13,
3.4.c] we observe that $M$ is maximal Cohen-Macaulay. Without loss of
generality we may and do assume that the ring is complete. Over Cohen-Macaulay
and complete ring the canonical module exists, as it is unique, we denoted it
by $\omega_{R}$. Since $M$ is maximal Cohen-Macaulay and of finite injective
dimension, we use a result of Sharp [22] to present the identification
$M=\oplus\omega_{R}$ (also, see [10, 3.3.28]). Since $\mu(M)=2$ and
$\mu(\omega_{R})=r(R)=2$, we deduce that $M=\omega_{R}$. By $(-)^{\ast}$ we
mean $\operatorname{Hom}_{R}(-,R)$. Thanks to [16, Appendix], there is an
exact sequence

$0\longrightarrow\omega^{\ast}\longrightarrow
R^{2}\longrightarrow\omega_{R}\longrightarrow 0\quad(\ast)$

Recall that $\operatorname{Ext}^{1}_{R}(R^{2},R)=0$, and apply
$\operatorname{Hom}_{R}(-,R)$ to the displayed short exact sequence $(\ast)$,
gives us the following:

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>\omega^{\ast}@>{}>{}>R^{2}@>{}>{}>\omega^{\ast\ast}@>{}>{}>\operatorname{Ext}^{1}_{R}(\omega_{R},R)@>{}>{}>0\\\
=@A{}A{}A=@A{}A{}A=@A{}A{}A\\\
0@>{}>{}>\omega^{\ast}@>{}>{}>R^{2}@>{}>{}>\omega @>{}>{}>0\end{CD}$

From this we conclude that $\operatorname{Ext}^{1}_{R}(\omega_{R},R)=0$. Now,
we are going to apply [2, 5.3] and to deduce that $R$ is Gorenstein. ∎

######  Proposition 3.8.

Let $(R,\mathfrak{m})$ be quasi-normal of type at most two, and let $M$ be an
indecomposable module such that

  1. i)

$\operatorname{id}_{R}(M)<\infty$,

  2. ii)

$M$ is reflexive,

  3. iii)

$\mu(M)\leq 4$.

Then $R$ is Gorenstein.

###### Proof.

We may assume that $\mu(M)>3$. Also, we may assume that $r(R)=2$. By the above
argument, $2\mu_{d}(M)=\mu_{d}(M)r(R)=\mu(M)$. From this, $\mu(M)$ is even. In
fact $\mu(M)=4$ and so $\mu_{d}(M)=2$. Recall that $M$ is $(S_{2})$. According
to [1, Theorem 3] we know $M$ is Cohen-Macaulay, and so maximal Cohen-
Macaulay. Since it is of finite injective dimension, $M=\oplus\omega_{R}$.
Since $M$ is indecomposable, $M=\omega_{R}$. But,

$4=\mu(M)=\mu(\omega_{R})=r(R)=2.$

This contradiction completes the proof. ∎

######  Corollary 3.9.

Let $(R,\mathfrak{m})$ be quasi-normal of type two, and let $M$ be a module
such that $\operatorname{id}_{R}(M)<\infty$. Then $\mu(M)$ is even.

The following is dual to [25, 6.2.4]:

######  Observation 3.10.

Suppose there is a finite length module $A$ of finite injective dimension.
Then $R$ is Cohen-Macaulay.

###### Proof.

We may assume the ring is complete. By Matlis,
$M:=\operatorname{Hom}_{R}(A,E_{R}(k))$ is of finite length and of finite flat
dimension. By [14], $\operatorname{pd}_{R}(M)<\infty$. Since $\ell(M\otimes
R)<\infty$, and by intersection theorem (see for example [10, Theorem 9.4.5]),
$\dim R\leq\operatorname{pd}_{R}(M)=0pt(R)-0pt(M)=0pt(R)$, as claimed. ∎

The following slightly extends [25, 6.2.4]:

######  Observation 3.11.

Suppose there is an artinian module $A$ of finite projective dimension. Then
$R$ is Cohen-Macaulay.

###### Proof.

This is similar to the previous item. ∎

The following drops the Cohen-Macaulay assumption from [27, Proposition 2.40]:

######  Corollary 3.12.

Let $(R,\mathfrak{m})$ be a 2-dimensional ring, $I$ an ideal of grade two and
finite projective dimension. Then $r(R/I)=r(R)(\mu(I)-1)$.

###### Proof.

Recall from $2=\dim R\geq\operatorname{ht}(I)\geq\operatorname{grade}(I,R)=2$
that $I$ is $\mathfrak{m}$-primary. From $0\to I\to R\to R/I\to 0$ we deduce
that the artinian module $R/I$ is of finite projective dimension. By
Observation 3.11, $R$ is Cohen-Macaulay. The exact sequence $\ldots\to
H^{i}_{\mathfrak{m}}(I)\to H^{i}_{\mathfrak{m}}(R)\to
H^{i}_{\mathfrak{m}}(R/I)\to\ldots$ gives us $0pt(I)=1$. By the Cohen-Macaulay
property and Auslander-Buchsbaum formula,
$\operatorname{pd}_{R}(I)=0pt(R)-0pt(I)=1$. These allow us to apply [27,
Proposition 2.40] and conclude $r(R/I)=r(R)(\mu(I)-1)$. ∎

Peskine-Szpiro [23, Corollary II.5.7] study the case $\mu(I)=2$. Here, is the
case $\mu(I)=3$:

######  Corollary 3.13.

Let $(R,\mathfrak{m})$ be a 2-dimensional ring, $I$ an ideal of grade two and
finite injective dimension. If $\mu(I)=3$, then:

  1. i)

$R$ is Gorenstein,

  2. ii)

$R/I$ is Cohen-Macaulay and of type two.

###### Proof.

i) Recall that $R$ is Cohen-Macaulay. Let $\underline{x}=x_{1},x_{2}$ be a
parameter sequence. Then
$\operatorname{Ext}^{2}_{R}(\operatorname{Hom}_{R}(k,R/\underline{x}R),I)\cong\frac{I}{\mathfrak{m}I}.$
This gives $r(R)\mu_{2}(I)=3$. Recall that $0pt(I)=1<2=\dim(I)$. Thanks to
[13, Proposition 3.8] we know that $\mu_{2}(I)\geq 2$. Plugging this in the
previous equality yields that $r(R)=1$, i.e., $R$ is Gorenstein.

ii) Since $R/I$ is zero-dimensional, it is Cohen-Macaulay. Due to the first
item we know $R$ is Gorenstein. This yields that
$\operatorname{pd}_{R}(I)<\infty$. By Corollary 3.12 $R/I$ is of type two. ∎

##  4\. Auslander’s zero-divisor conjecture

As a basic tool, we prepare a version of Auslander’s zero-divisor conjecture
for modules of finite injective dimension. This has its own importance.

######  Proposition 4.1.

Let $M$ be finitely generated, and of finite injective dimension. Then any
$M$-regular element is $R$-regular.

###### Proof.

By [12, Proposition 3.6] it is enough to show that

$\operatorname{grade}(\mathfrak{p},M)\leq\operatorname{grade}(\mathfrak{p},R)\quad\forall\mathfrak{p}\in\operatorname{Spec}(R).$

Following Bass’ conjecture, $R$ is Cohen-Macaulay. Then

$\begin{array}[]{ll}\operatorname{grade}(\mathfrak{p},M)&\stackrel{{\scriptstyle(1)}}{{\leq}}0pt(M_{\mathfrak{p}})\\\
&\stackrel{{\scriptstyle(2)}}{{\leq}}\dim(M_{\mathfrak{p}})\\\
&\stackrel{{\scriptstyle(3)}}{{\leq}}\operatorname{id}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\\\
&\stackrel{{\scriptstyle(4)}}{{=}}0pt(R_{\mathfrak{p}})\\\
&\stackrel{{\scriptstyle(5)}}{{=}}\operatorname{grade}(\mathfrak{p},R),\end{array}$

where

  1. 1):

see [18, Exercise 16.5];

  2. 2):

see [10, Proposition 1.2.12];

  3. 3),

4): these are in [10, Theorem 3.1.17];

  4. 5):

see [10, Theorem 2.1.3(b)] and recall that $R$ is Cohen-Macaulay.

The proof is now complete. ∎

Let us extend Proposition 4.1 by a new argument:

######  Proposition 4.2.

Let $M$ be finitely generated, and of finite injective dimension. Then any
$M$-regular element is $R$-regular.

###### Proof.

Let $x\in\mathfrak{m}$ be $M$-regular. Then $x$ is regular over
$M\otimes_{R}\widehat{R}$, as the completion is flat. It easy to see
$\operatorname{id}_{\widehat{R}}(M\otimes_{R}\widehat{R})<\infty$. In sum,
$\widehat{R}$ is Cohen-Macaulay and accepts the canonical module. Recall that

$\begin{array}[]{ll}\operatorname{Ass}_{\widehat{R}}\big{(}\operatorname{Hom}_{\widehat{R}}(\omega_{\widehat{R}},M\otimes_{R}\widehat{R})\big{)}&\stackrel{{\scriptstyle(1)}}{{=}}\operatorname{Supp}(\omega_{\widehat{R}})\cap\operatorname{Ass}_{\widehat{R}}(M\otimes_{R}\widehat{R})\\\
&\stackrel{{\scriptstyle(2)}}{{=}}\operatorname{Spec}({\widehat{R}})\cap\operatorname{Ass}_{\widehat{R}}(M\otimes_{R}\widehat{R})\\\
&\stackrel{{\scriptstyle(3)}}{{=}}\operatorname{Ass}_{\widehat{R}}(M\otimes_{R}\widehat{R}),\end{array}$

where

  1. 1):

see [10, 1.2.27];

  2. 2):

see [10, Theorem 3.3.5(b)];

  3. 3):

trivial.

Since

$\begin{array}[]{ll}\operatorname{Zd}(\operatorname{Hom}_{\widehat{R}}(\omega_{\widehat{R}},M\otimes_{R}\widehat{R}))&\stackrel{{\scriptstyle}}{{=}}\cup_{\mathfrak{q}\in\operatorname{Ass}(\operatorname{Hom}_{\widehat{R}}(\omega_{\widehat{R}},M\otimes_{R}\widehat{R}))}\mathfrak{q}\\\
&\stackrel{{\scriptstyle}}{{=}}\cup_{\mathfrak{q}\in\operatorname{Ass}(M\otimes_{R}\widehat{R})}\mathfrak{q}\\\
&\stackrel{{\scriptstyle}}{{=}}\operatorname{Zd}(M\otimes_{R}\widehat{R}),\end{array}$

and $x\notin\operatorname{Zd}(M\otimes_{R}\widehat{R})$, we deduce that $x$ is
regular over
$\operatorname{Hom}_{\widehat{R}}(\omega_{\widehat{R}},M\otimes_{R}\widehat{R})$.
It is easy to see from [22] that
$\operatorname{pd}_{\widehat{R}}\big{(}\operatorname{Hom}_{\widehat{R}}(\omega_{\widehat{R}},M\otimes_{R}\widehat{R})\big{)}$
is finite. Also, see [10, 9.6.5(b)]. Following Auslander’s zero-divisor, $x$
is $\widehat{R}$-regular, see [10, Theorem 9.4.7]. Since
$R\subseteq\widehat{R}$, $x$ is $R$-regular. This is what we want to prove. ∎

Let us present a modern proof of Proposition 2.2:

######  Corollary 4.3.

Suppose $\min(R)$ is singleton. If for some
$\mathfrak{p}\in\operatorname{Spec}(R)$ one has
$\operatorname{id}_{R}(\mathfrak{p})<\infty$, then $R$ is an integral domain.

###### Proof.

We know $R$ is Cohen-Macaulay, and in particular,
$\min(R)=\operatorname{Ass}(R)$. Let $\min(R)=\\{P\\}$. Let $\pi:R\to
R_{\mathfrak{p}}$ be the natural localization map, sending $r$ to $r/1$. Let
$r\in\ker(\pi)$. By definition, there is $x\in R\setminus\mathfrak{p}$ such
that $rx=0$. We claim that $x$ is $\mathfrak{p}$-regular. Indeed, if not, then

$x\in\operatorname{Zd}(\mathfrak{p})=\cup_{\mathfrak{q}\in\operatorname{Ass}(\mathfrak{p})}\mathfrak{q}\subseteq\cup_{\mathfrak{q}\in\operatorname{Ass}(R)}\mathfrak{q}=P$

Since $\min(R)=\\{P\\}$, we see $P\subseteq\mathfrak{p}$. So,
$x\in\mathfrak{p}$. This contradiction shows that $x$ is
$\mathfrak{p}$-sequence. Now, we apply the assumption along with Proposition
4.2 to observe that $x$ is $R$-sequence. From this $r=0$, i.e., $\pi$ is
injective. Now recall from $\operatorname{id}_{R}(\mathfrak{p})<\infty$ that
$R_{\mathfrak{p}}$ is regular and local. Regular local rings are domain (see
[10, Proposition 2.2.3]). From this, $R_{\mathfrak{p}}$ is an integral domain.
Since $R\subseteq R_{\mathfrak{p}}$ we get to the desired claim. ∎

Let us reprove Auslander’s zero-divisor conjecture, over Cohen-Macaulay rings:

######  Observation 4.4.

Let $(R,\mathfrak{m})$ be Cohen-Macaulay, $M$ be finitely generated with
$\operatorname{pd}_{R}(M)<\infty$. If $x$ is $M$-regular, then $x$ is
$R$-regular.

###### Proof.

We may and do assume that $R$ is complete. In particular, $\omega_{R}$ exists.
The assumption $\operatorname{pd}_{R}(M)<\infty$ implies that
$\operatorname{id}_{R}(M\otimes_{R}\omega_{R})<\infty$. We proved in the
previous proposition that
$\operatorname{Ass}(M\otimes_{R}\omega_{R})=\operatorname{Ass}(\operatorname{Hom}_{R}(\omega_{R},M\otimes_{R}\omega_{R}))$.
We are going to use [22, Theorem 2.9] to deduce that
$\operatorname{Hom}_{R}(\omega_{R},M\otimes_{R}\omega_{R})\cong M$. Combining
this along with the previous observation, we know that
$\operatorname{Ass}(M\otimes_{R}\omega_{R})=\operatorname{Ass}(M)$. So, $x$ is
$(M\otimes_{R}\omega_{R})$-regular. Thanks to Proposition 4.1, $x$ is
$R$-regular, as claimed. ∎

######  Remark 4.5.

i) Recall that [4] asks the computation of associated prime ideals of tensor
product, as a sample, we proved in the setting of Observation 4.4 that

$\operatorname{Ass}(M\otimes_{R}\omega_{R})=\operatorname{Ass}(M).$

In particular, we reproved [4, Corollary 4.6] by a different argument.

ii) By symmetry, and in the proof of Observation 4.4, one may use [4,
Corollary 4.6] instead of [22, Theorem 2.9].

Recall from [7] module $T$ is said to be tor-rigid if there is a non-negative
integer $n$ such that for every finitely generated $R$-module $M$, vanishing
of $\operatorname{Tor}_{n}^{R}(T,M)$ implies
$\operatorname{Tor}_{n+i}^{R}(T,M)$ vanishes for all $i\geq 0$. By Auslander
[7, 4.3], this implies the zero-divisor property.

######  Remark 4.6.

Despite to above useful duality between modules of finite projective dimension
and modules of finite injective dimension, let us present a different
situation. It is easy to see modules of projective dimension one are tor-
rigid. But modules of injective dimension one are not necessarily tor-rigid.
This may appear even over 1-dimensional integral domains of type two. For
instance, let $R:=k[[x^{3},x^{4},x^{5}]]$. Then $(x^{3},x^{4})$ is of
injective dimension one, but not tor-rigid.

For more connections to these topics, we cite [4].

##  5\. Almost complete-intersection

In this section $(S,\mathfrak{n})$ is regular, $I\lhd S$ is an ideal, and
$R:=S/I$. We study the following:

######  Question 5.1.

(See [15]) Suppose $\operatorname{id}_{R}(I/I^{2})$ is finite. Is $I$
generated by a regular sequence?

Here, we support it by the following two observations:

######  Observation 5.2.

If $R$ is zero-dimensional, then $I$ is generated by a regular sequence.

###### Proof.

The $R$-module $I/I^{2}$ is injective, as injective dimension is bounded by
depth. By Matlis decomposition, $I/I^{2}=\oplus\omega_{R}$. Now, let $n\geq 0$
be such that $\mathfrak{m}^{n}\neq 0$ but $\mathfrak{m}^{n+1}=0$. Then

$\mathfrak{m}^{n}(I+I^{2})=(\mathfrak{m}^{n}I+I^{2})\subseteq\mathfrak{m}^{n+1}+I^{2}=0.$

From this, $\mathfrak{m}^{n}\omega_{R}=0.$ As $\omega_{R}$ is faithful, we
conclude that $n=0$. In other words, $R$ is field. So, $I$ is generated by a
regular sequence. ∎

######  Corollary 5.3.

Question 5.1 reduces to locally complete-intersection ideals over the
punctured spectrum.

###### Proof.

We proceed by induction on $d:=\dim R$. The case $d=0$ is subject of
Observation 5.2. So, we may assume $d>0$ and suppose the desired claim holds
for rings of dimension $<d$. The assumptions of Question 5.1 behave well with
respect to localization for all
$\mathfrak{q}\in\operatorname{Var}(I)\setminus\\{\mathfrak{n}\\}$, and note
that $\dim(R_{\mathfrak{q}})<d$. These allow us to apply the inductive step
and assume in addition that $I$ is locally complete-intersection over the
punctured spectrum. This completes the proof. ∎

######  Observation 5.4.

Suppose $I$ is prime, $\mu(I)\leq\operatorname{ht}(I)+1$ and
$\operatorname{id}_{R}(I/I^{2})<\infty$. Then $I$ is generated by a regular
sequence.

###### Proof.

By Bass’ conjecture, $R$ is Cohen-Macaulay. Then $K_{R}=\omega_{R}$ is of
finite injective dimension. Suppose on the way of contradiction that
$\mu(I)=\operatorname{ht}(I)+1$. In the light of [20, Proposition 1] there is
an exact sequence

$0\longrightarrow K_{R}\longrightarrow R^{n}\longrightarrow
I/I^{2}\longrightarrow 0,$

which show that $\operatorname{id}_{R}(R)<\infty$. Over Gorenstein rings,
finite injective dimension implies finite projective dimension. So,
$\operatorname{pd}_{R}(I/I^{2})$ is finite. Follows from [28] that $I$ is
generated by a regular sequence, as claimed. ∎

Let us present a new proof of [17]:

######  Theorem 5.5.

(Kunz) Almost complete intersection domains are not Gorenstein rings.

###### Proof.

We adopt the previous notation. Suppose on the way of contradiction that $R$
is Gorenstein. Then from $0\to K_{R}\to R^{n}\to I/I^{2}\to 0,$ and finiteness
of $\operatorname{id}_{R}(R)$, we conclude that
$\operatorname{id}_{R}(I/I^{2})<\infty$. By the previous result,
$\operatorname{ht}(I)=\mu(I)$, a contradiction. ∎

######  Corollary 5.6.

Suppose $I$ is prime and generated by $3$ elements. If
$\operatorname{id}_{R}(I/I^{2})<\infty$, then $I$ is generated by a regular
sequence.

###### Proof.

We have $\operatorname{ht}(I)\leq 3$. In the case $\operatorname{ht}(I)=1$
there is nothing to prove, because it becomes principal as $S$ is
$\operatorname{UFD}$. In the case $\operatorname{ht}(I)=3$ the claim is in
[10, 1.2.21]. Then we may assume that $\operatorname{ht}(I)=2$. The desired
claim follows by Observation 5.4. ∎

######  Observation 5.7.

If $R$ is Gorenstein and $\operatorname{id}_{R}(I/I^{2})<\infty$, then $I$ is
generated by a regular sequence.

###### Proof.

Since $R$ is Gorenstein, $\operatorname{pd}_{R}(I/I^{2})<\infty$. In view of
[8, main result] $I$ generated by a regular sequence. ∎

For more connections to [15], we cite [3].

##  6\. Matlis and canonical module

A ring $R$ of dimension $d:=\dim(R)$ is called quasi-Gorenstein, if
$H^{d}_{\mathfrak{m}}(R)\cong E_{R}(k)$. When is
$H^{d}_{\mathfrak{m}}(R)\twoheadrightarrow E_{R}(k)$ surjective? The following
may regard as a higher version of [19, 15.17]:

######  Proposition 6.1.

Suppose $R$ is complete, generically Gorenstein and Cohen-Macaulay. Then
$H^{\dim R}_{\mathfrak{m}}(R)\twoheadrightarrow E_{R}(k)$ is surjective.

###### Proof.

Let $d:=\dim(R)$. Since the ring is complete and Cohen-Macaulay, we know
$\omega_{R}$ exists. Since $R$ is generically Gorenstein, $\omega_{R}\lhd R$.
Suppose first that $d:=\dim R=0$. Then $R$ is Gorenstein, and so
$\omega_{R}=E_{R}(k)=R$. To see
$H^{d}_{\mathfrak{m}}(R)\stackrel{{\scriptstyle}}{{\twoheadrightarrow}}E_{R}(k)$,
it remains to note that $H^{d}_{\mathfrak{m}}(R)=R$. So, we may assume that
$d>0$. Either $\omega_{R}=R$, or $\omega_{R}\lhd R$ is of height one. Without
loss of the generality we may assume that $\omega_{R}\lhd R$ is of height one.
The Cohen-Macaulay assumption says
$1=\operatorname{ht}(\omega_{R})=\operatorname{grade}(\omega_{R},R)$. Let
$x\in\omega_{R}$ be a regular element. This gives an embedding
$R\hookrightarrow Rx\subseteq\omega_{R}$. Apply
$\operatorname{Hom}_{R}(-,E_{R}(k))$ to it, and use local duality, we observe
that

$\begin{CD}\omega^{\vee}@>{}>{}>R^{\vee}@>{}>{}>0\\\ @A{}A{}A\cong @A{}A{}A\\\
H^{d}_{\mathfrak{m}}(R)@>{}>{}>E_{R}(k).\\\ \end{CD}$

In other words, the induced map $H^{d}_{\mathfrak{m}}(R)\twoheadrightarrow
E_{R}(k)$ is surjective, as claimed. ∎

######  Proposition 6.2.

$d:=\dim(R)$. Suppose $H^{d}_{\mathfrak{m}}(R)\twoheadrightarrow E_{R}(k)$. If
$d=0$, then $R$ is Gorenstein.

###### Proof.

Recall that

$\begin{CD}H^{d}_{\mathfrak{m}}(R)@>{}>{}>E_{R}(k)@>{}>{}>0\\\
=@A{}A{}A=@A{}A{}A\\\ R@>{f}>{}>E_{R}(k).\\\ \end{CD}$

Let $I:=\ker(f)$. So, $R/I=E_{R}(k)$. Recall that

$R\stackrel{{\scriptstyle\subseteq}}{{\longrightarrow}}E(R)=\oplus
E_{R}(k)=\oplus R/I.$

From this, $I=0$. Thus, $R=E_{R}(k)$. Consequently, $R$ is Gorenstein. ∎

Let $R$ be a 1-dimensional integral domain and let $Q$ be the fraction field.
Matlis proved that $Q/\omega_{R}=E_{R}(k)$. Here, is the higher version
(recall that $Q/\omega_{R}=H^{1}_{\mathfrak{m}}(\omega_{R})$):

######  Proposition 6.3.

Suppose $R$ is complete, integral domain and Cohen-Macaulay. Let $I$ be a
nonzero Cohen-Macaulay ideal. Let $d:=\dim(R)$. The following are equivalent:

  1. i)

$\operatorname{id}_{R}(I)<\infty$,

  2. ii)

$I=\omega_{R}$,

  3. iii)

$H^{d}_{\mathfrak{m}}(I)=E_{R}(k)$.

###### Proof.

$i)\Rightarrow ii)$: Since $\dim I=d$, the ideal $I$ becomes maximal Cohen-
Macaulay. The desired claim is in [10, 3.3.28].

$ii)\Rightarrow iii)$: Due to the right exactness,
$H^{d}_{\mathfrak{m}}(\omega_{R})=H^{d}_{\mathfrak{m}}(R)\otimes_{R}\omega_{R}$.
Recall that $H^{d}_{\mathfrak{m}}(\omega_{R})$ is artinian, and apply Matlis
duality to it,

$\begin{array}[]{ll}H^{d}_{\mathfrak{m}}(\omega_{R})&\stackrel{{\scriptstyle}}{{=}}H^{d}_{\mathfrak{m}}(\omega_{R})^{\vee\vee}\\\
&=\operatorname{Hom}_{R}\big{(}H^{d}_{\mathfrak{m}}(R)\otimes_{R}\omega_{R},E_{R}(k)\big{)}^{\vee}\\\
&=\operatorname{Hom}_{R}\big{(}\omega_{R},\operatorname{Hom}_{R}(H^{d}_{\mathfrak{m}}(R),E_{R}(k)\big{)}^{\vee}\\\
&=\operatorname{Hom}_{R}(\omega_{R},\omega_{R})^{\vee}\\\ &=R^{\vee}\\\
&=E_{R}(k).\end{array}$

$iii)\Rightarrow i)$: Let $D(-):=\operatorname{Hom}_{R}(-,\omega_{R})$. We
apply local duality along with the assumption to deduce that:

$D(I)=H^{d}_{\mathfrak{m}}(I)^{\vee}=E_{R}(k)^{\vee}=\widehat{R}=R.$

Over maximal Cohen-Macaulay, $D^{2}(-)=\emph{Id}(-)$. Then $I\cong
D^{2}(I)\cong D(R)\cong\omega_{R}.$ Since $\omega_{R}$ is of finite injective
dimension, $I$ is as well. ∎

######  Corollary 6.4.

Let $R$ be a 1-dimensional integral domain and let $Q$ be the fraction field.
Let $I\lhd R$ be nonzero. Then $Q/I=E_{R}(k)$ iff $I=\omega_{R}$.

The following was proved by Matlis in 1-dimensional complete Gorenstein domain
case, and by Auslander [5] in the $\mathfrak{m}$-adic complete case. Since
being complete in $\mathfrak{m}$-adic topology implies completeness in
$R$-topology, so it extends both:

######  Observation 6.5.

Let $R$ be a quasi-local domain complete in $R$-topology. Then
$\operatorname{Ext}^{1}_{R}(Q/R,R)\cong R$.

###### Proof.

Completeness in $R$-topology implies $\operatorname{Ext}^{1}_{R}(Q,R)=0$.
Also, $\operatorname{Hom}_{R}(Q,R)=0$. The sequence $0\to R\to Q\to Q/R\to 0$
implies

$0=\operatorname{Hom}_{R}(Q,R)\to\operatorname{Hom}_{R}(R,R)\to\operatorname{Ext}^{1}_{R}(Q/R,R)\to\operatorname{Ext}^{1}_{R}(Q,R)=0,$

i.e., the claim follows.∎

Noetherian local rings are reduced***Sorry, we used “reduced” for different
concepts. by Krull’s intersection theorem. So, the following is converse to
Auslander [5]:

######  Observation 6.6.

Let $R$ be a quasi-local reduced domain such that
$\operatorname{Ext}^{1}_{R}(Q/R,R)\cong R$. Then $R$ is complete in
$R$-topology.

###### Proof.

The sequence $0\to R\to Q\to Q/R\to 0$ implies

$0=\operatorname{Hom}_{R}(Q,R)\to\operatorname{Hom}_{R}(R,R)\to\operatorname{Ext}^{1}_{R}(Q/R,R)\to\operatorname{Ext}^{1}_{R}(Q,R)\to\operatorname{Ext}^{1}_{R}(R,R)=0.$

Recall that $\operatorname{Ext}^{1}_{R}(Q,R)=\oplus_{I}Q$ is a $Q$-vector
space. We localize the above sequence and use the assumption
$\operatorname{Ext}^{1}_{R}(Q/R,R)\cong R$ to deduce that
$\operatorname{Ext}^{1}_{R}(Q,R)=\oplus_{I}Q=0$. The reduced assumption
implies that $0\to R\to\operatornamewithlimits{\varprojlim}_{r\in
R}R/rR\to\operatorname{Ext}^{1}_{R}(Q,R)\to 0$ is exact. Since
$\operatorname{Ext}^{1}_{R}(Q,R)=0$, we have
$R=\operatornamewithlimits{\varprojlim}_{r\in R}R/rR$, as claimed.∎

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