# Rings with $q$-torsionfree canonical modules

Naoki Endo  School of Political Science and Economics, Meiji University, 1-9-1
Eifuku, Suginami-ku, Tokyo 168-8555, Japan<EMAIL_ADDRESS>https://www.isc.meiji.ac.jp/ endo/ ,  Laura Ghezzi  Department of Mathematics,
New York City College of Technology and the Graduate Center, The City
University of New York, 300 Jay Street, Brooklyn, NY 11201, U.S.A.; 365 Fifth
Avenue, New York, NY 10016, U.S.A<EMAIL_ADDRESS>,  Shiro Goto
Department of Mathematics, School of Science and Technology, Meiji University,
1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan<EMAIL_ADDRESS>,
Jooyoun Hong  Department of Mathematics, Southern Connecticut State
University, 501 Crescent Street, New Haven, CT 06515-1533, U.S.A.
<EMAIL_ADDRESS>,  Shin-ichiro Iai  Mathematics laboratory, Sapporo
College, Hokkaido University of Education, 1-3 Ainosato 5-3, Kita-ku, Sapporo
002-8502, Japan<EMAIL_ADDRESS>,  Toshinori Kobayashi
Department of Mathematics, School of Science and Technology, Meiji University,
1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
<EMAIL_ADDRESS>,  Naoyuki Matsuoka  Department of Mathematics,
School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-
ku, Kawasaki 214-8571, Japan<EMAIL_ADDRESS>and  Ryo Takahashi
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a Noetherian local ring with canonical module $\mathrm{K}_{A}$. We
characterize $A$ when $\mathrm{K}_{A}$ is a torsionless, reflexive, or
$q$-torsionfree module for an integer $q\geq 3$. If $A$ is a Cohen-Macaulay
ring, H.-B. Foxby proved in 1974 that the $A$-module $\mathrm{K}_{A}$ is
$q$-torsionfree if and only if the ring $A$ is $q$-Gorenstein. With mild
assumptions, we provide a generalization of Foxby’s result to arbitrary
Noetherian local rings admitting the canonical module. In particular, since
the reflexivity of the canonical module is closely related to the ring being
Gorenstein in low codimension, we also explore quasi-normal rings, introduced
by W. V. Vasconcelos. We provide several examples as well.

2020 Mathematics Subject Classification. 13H10, 13A02, 13A15.

Key words and phrases. Canonical module, Gorenstein ring, Cohen-Macaulay ring,
$q$-torsionfree module, $q$-Gorenstein ring, quasi-normal ring

N. Endo was partially supported by JSPS Grant-in-Aid for Young Scientists
20K14299. L. Ghezzi was partially supported by the Fellowship Leave from the
New York City College of Technology-CUNY (Fall 2022-Spring 2023) and by a
grant from the City University of New York PSC-CUNY Research Award Program
Cycle 53. S. Goto was partially supported by JSPS Grant-in-Aid for Scientific
Research (C) 21K03211. J. Hong was partially supported by the Sabbatical Leave
Program at Southern Connecticut State University (Spring 2022). T. Kobayashi
was partly supported by JSPS Grant-in-Aid for JSPS Fellows 21J00567. N.
Matsuoka was partially supported by JSPS Grant-in-Aid for Scientific Research
(C) 18K03227. R. Takahashi was partially supported by JSPS Grant-in-Aid for
Scientific Research (C) 19K03443.

Dedicated to the memory of Wolmer V. Vasconcelos

##  1\. Introduction

This paper investigates the question of the structure of a Noetherian local
ring $A$ if its canonical module $\mathrm{K}_{A}$ is a torsionless, reflexive,
or more generally, $q$-torsionfree $A$-module for an integer $q\geq 3$. The
notion of $q$-torsionfree modules was one of the central contributions of the
famous research of M. Auslander and M. Bridger [1], which was succeeded by
H.-B. Foxby [11] to be a striking study of $q$-Gorenstein rings. Among many
interesting results, Foxby settled the above question in the case where $A$ is
a Cohen-Macaulay ring. More precisely, the $A$-module $\mathrm{K}_{A}$ is
$q$-torsionfree if and only if the ring $A$ is $q$-Gorenstein, i.e.,
$A_{\mathfrak{p}}$ is a Gorenstein ring for every
$\mathfrak{p}\in\operatorname{Spec}A$ with
$\operatorname{depth}A_{\mathfrak{p}}<q$ (see [10, Proposition 3.2]). It
remains unclear what happens if we do not assume the ring $A$ is Cohen-
Macaulay. The theory of canonical modules nowadays has been developed mainly
over Cohen-Macaulay rings in connection with the Gorenstein property; see
e.g., [6, 13, 14, 15, 19]. However, over Noetherian local (not necessarily
Cohen-Macaulay) rings, there are also remarkable preceding researches on
canonical modules, including the study of their endomorphism algebras; see [2,
3, 5]. Therefore, behaviors of canonical modules, even for non-Cohen-Macaulay
rings, are interesting and the $q$-torsionfree property is well worth
studying. The motivation for the present research started with this question
that arose while the second and fourth authors were writing the last paper
with Vasconcelos concerning (torsionless) canonical modules [4].

To explain our results more precisely, let us start from definitions which we
will use throughout this paper. For a Noetherian local ring $A$ of dimension
$d$ with maximal ideal $\mathfrak{m}$, a canonical module $K$ of $A$ is a
finitely generated $A$-module satisfying

$\widehat{A}\otimes_{A}K\cong\operatorname{Hom}_{\widehat{A}}(\mathrm{H}^{d}_{\widehat{\mathfrak{m}}}(\widehat{A}),\widehat{E})$

where $\mathrm{H}^{d}_{\widehat{\mathfrak{m}}}(\widehat{A})$ denotes the
$d^{\mbox{\tiny th}}$ local cohomology module of the $\mathfrak{m}$-adic
completion $\widehat{A}$ of $A$ with respect to its maximal ideal
${\displaystyle\widehat{\mathfrak{m}}}$ and ${\displaystyle\widehat{E}}$ is
the injective hull of the $\widehat{A}$-module
$\widehat{A}/\widehat{\mathfrak{m}}$ ([14, Definition 5.6]). Equivalently, a
finitely generated $A$-module $K$ is a canonical module of $A$ if
${\displaystyle\operatorname{Hom}_{A}(K,E)\cong\mathrm{H}^{d}_{\mathfrak{m}}(A)}$,
where $\mathrm{H}^{d}_{\mathfrak{m}}(A)$ is the $d^{\mbox{\tiny th}}$ local
cohomology module of $A$ with respect to $\mathfrak{m}$ and ${\displaystyle
E}$ is the injective hull of $A/\mathfrak{m}$ ([5, Definition 12.1.2, Remarks
12.1.3]). The canonical module $\mathrm{K}_{A}$ is uniquely determined up to
isomorphisms ([2, (1.5)], see also [14, Lemma 5.8]) if it exists. Although the
existence is not guaranteed even for Cohen-Macaulay local domains, provided
$A$ is Cohen-Macaulay, the ring $A$ admits the canonical module if and only if
$A$ is a homomorphic image of a Gorenstein ring ([16, 18]). The fundamental
theory of canonical modules over Cohen-Macaulay rings was developed in the
monumental book [14] of J. Herzog and E. Kunz. We shall in this paper freely
refer to [14] for basic results on canonical modules (see [6, Chapter 3]
also).

We now continue to state our setup. Let $R$ be a Noetherian (not necessarily
local) ring. For an $R$-module $M$, we have a canonical homomorphism

$\varphi:M\rightarrow M^{**}$

defined by ${\displaystyle\big{[}\varphi(x)\big{]}(f)=f(x)}$ for each
${\displaystyle f\in M^{*}}$ and $x\in M$, where
$(-)^{*}=\operatorname{Hom}_{R}(-,R)$ denotes the $R$-dual functor. We say
that $M$ is torsionless (resp. reflexive) if $\varphi$ is injective (resp.
bijective). Torsionless modules are torsionfree, i.e., there is no nonzero
torsion elements, and the converse holds if the total ring of fractions
$\mathrm{Q}(R)$ of $R$ is Gorenstein ([21, Theorem (A.1)]). Moreover, the
$R$-module $M$ is torsionless (resp. reflexive) if and only if $\mbox{\rm
Ext}_{R}^{i}(\mathrm{D}(M),R)=(0)$ for $i=1$ (resp. $i=1,2$), where
$\mathrm{D}(M)$ denotes the Auslander transpose of $M$ ([1]). From this point
of view, Auslander and Bridger introduced a $q$-torsionfree module $M$ to be
$\mbox{\rm Ext}_{R}^{i}(\mathrm{D}(M),R)=(0)$ for all $i=1,2,\ldots,q$. In
addition, for an integer $n$, we say that

  * •

$M$ satisfies $(\mathrm{S}_{n})$ if
${\displaystyle\operatorname{depth}M_{\mathfrak{p}}\geq\min\left\\{n,\,\dim
R_{\mathfrak{p}}\right\\}}$ for every $\mathfrak{p}\in\operatorname{Spec}R$,

  * •

$M$ satisfies $(\widetilde{\mathrm{S}}_{n})$ if
${\displaystyle\operatorname{depth}M_{\mathfrak{p}}\geq\min\left\\{n,\,\operatorname{depth}R_{\mathfrak{p}}\right\\}}$
for every $\mathfrak{p}\in\operatorname{Spec}R$,

  * •

$R$ satisfies $(\mathrm{G}_{n})$ if $R_{\mathfrak{p}}$ is Gorenstein for every
$\mathfrak{p}\in\operatorname{Spec}R$ with ${\displaystyle\dim
R_{\mathfrak{p}}\leq n}$,

  * •

$R$ satisfies $(\widetilde{\mathrm{G}}_{n})$ if $R_{\mathfrak{p}}$ is
Gorenstein for every $\mathfrak{p}\in\operatorname{Spec}R$ with
${\displaystyle\operatorname{depth}R_{\mathfrak{p}}\leq n}$.

The condition $(\mathrm{S}_{n})$ is known as Serre’s condition. A Noetherian
ring satisfying $(\widetilde{\mathrm{G}}_{n})$ coincides with
$(n+1)$-Gorenstein ring in earlier publications such as [1, 11]. The condition
$(\widetilde{\mathrm{G}}_{n})$ is equivalent to saying that the ring satisfies
both $(\mathrm{S}_{n+1})$ and $(\mathrm{G}_{n})$.

Let us now state our results, explaining how this paper is organized. In
Section 2, after recalling the necessary definitions and preliminaries, we
give a criterion for a Noetherian local ring $A$ to have the torsionless
canonical module. We show that the $A$-module $\mathrm{K}_{A}$ is torsionless
if and only if $A_{\mathfrak{p}}$ is Gorenstein for every
$\mathfrak{p}\in\operatorname{Assh}A$, where
$\operatorname{Assh}A=\\{\mathfrak{p}\in\operatorname{Spec}A\mid\dim
A/\mathfrak{p}=\dim A\\}=\operatorname{Ass}_{A}\mathrm{K}_{A}$ (Proposition
2.3). Section 3 is devoted to the characterizations of local rings $A$ with
reflexive canonical modules. When $\dim A=1$, this is exactly the case where
$A$ is a Gorenstein ring (Proposition 3.3). We elaborate on the one-
dimensional case in Section 5. For the higher dimensional case, the
reflexivity of $\mathrm{K}_{A}$ is characterized by the local ring
$A_{\mathfrak{p}}$ being Gorenstein for every
$\mathfrak{p}\in\operatorname{Supp}_{A}K_{A}$ with $\dim A_{\mathfrak{p}}\leq
1$ and $\operatorname{Ass}A\cap V(U)=\operatorname{Assh}A$, where $U$ denotes
the unmixed component of $(0)$ in $A$ and $V(U)$ is the set of all prime
ideals in $A$ containing $U$ (Theorem 3.6). This lead us to obtain Corollary
3.8, which claims that $\mathrm{K}_{A}$ is reflexive if and only if $A$
satisfies $({\mathrm{G}}_{1})$, provided
$\operatorname{Assh}A=\operatorname{Ass}A$. This indicates that the
reflexivity of canonical modules is deeply related to the ring being
Gorenstein in low codimension. Thus Section 4 is dedicated to quasi-normal
rings, i.e., rings with $(\mathrm{S}_{2})$ and $(\mathrm{G}_{1})$, which have
been introduced by Vasconcelos. In Section 6, we generalize Foxby’s result on
$q$-torsionfree canonical modules to arbitrary Noetherian local rings $A$
admitting a canonical module. Our results of Sections 2 and 3 provide a
complete generalization in case $q=1,2$. When $q\geq 3$, Theorem 6.6 states
that the $A$-module $\mathrm{K}_{A}$ is $q$-torsionfree if and only if the
ring $A$ satisfies $(\mathrm{G}_{q-1})$ and $(\mathrm{S}_{q-1})$ on
$\operatorname{Supp}_{A}\mathrm{K}_{A}$, provided that $\mathrm{K}_{A}$
satisfies $(\mathrm{S}_{q})$. In the final section we provide concrete
examples of Cohen-Macaulay and $q$-Gorenstein rings in order to illustrate our
theorems.

##  2\. Torsionless canonical modules

Throughout the section, let $(A,\mathfrak{m})$ be a Noetherian local ring of
dimension $d$. We begin with some preliminaries. Let
${\displaystyle(0)=\bigcap_{\mathfrak{p}\in\operatorname{Ass}A}Q(\mathfrak{p})}$
denote a primary decomposition of $(0)$ in $A$. We set

$\operatorname{Assh}A=\\{\mathfrak{p}\in\operatorname{Spec}A\mid\dim
A/\mathfrak{p}=d\\}\quad\mbox{and}\quad
U=\bigcap_{\mathfrak{p}\in\operatorname{Assh}A}Q(\mathfrak{p})$

where $U$ is called the unmixed component of $(0)$ in $A$. Let $V(U)$ denote
the set of all prime ideals of $A$ containing $U$.

######  Lemma 2.1.

There is an embedding ${\displaystyle 0\rightarrow A/U\rightarrow A}$ of
$A$-modules.

###### Proof.

We may assume that $U\neq(0)$. Then
$\operatorname{Assh}A\subsetneq\operatorname{Ass}A$. Let

$L=\bigcap_{\tiny\mathfrak{p}\in\operatorname{Ass}A\setminus\operatorname{Assh}A}Q(\mathfrak{p}).$

We then have ${\displaystyle
L\not\subseteq\bigcup_{\tiny\mathfrak{p}\in\operatorname{Assh}A}\mathfrak{p}}$.
Choose an element $a\in L$ but ${\displaystyle
a\not\in\bigcup_{\mathfrak{p}\in\operatorname{Assh}A}\mathfrak{p}}$. Since $a$
is a non-zerodivisor on $A/U$ and $aU\subseteq L\cap U=(0)$, we obtain
$((0):_{A}a)=U$. Then $U$ is the kernel of the homomorphism
$\varphi:A\rightarrow A$ given by $\varphi(1)=a$. Thus,
$A/U\cong\operatorname{Im}(\varphi)\hookrightarrow A$. ∎

In the rest of this section, we assume the ring $A$ admits the canonical
module $\mathrm{K}_{A}$. We recall several known facts about $\mathrm{K}_{A}$
which we will use throughout this article; see [2, (1.6), (1.7), (1.8), (1.9),
(1.10), Theorem 3.2] and [14, Korollar 6.3] (also [5, Chapter 12]) for the
proofs.

######  Proposition 2.2.

The following assertions hold true.

  1. $(1)$

The annihilator of $\mathrm{K}_{A}$ is $U$. In particular,
${\dim_{A}\mathrm{K}_{A}=d}$ and
${\operatorname{Ass}_{A}\mathrm{K}_{A}=\operatorname{Assh}A}$.

  2. $(2)$

If ${\displaystyle a\in\mathfrak{m}}$ is $A$-regular, then $a$ is
$\mathrm{K}_{A}$-regular.

  3. $(3)$

$V(U)=\operatorname{Supp}_{A}\mathrm{K}_{A}=\\{\mathfrak{p}\in\operatorname{Spec}A\mid\dim
A=\dim A/\mathfrak{p}+\operatorname{ht}_{A}\mathfrak{p}\\}$.

  4. $(4)$

Both $\mathrm{K}_{A}$ and
${\displaystyle\operatorname{Hom}_{A}(\mathrm{K}_{A},\mathrm{K}_{A})}$ satisfy
$(\mathrm{S}_{2})$.

  5. $(5)$

${\displaystyle\mathrm{K}_{A_{\mathfrak{p}}}=\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}}$
for every
${\displaystyle\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}}$.

  6. $(6)$

$\operatorname{Supp}_{A}\mathrm{K}_{A}=\operatorname{Spec}A$ if and only if
$\operatorname{Min}A=\operatorname{Assh}A$.

  7. $(7)$

$\operatorname{Ass}A=\operatorname{Assh}A$ if and only if $\mathrm{K}_{A}$ is
a faithful $A$-module.

  8. $(8)$

Suppose $A$ is Cohen-Macaulay. If $a\in\mathfrak{m}$ is $A$-regular, then
$\mathrm{K}_{A/(a)}$ exists and
$\mathrm{K}_{A/(a)}\cong\mathrm{K}_{A}/a\mathrm{K}_{A}$.

  9. $(9)$

Suppose that $\mathrm{K}_{A/U}$ exists. Then $\mathrm{K}_{A/U}$, as an
$A$-module, is the canonical module of $A$.

By Proposition 2.2-(2), the canonical module $\mathrm{K}_{A}$ is torsionfree
as an $A$-module. In general, torsionless modules are torsionfree, and the
converse holds if and only if $A_{\mathfrak{p}}$ is a Gorenstein local ring
for every $\mathfrak{p}\in\operatorname{Ass}A$ ([21, Theorem (A.1)]).
Therefore, if the total ring of fractions $\mathrm{Q}(A)$ of $A$ is
Gorenstein, then $\mathrm{K}_{A}$ is torsionless. The following proposition
shows the case where $\mathrm{K}_{A}$ is torsionless without assuming
$\mathrm{Q}(A)$ is Gorenstein. It is also a generalization of [4, Proposition
3.2].

######  Proposition 2.3.

The following conditions are equivalent :

  1. (1)

$\mathrm{K}_{A}$ is a torsionless $A$-module ;

  2. (2)

$A_{\mathfrak{p}}$ is a Gorenstein ring for every
$\mathfrak{p}\in\operatorname{Assh}A$ ;

  3. (3)

$\mathrm{K}_{A}\cong I$ for some ideal $I$ of $A$.

###### Proof.

(1) $\Rightarrow$ (2) Since $\mathrm{K}_{A}$ is torsionless, there exists an
exact sequence $0\rightarrow\mathrm{K}_{A}\rightarrow F$ of $A$-modules, where
$F$ is a finitely generated free $A$-module. Let
${\mathfrak{p}\in\operatorname{Assh}A\subseteq\operatorname{Supp}_{A}\mathrm{K}_{A}}$.
Then $A_{\mathfrak{p}}$ is Artinian and
$\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}$ is the canonical module of
$A_{\mathfrak{p}}$. Therefore, we may assume that
$\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}$ is the injective hull of
$A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$. The splitting monomorphism
$0\rightarrow\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}\rightarrow
F_{\mathfrak{p}}$ induces that $\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}$
is a direct summand of the free $A_{\mathfrak{p}}$-module $F_{\mathfrak{p}}$.
Since $A_{\mathfrak{p}}$ is Artinian, by the Matlis duality, we have
$\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}\cong A_{\mathfrak{p}}$. Hence
$A_{\mathfrak{p}}$ is a Gorenstein ring.

(2) $\Rightarrow$ (3) Let
${W=A\setminus\bigcup_{\mathfrak{p}\in\operatorname{Assh}A}\mathfrak{p}}$. By
assumption, $\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}\cong
A_{\mathfrak{p}}$ for every $\mathfrak{p}\in\operatorname{Assh}A$. Thus,
${\displaystyle W^{-1}\mathrm{K}_{A}\cong W^{-1}A}$. Moreover, we have
${\displaystyle W^{-1}A\cong W^{-1}(A/U)}$ because $W^{-1}U=(0)$ by the proof
of Lemma 2.1. Since every element of $W$ is a non-zerodivisor on both
$\mathrm{K}_{A}$ and $A/U$, the isomorphism ${\displaystyle
W^{-1}\mathrm{K}_{A}\cong W^{-1}(A/U)}$ induces the embedding
${\displaystyle\mathrm{K}_{A}\hookrightarrow A/U}$. By Lemma 2.1, there is an
embedding ${\displaystyle\mathrm{K}_{A}\hookrightarrow A}$.

(3) $\Rightarrow$ (1) is clear. ∎

If $A$ is reduced, which means there are no nonzero nilpotents, then the local
ring $A_{\mathfrak{p}}$ is a field for every
$\mathfrak{p}\in\operatorname{Ass}A$. Hence we obtain the following.

######  Corollary 2.4.

If $A$ is a reduced ring, then $\mathrm{K}_{A}\cong I$ for some ideal $I$ of
$A$.

We recall that if $A$ is Cohen-Macaulay, the canonical module $\mathrm{K}_{A}$
has rank one if and only if the ring $A$ is generically Gorenstein, i.e.,
$A_{\mathfrak{p}}$ is a Gorenstein local ring for every
$\mathfrak{p}\in\operatorname{Min}A$. When one of the equivalent conditions of
[6, Proposition 3.3.18] is satisfied, the canonical module can be identified
with an ideal of $A$ (see also [14, Satz 6.21]). The example below shows that
the assumption $\mathfrak{p}\in\operatorname{Assh}A$ is necessary for
Proposition 2.3.

######  Example 2.5.

Let $S=k[\\![X,Y,Z]\\!]$ be the formal power series ring over a field $k$ and
set $A=S/[(X)\cap(Y,Z)^{2}]$. Let $x,y,z$ denote the images of $X,Y,Z$ in $A$,
respectively. Then we have

$U=(x)$ and $\mathrm{K}_{A}\cong A/(x)\cong y^{2}A$.

By Proposition 2.3, $\mathrm{K}_{A}$ is torsionless. However, $A$ is not
generically Gorenstein. In fact, $A_{\mathfrak{q}}$ is not a Gorenstein ring
for $\mathfrak{q}=(y,z)\in\operatorname{Min}A$.

##  3\. Reflexive canonical modules

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ admitting
the canonical module $\mathrm{K}_{A}$. We denote by $U$ the unmixed component
of $(0)$ in $A$. In this section we will show how the reflexivity of the
canonical module is related to the Gorensteinness of the ring. We begin with
the following simple but effective lemma.

######  Lemma 3.1.

Suppose that $\mathrm{K}_{A}$ is reflexive. Then $\operatorname{Ass}A\cap
V(U)=\operatorname{Assh}A$. In particular, if $\mathrm{K}_{A}$ is reflexive
and $\operatorname{depth}A=0$, then $\dim A=0$.

###### Proof.

The assertion follows from

$V(U)\cap\operatorname{Ass}A=\operatorname{Supp}_{A}\mathrm{K}_{A}^{*}\cap\operatorname{Ass}A=\operatorname{Ass}_{A}\operatorname{Hom}_{A}(\mathrm{K}_{A}^{*},A)=\operatorname{Ass}_{A}\mathrm{K}_{A}^{**}=\operatorname{Ass}_{A}\mathrm{K}_{A}=\operatorname{Assh}A.\qed$

The example below shows that the reflexivity of $\mathrm{K}_{A}$ may require a
rather strong restriction on $A$.

######  Example 3.2.

Let $S=k[\\![X,Y]\\!]$ be the formal power series ring over a field $k$ and
set $A=S/[(X)\cap(X^{2},Y)]$. Let $x,y$ denote the images of $X,Y$ in $A$,
respectively. Let $\mathfrak{m}=(x,y)$ be the maximal ideal of $A$. Then we
have ${\displaystyle\operatorname{Assh}A=\left\\{(x)\right\\}}$,
${\displaystyle U=(x)}$, and ${\displaystyle\mathrm{K}_{A}=A/U}$.

  1. (1)

Let $\mathfrak{p}=(x)$. Since $A_{\mathfrak{p}}$ is a field, by Proposition
2.3, $\mathrm{K}_{A}$ is torsionless.

  2. (2)

Since $\operatorname{depth}A=0$ and $\dim A=1$, by Lemma 3.1, $\mathrm{K}_{A}$
is not reflexive.

######  Proposition 3.3.

Suppose $d=1$. Then $\mathrm{K}_{A}$ is a reflexive $A$-module if and only if
$A$ is a Gorenstein ring.

###### Proof.

Suppose that $\mathrm{K}_{A}$ is reflexive. By Lemma 3.1, $A$ is Cohen-
Macaulay. Since $\mathrm{K}_{A}$ is reflexive, there exists an exact sequence
${\displaystyle 0\rightarrow\mathrm{K}_{A}\rightarrow F_{1}\rightarrow
F_{0}}$, where $F_{0},F_{1}$ are finite free $A$-modules [11, Proposition
2.1]. Let $a\in\mathfrak{m}$ be an $A$-regular element. Since $A$ is Cohen-
Macaulay, $\mathrm{K}_{A}/a\mathrm{K}_{A}$ is the canonical module of $A/aA$.
Moreover, the embedding ${\displaystyle
0\rightarrow\mathrm{K}_{A}/a\mathrm{K}_{A}\rightarrow F_{1}/aF_{1}}$ proves
that $K_{A/aA}$ is torsionless. Therefore, by Proposition 2.3, $A/aA$ is a
Gorenstein ring. Thus, $A$ is a Gorenstein ring. The converse is clear. ∎

######  Remark 3.4.

There exist non-Cohen-Macaulay local rings with reflexive canonical module.
Example 6.2 shows a two-dimensional non-Cohen-Macaulay local ring $A$ with
$\mathrm{K}_{A}$ reflexive. The example also shows that, even if
$\mathrm{K}_{A}$ is reflexive, the equality
$\operatorname{Ass}A=\operatorname{Assh}A$ does not hold true in general.

Recall that a finitely generated $A$-module $M$ is reflexive, i.e., the
canonical map $\varphi:M\to M^{**}$ is an isomorphism, if and only if there is
at least one isomorphism $M\cong M^{**}$ of $A$-modules.

######  Lemma 3.5.

Suppose that there is an exact sequence

$0\to\mathrm{K}_{A}\to\mathrm{K}_{A}^{**}\to C\to 0$

of $A$-modules. If $C\neq(0)$, then $A_{\mathfrak{p}}$ is a Cohen-Macaulay
ring with $\dim A_{\mathfrak{p}}=1$ for every
$\mathfrak{p}\in\operatorname{Ass}_{A}C$ with
$\operatorname{depth}A_{\mathfrak{p}}\geq 1$. In particular, $\mathfrak{p}\in
V(U)$ and $U_{\mathfrak{p}}=(0)$.

###### Proof.

Let $\mathfrak{p}\in\operatorname{Ass}_{A}C$ such that
$\operatorname{depth}A_{\mathfrak{p}}\geq 1$. Then
$\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}=V(U)$. Since
$\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}\cong\mathrm{K}_{A_{\mathfrak{p}}}$,
by passing to the ring $A_{\mathfrak{p}}$, we may assume
$\operatorname{depth}A>0$ and $\operatorname{depth}_{A}C=0$. Since
$\operatorname{Ass}_{A}\mathrm{K}_{A}^{**}\subseteq\operatorname{Ass}A$, we
have $\operatorname{depth}_{A}\mathrm{K}_{A}^{**}\geq 1$. From the exact
sequence ${\displaystyle 0\to\mathrm{K}_{A}\to\mathrm{K}_{A}^{**}\to C\to 0}$,
we obtain

$0=\operatorname{depth}_{A}C\geq\min\\{\operatorname{depth}_{A}\mathrm{K}_{A}-1,\;\operatorname{depth}_{A}\mathrm{K}_{A}^{**}\\}.$

Thus, $\operatorname{depth}_{A}\mathrm{K}_{A}=1$. Since $\mathrm{K}_{A}$
satisfies $(\mathrm{S}_{2})$, we have

$1=\operatorname{depth}_{A}\mathrm{K}_{A}\geq\min\\{2,\;\dim A\\}.$

Therefore, $A$ is a Cohen-Macaulay ring of dimension $1$. In particular,
$U=(0)$. ∎

Now we aim to generalize Proposition 3.3.

######  Theorem 3.6.

The following conditions are equivalent :

  1. (1)

$\mathrm{K}_{A}$ is a reflexive $A$-module ;

  2. (2)

$\operatorname{Ass}A\cap V(U)=\operatorname{Assh}A$, and $A_{\mathfrak{p}}$ is
Gorenstein for every $\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}$
with $\operatorname{ht}_{A}\mathfrak{p}\leq 1$.

###### Proof.

$(1)\Rightarrow(2)$ By Lemma 3.1, we have $\operatorname{Ass}A\cap
V(U)=\operatorname{Assh}A$. Let
$\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}$ with
$\operatorname{ht}_{A}\mathfrak{p}\leq 1$. If
$\mathfrak{p}\in\operatorname{Assh}A$, then $A_{\mathfrak{p}}$ is Gorenstein
by Proposition 2.3. Otherwise, we have $\dim A_{\mathfrak{p}}=1$. Since
$\mathrm{K}_{A_{\mathfrak{p}}}$ is a reflexive $A_{\mathfrak{p}}$-module, the
ring $A_{\mathfrak{p}}$ is Gorenstein by Proposition 3.3.

$(2)\Rightarrow(1)$ Since $A_{\mathfrak{p}}$ is Gorenstein for every
$\mathfrak{p}\in\operatorname{Assh}A$, by Proposition 2.3, $\mathrm{K}_{A}$ is
torsionless. Hence we have the exact sequence

$0\rightarrow\mathrm{K}_{A}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\mathrm{K}_{A}^{**}\rightarrow
C\rightarrow 0$

of $A$-modules, where $\varphi$ is the canonical homomorphism. Suppose that
$C\neq(0)$. Let $\mathfrak{p}\in\operatorname{Ass}_{A}C$. Note that
$\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}$ and
$\big{[}\mathrm{K}_{A}\big{]}_{\mathfrak{p}}\cong\mathrm{K}_{A_{\mathfrak{p}}}$.
If $\operatorname{ht}_{A}\mathfrak{p}\leq 1$, then by assumption
$A_{\mathfrak{p}}$ is Gorenstein. Then $C_{\mathfrak{p}}=(0)$, which is a
contradiction. Thus, $\operatorname{ht}_{A}\mathfrak{p}\geq 2$. Since
$\operatorname{Ass}A\cap V(U)=\operatorname{Assh}A$, we have
$\operatorname{depth}A_{\mathfrak{p}}\geq 1$. This shows, by Lemma 3.5, that
$A_{\mathfrak{p}}$ is a Cohen-Macaulay ring with $\dim A_{\mathfrak{p}}=1$,
which is a contradiction. Therefore $C=(0)$ and $\mathrm{K}_{A}$ is a
reflexive $A$-module. ∎

We summarize some consequences of Theorem 3.6. Note that $A$ satisfies
$(\mathrm{S}_{1})$ if and only if $\operatorname{Ass}A=\operatorname{Min}A$,
and the latter condition implies $\operatorname{Ass}A\cap
V(U)=\operatorname{Assh}A$.

######  Corollary 3.7.

If $A$ satisfies $(\mathrm{S}_{1})$ and $(\mathrm{G}_{1})$, then
$\mathrm{K}_{A}$ is a reflexive $A$-module.

Recall that if $\operatorname{Assh}A=\operatorname{Ass}A$, then
${\displaystyle\operatorname{Spec}A=\operatorname{Supp}_{A}\mathrm{K}_{A}}$.
Thus, we obtain the following as another direct consequence of Theorem 3.6.

######  Corollary 3.8.

Suppose that $\operatorname{Assh}A=\operatorname{Ass}A$. Then $\mathrm{K}_{A}$
is a reflexive $A$-module if and only if $A$ satisfies $(\mathrm{G}_{1})$.

If $A$ is a Cohen-Macaulay ring, the above corollary recovers [14, Korollar
7.29]. Recall that $A$ is a generalized Cohen-Macaulay ring, if the
$i^{\mbox{\tiny th}}$ local cohomology module
$\mathrm{H}_{\mathfrak{m}}^{i}(A)$ is a finitely generated $A$-module for
every $i\neq d$.

######  Corollary 3.9.

Suppose that $A$ is a generalized Cohen-Macaulay ring and $d>0$. Then
$\mathrm{K}_{A}$ is a reflexive $A$-module if and only if
$\operatorname{depth}A>0$ and $A$ satisfies $(\mathrm{G}_{1})$.

###### Proof.

By assumption, we have
${\displaystyle\operatorname{Ass}A\setminus\\{\mathfrak{m}\\}\subseteq\operatorname{Assh}A}$.
If $\mathrm{K}_{A}$ is reflexive, then by Lemma 3.1 we see that
$\operatorname{depth}A>0$. Without loss of generality, we may assume
$\operatorname{depth}A>0$. Hence
${\displaystyle\operatorname{Ass}A=\operatorname{Assh}A}$. The assertion
follows from Corollary 3.8. ∎

It seems natural to ask for the relation between the reflexivity of the
$A$-module $\mathrm{K}_{A}$ and that of the $A/U$-module $\mathrm{K}_{A/U}$.
As for this question, we have the following.

######  Theorem 3.10.

The following conditions are equivalent :

  1. (1)

$\mathrm{K}_{A}$ is a reflexive $A$-module ;

  2. (2)

$\mathrm{K}_{A/U}$ is a reflexive $A/U$-module and $\operatorname{Ass}A\cap
V(U)=\operatorname{Assh}A$.

###### Proof.

Let $B=A/U$. Then ${\displaystyle\mathrm{K}_{A}=\mathrm{K}_{B}}$ ([2, 1.8]).
Also note that $\operatorname{Ass}B=\operatorname{Assh}B$.

(1) $\Rightarrow$ (2) By Lemma 3.1, we have $\operatorname{Ass}A\cap
V(U)=\operatorname{Assh}A$. By Corollary 3.8, it suffices to show that $B$
satisfies $(\mathrm{G}_{1})$. Let
${\displaystyle\mathfrak{P}\in\operatorname{Spec}B}$ be a prime with
${\displaystyle\operatorname{ht}_{B}\mathfrak{P}\leq 1}$. We write
${\displaystyle\mathfrak{P}=\mathfrak{p}/U}$ for some $\mathfrak{p}\in V(U)$.
Then $\operatorname{ht}_{A}\mathfrak{p}=\operatorname{ht}_{B}\mathfrak{P}\leq
1$. Moreover, $\mathrm{K}_{A_{\mathfrak{p}}}$ is a reflexive
$A_{\mathfrak{p}}$-module. By Propositions 2.3 and 3.3, $A_{\mathfrak{p}}$ is
Gorenstein. Since ${\displaystyle
U_{\mathfrak{p}}=(0):_{A_{\mathfrak{p}}}\mathrm{K}_{A_{\mathfrak{p}}}=(0)}$,
we obtain ${\displaystyle B_{\mathfrak{P}}=A_{\mathfrak{p}}}$. Thus,
${\displaystyle B_{\mathfrak{P}}}$ is a Gorenstein ring.

$(2)\Rightarrow(1)$ Let $\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}$
with $\operatorname{ht}_{A}\mathfrak{p}\leq 1$. By Theorem 3.6, it is enough
to show that $A_{\mathfrak{p}}$ is Gorenstein. Let
$\mathfrak{P}=\mathfrak{p}/U$. Then by Corollary 3.8, $B_{\mathfrak{P}}$ is a
Gorenstein ring. Since $\operatorname{Ass}A\cap V(U)=\operatorname{Assh}A$,
the ring $A_{\mathfrak{p}}$ is Cohen-Macaulay. In particular,
$U_{\mathfrak{p}}=(0)$ and ${\displaystyle
A_{\mathfrak{p}}=B_{\mathfrak{P}}}$. Therefore $A_{\mathfrak{p}}$ is
Gorenstein. ∎

######  Corollary 3.11.

Suppose that $A/U$ is a Gorenstein ring. Then the following assertions hold
true.

  1. (1)

$\mathrm{K}_{A}$ is a reflexive $A$-module if and only if
${\displaystyle\operatorname{Ass}A\cap V(U)=\operatorname{Assh}A}$.

  2. (2)

If $A$ satisfies $(\mathrm{S}_{1})$, then $\mathrm{K}_{A}$ is reflexive.

###### Proof.

Note that (1) follows directly from Theorem 3.10. To prove (2), it is enough
to show that ${\displaystyle\operatorname{Ass}A\cap
V(U)\subseteq\operatorname{Assh}A}$. Let
${\displaystyle\mathfrak{p}\in\operatorname{Ass}A\cap V(U)}$. Since $A$
satisfies $(\mathrm{S}_{1})$, we have $\operatorname{ht}_{A}\mathfrak{p}=0$.
Since $\mathfrak{p}\in V(U)$, we have ${\displaystyle\dim A=\dim
A/\mathfrak{p}+\operatorname{ht}_{A}\mathfrak{p}=\dim A/\mathfrak{p}}$.
Therefore $\mathfrak{p}\in\operatorname{Assh}A$. ∎

Closing this section, we provide the examples of (not necessarily Cohen-
Macaulay) local rings admitting reflexive canonical modules.

######  Example 3.12.

Let $S=k[\\![X,Y_{1},Y_{2},\ldots,Y_{n}]\\!]$ $(n\geq 2)$ be the formal power
series ring over a field $k$ and let $A=S/[(X^{m})\cap J]$ where $m\geq 1$ and
$J$ is a $(Y_{1},Y_{2},\ldots,Y_{n})$-primary ideal of $S$. Let $x$ denote the
image of $X$ in $A$. Then $U=(x^{m})$, $\operatorname{Assh}A=\\{(x)\\}$, and
$A/U$ is a Gorenstein ring. By Corollary 3.11, the $A$-module $\mathrm{K}_{A}$
is reflexive.

######  Example 3.13.

Let $k$ be a field and $R=k[\Delta]$ be the Stanley-Reisner ring of a
simplicial complex $\Delta$. Since $R$ is reduced, the graded canonical module
$\mathrm{K}_{R}$ (see [13, 20]) is torsionless. Moreover, if
$\\#(\operatorname{Assh}R)=1$, the ring $R/U$ is Gorenstein, so that
$\mathrm{K}_{R}$ is reflexive as an $R$-module, where $U$ stands for the
unmixed component of $(0)$ in $R$ and
$\operatorname{Assh}R=\\{\mathfrak{p}\in\operatorname{Spec}R\mid\dim
R/\mathfrak{p}=\dim R\\}$.

##  4\. Quasi-normal rings

Quasi-normal rings were introduced by Vasconcelos ([21, Definition 1.2]) and
they are exactly $2$-Gorenstein rings.

######  Definition 4.1.

A Noetherian ring $R$ is said to be quasi-normal if $R$ satisfies
$(\mathrm{S}_{2})$ and $(\mathrm{G}_{1})$.

The following is a direct consequence of [11, Proposition 2.3] (for a local
ring case, see also [8, Theorem 3.8]). Here we include our alternative proof
specifically for quasi-normal rings.

######  Proposition 4.2.

Let $R$ be a quasi-normal ring and let $M$ be a finitely generated $R$-module.
If
${\displaystyle\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq\min\\{2,\;\dim
R_{\mathfrak{p}}\\}}$ for every $\mathfrak{p}\in\operatorname{Spec}R$, then
$M$ is reflexive.

###### Proof.

Consider the exact sequence of $R$-modules

$0\rightarrow X\rightarrow
M\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}M^{**}\rightarrow
C\rightarrow 0,$

where $\varphi$ denotes the canonical homomorphism. Suppose $X\neq(0)$ and
choose $\mathfrak{p}\in\operatorname{Ass}_{R}X$. By assumption, we have $\dim
R_{\mathfrak{p}}=0$. Since $R$ satisfies $(\mathrm{G}_{1})$, the local ring
${\displaystyle R_{\mathfrak{p}}}$ is Gorenstein, whence $M_{\mathfrak{p}}$ is
reflexive. Hence $X_{\mathfrak{p}}=(0)$, which is a contradiction. So $X=(0)$,
and we have the exact sequence

$0\rightarrow M\rightarrow M^{**}\rightarrow C\rightarrow 0.$

Suppose $C\neq(0)$. Let
${\displaystyle\mathfrak{p}\in\operatorname{Ass}_{R}C}$. If $\dim
R_{\mathfrak{p}}=0$, then $M_{\mathfrak{p}}$ is reflexive, so
$C_{\mathfrak{p}}=(0)$. This is a contradiction. Thus ${\displaystyle\dim
R_{\mathfrak{p}}\geq 1}$. As $R$ satisfies $(\mathrm{S}_{2})$, we have
${\displaystyle\operatorname{depth}R_{\mathfrak{p}}\geq\min\\{2,\;\dim
R_{\mathfrak{p}}\\}}$. Hence
${\displaystyle\operatorname{depth}R_{\mathfrak{p}}\geq 1}$. Since
${\displaystyle\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}^{**}\geq\min\\{2,\;\operatorname{depth}R_{\mathfrak{p}}\\}}$,
we then have
${\displaystyle\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}^{**}\geq
1}$. The exact sequence

$0\rightarrow M_{\mathfrak{p}}\rightarrow M_{\mathfrak{p}}^{**}\rightarrow
C_{\mathfrak{p}}\rightarrow 0$

gives that

$0=\operatorname{depth}_{R_{\mathfrak{p}}}C_{\mathfrak{p}}\geq\min\\{\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}-1,\;\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}^{**}\\}.$

Hence
${\displaystyle\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\leq
1}$. By assumption, we have

$1\geq\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq\min\\{2,\;\dim
R_{\mathfrak{p}}\\}.$

Therefore ${\displaystyle\dim R_{\mathfrak{p}}=1}$ and
${\displaystyle\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}=1}$.
Since $R$ satisfies $(\mathrm{G}_{1})$, the ring ${\displaystyle
R_{\mathfrak{p}}}$ is Gorenstein. By [21, Corollary 2.3], we see that
$M_{\mathfrak{p}}$ is reflexive. Hence $C_{\mathfrak{p}}=(0)$, which is a
contradiction. ∎

A finitely generated $R$-module $\omega_{R}$ is a canonical module of $R$, if
${\displaystyle(\omega_{R})_{\mathfrak{m}}}$ is the canonical module of
$R_{\mathfrak{m}}$ for all maximal ideals $\mathfrak{m}$ of $R$. In contrast
to the local case, the canonical module is in general not unique up to
isomorphisms; see e.g., [6, Remark 3.3.17].

######  Corollary 4.3.

Let $R$ be a Noetherian ring with $d=\dim R>0$. Suppose that there exists a
canonical module $\omega_{R}$ and
${\displaystyle\operatorname{Ass}R_{\mathfrak{m}}=\operatorname{Assh}R_{\mathfrak{m}}}$
for every maximal ideal $\mathfrak{m}$. Then $R$ is quasi-normal if and only
if $R$ satisfies $(\mathrm{S}_{2})$ and $\omega_{R}$ is reflexive.

###### Proof.

Suppose $R$ is quasi-normal. Since $\omega_{R}$ satisfies $(\mathrm{S}_{2})$
and ${\displaystyle\dim_{R_{\mathfrak{p}}}[\omega_{R}]_{\mathfrak{p}}=\dim
R_{\mathfrak{p}}}$ for every $\mathfrak{p}\in\operatorname{Spec}R$, by
Proposition 4.2, we conclude that $\omega_{R}$ is reflexive. For the converse,
it remains to show that $R$ satisfies $(\mathrm{G}_{1})$. Let
$A=R_{\mathfrak{m}}$, where $\mathfrak{m}$ is a maximal ideal of $R$. Then
$\mathrm{K}_{A}=(\omega_{R})_{\mathfrak{m}}$ is reflexive. Therefore we have
${\displaystyle\operatorname{Ass}A=\operatorname{Assh}A}$. By Corollary 3.8,
$A$ satisfies $(\mathrm{G}_{1})$. Thus $R$ satisfies $(\mathrm{G}_{1})$. ∎

We summarize some examples. First, we note examples of quasi-normal rings
which are not normal. The simplest ones are non-normal Gorenstein rings.

######  Example 4.4.

Suppose that $R$ is a Cohen-Macaulay ring with canonical module $\omega_{R}$.
We set $T=R\ltimes\omega_{R}$ to be the idealization of $\omega_{R}$ over $R$.
Then $T$ is a Gorenstein ring ([16]), but not normal because it is never a
reduced ring.

For a commutative ring $R$, we denote by $\overline{R}$ the integral closure
of $R$ in $\mathrm{Q}(R)$. We refer to [6, p. 178] for background on numerical
semigroups.

######  Example 4.5.

Let $H=\left<a_{1},a_{2},\ldots,a_{\ell}\right>$ be a symmetric numerical
semigroup. We consider $R=k[s,t^{a_{1}},t^{a_{2}},\ldots,t^{a_{\ell}}]$, where
$s,t$ are indeterminates and $k$ is a field. Then $R$ is a two-dimensional
Gorenstein ring with $\overline{R}=k[s,t]$, so that $R$ is not normal if
$1\not\in H$. As a special case, the ring $R=k[s,t^{2},t^{3}]$ is quasi-
normal, but not normal.

Next, we note examples of quasi-normal but non-normal Cohen-Macaulay rings
which are moreover not Gorenstein.

######  Example 4.6.

Let $k$ be a field and $k[X,Y]$ the polynomial ring over $k$. Let
$H=\left<a_{1},a_{2},\ldots,a_{\ell}\right>$ be a symmetric numerical
semigroup such that $1\not\in H$ and let
$k[H]=k[t^{a_{1}},t^{a_{2}},\ldots,t^{a_{\ell}}]$ denote the semigroup ring of
$H$ over $k$, where $t$ is an indeterminate. Let
$T=k[X^{n},X^{n-1}Y,\ldots,XY^{n-1},Y^{n}]$, where $n\geq 3$ is an integer. We
set $R=T\otimes_{k}k[H]$. Then $R$ is a quasi-normal Cohen-Macaulay ring with
$\dim R=3$, which is neither Gorenstein nor normal. Indeed, because
$\overline{R}=T\otimes_{k}k[t]$ and $k[H]\neq k[t]$, the ring $R$ is not
normal. As $T$ is normal, we see that $R$ is a quasi-normal ring (see
Proposition 7.3 (2)). Moreover, $R$ is not a Gorenstein ring because $T$ is
not Gorenstein. The simplest example in this class is
$R=k[X^{3},X^{2}Y,XY^{2},Y^{3}]\otimes_{k}k[t^{2},t^{3}]$.

######  Example 4.7.

Let $T=k[X,Y,Z,V]$ be the polynomial ring over a field $k$. We denote by
$\mathbf{I}_{2}(\mathbb{N})$ the ideal of $T$ generated by all the $2\times 2$
minors of a matrix $\mathbb{N}$. Let $I=\mathbf{I}_{2}(\mathbb{M})$ where
$\mathbb{M}=\left(\begin{smallmatrix}X^{a}&Y^{b}+V&Z^{c}\\\
Y^{b^{\prime}}&Z^{c^{\prime}}&X^{a^{\prime}}\\\ \end{smallmatrix}\right)$ for
some integers $a,b,c,a^{\prime},b^{\prime},c^{\prime}\geq 1$. We set $R=T/I$.
Then $R$ is a Cohen-Macaulay ring of dimension $2$. Let $x,y,z,v$ denote the
images of $X,Y,Z,V$ in $R$, respectively. We first check the isomorphism
$\omega_{R}\cong(x^{a},y^{b^{\prime}})R$. In fact, by setting

$f=Z^{c+c^{\prime}}-X^{a^{\prime}}(Y^{b}+V),\ \
g=X^{a+a^{\prime}}-Y^{b^{\prime}}Z^{c},\ \ \text{and}\ \
h=-X^{a}Z^{c^{\prime}}+Y^{b^{\prime}}(Y^{b}+V),$

we can consider the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{t}\mathbb{M}}$$\textstyle{T^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\scriptsize\begin{pmatrix}f&g&h\end{pmatrix}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

of $T$-modules. By taking the $T$-dual, we get the presentation of
$\omega_{R}$ of the form $T^{3}\overset{\mathbb{M}}{\to}T^{2}\to\omega_{R}\to
0$. Therefore, the complex of $R$-modules

$\textstyle{R^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{M}}$$\textstyle{R^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\begin{smallmatrix}Y^{b^{\prime}}&-X^{a}\end{smallmatrix}\right)\quad\
}$$\textstyle{(x^{a},y^{b^{\prime}})R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

induces a natural epimorphism

$\varphi:\omega_{R}\twoheadrightarrow(x^{a},y^{b^{\prime}})R$

of $R$-modules. Moreover, $\varphi$ is an isomorphism because $\omega_{R}$ is
a torsionfree $R$-module of rank one and $x^{a}$ is a non-zerodivisor on $R$.
Hence $\omega_{R}\cong(x^{a},y^{b^{\prime}})R$, as claimed. We similarly have

$\omega_{R}\cong(y^{b}+v,z^{c^{\prime}})R\cong(z^{c},x^{a^{\prime}})R.$

We also note that the isomorphisms can be obtained by using the procedure of
[22, Section 6.1.2]. In particular, $R$ is not a Gorenstein ring, since the
type of $R$ is two.

Next, we show that $R$ is a quasi-normal ring. Let
$\mathfrak{p}\in\operatorname{Spec}R$ with
$\operatorname{ht}_{R}\mathfrak{p}\leq 1$. If $x\not\in\mathfrak{p}$, then
$[\omega_{R}]_{\mathfrak{p}}\cong(x^{a},y^{b^{\prime}})R_{\mathfrak{p}}=R_{\mathfrak{p}}$,
so that $R_{\mathfrak{p}}$ is a Gorenstein ring. Assume that
$x\in\mathfrak{p}$. Similarly, we may assume that $y,z\in\mathfrak{p}$. Then,
$v\notin\mathfrak{p}$, since $\operatorname{ht}_{R}\mathfrak{p}\leq 1$.
Therefore,
$[\omega_{R}]_{\mathfrak{p}}\cong(y^{b}+v,z^{c^{\prime}})R_{\mathfrak{p}}=R_{\mathfrak{p}}$,
so that $R_{\mathfrak{p}}$ is a Gorenstein ring. Hence $R$ is a quasi-normal
ring.

Finally, we prove that $R$ is a normal ring if and only if
$a^{\prime}=b^{\prime}=c=1$. Assume $a^{\prime}=b^{\prime}=c=1$ and consider
the ideal

$J=\mathbf{I}_{2}\begin{pmatrix}Z^{c^{\prime}}&(a+1)X^{a}&Y^{b}+V\\\
-(b+1)Y^{b}+V&-Z&bXY^{b-1}\\\
c^{\prime}X^{a}Z^{c^{\prime}-1}&-Y&-(c^{\prime}+1)Z^{c^{\prime}}\\\
-Y&0&X\end{pmatrix}.$

Then $J+I/I$ is the Jacobian ideal of $R$ over $k$. A direct computation shows
that $\sqrt{J+I}=(X,Y,Z,V)$ and hence, by the Jacobian criterion, the local
ring $R_{\mathfrak{p}}$ is regular for every
$\mathfrak{p}\in\operatorname{Spec}R\setminus\\{(x,y,z,v)\\}$. Hence $R$ is a
normal ring. Conversely, we assume $a^{\prime}\geq 2$, or $b^{\prime}\geq 2$,
or $c\geq 2$. By taking

$P=\begin{cases}(X,Y^{b}+V,Z)&\text{(if $c\geq 2$)}\\\ (X,Y,Z)&\text{(if
$c=1$)},\end{cases}$

we then have $J\subseteq P$. We set $\mathfrak{p}=PR$. Then
$\operatorname{ht}_{R}\mathfrak{p}=1$, but $R_{\mathfrak{p}}$ is not a DVR.
Indeed, because $\varepsilon=Y$ or $\varepsilon^{\prime}=Y^{b}+V$ is
invertible in $T_{P}$, we see that

$JT_{P}=\begin{cases}\left(Z^{c+c^{\prime}}-X^{a^{\prime}}(Y^{b}+V),\dfrac{X^{a+a^{\prime}}}{\varepsilon^{b^{\prime}}}-Z^{c},-\dfrac{X^{a}Z^{c^{\prime}}}{\varepsilon}+(Y^{b}+V)\right)\subseteq(Y^{b}+V)+(X,Z)^{2}&\text{(if
$c\geq 2$)}\\\
\left(\dfrac{Z^{c+c^{\prime}}}{\varepsilon^{\prime}}-X^{a^{\prime}},X^{a+a^{\prime}}-Y^{b^{\prime}}Z^{c},-\dfrac{X^{a}Z^{c^{\prime}}}{\varepsilon^{\prime}}+Y^{b^{\prime}}\right)\subseteq(Y)+(X,Z)^{2}&\text{(if
$c=1$)}\end{cases}$

in $T_{P}$. Thus $R_{\mathfrak{p}}=T_{P}/JT_{P}$ cannot be a DVR. Hence $R$ is
not a normal ring. As a special case,
$R=k[X,Y,Z,V]/\mathbf{I}_{2}\left(\begin{smallmatrix}X&Y+V&Z\\\ Y&Z&X^{2}\\\
\end{smallmatrix}\right)$ is a quasi-normal ring, but not normal.

##  5\. Reflexive canonical modules in dimension one

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with $\dim A=1$
admitting the canonical module $\mathrm{K}_{A}$. In this section, we explore
the question of when $A$ has a reflexive canonical module. We denote by
$\mathrm{Q}(A)$ the total ring of fractions of $A$. Throughout this section,
we assume that there exists an $A$-submodule $K$ of $\mathrm{Q}(A)$ such that
$A\subseteq K\subseteq\overline{A}$ and $K\cong\mathrm{K}_{A}$ as an
$A$-module, where $\overline{A}$ denotes the integral closure of $A$ in
$\mathrm{Q}(A)$. Note that the assumption is automatically satisfied if
$\mathrm{Q}(A)$ is Gorenstein and the residue class field $A/\mathfrak{m}$ is
infinite; see [12, Corollaries 2.8, 2.9]. For $A$-submodules $X$ and $Y$ of
$\mathrm{Q}(A)$, let $X:Y=\\{a\in\mathrm{Q}(A)\mid aY\subseteq X\\}$. If we
consider ideals $I,J$ of $A$, we set $I:_{A}J=\\{a\in A\mid aJ\subseteq I\\}$;
hence $I:_{A}J=(I:J)\cap A$.

######  Proposition 5.1.

The following conditions are equivalent :

  1. $(1)$

$A$ is a Gorenstein ring ;

  2. $(2)$

$K^{2}:K=K$ ;

  3. $(3)$

$\mathrm{K}_{A}$ is a reflexive $A$-module.

###### Proof.

(1) $\Leftrightarrow$ (3) See Proposition 3.3.

(3) $\Leftrightarrow$ (2) Since $A:K=[K:K]:K=K:K^{2}$ ([14, Bemerkung 2.5]),
we have

$A:(A:K)=(K:K):(K:K^{2})=[K:(K:K^{2})]:K=K^{2}:K.$

Therefore, $K^{2}:K=K$ if and only if $A:(A:K)=K$, that is $\mathrm{K}_{A}$ is
a reflexive $A$-module. ∎

Recall that an ideal $I$ of $A$ is called a canonical ideal of $A$, if $I\neq
A$ and $I\cong\mathrm{K}_{A}$ as an $A$-module. By [12, Corollary 2.8], there
exists a canonical ideal $I$ of $A$. We then have the following.

######  Theorem 5.2.

Let $I$ be a canonical ideal of $A$. Then the following conditions are
equivalent :

  1. $(1)$

$A$ is a Gorenstein ring ;

  2. $(2)$

$I^{2}:_{A}I=I$ ;

  3. $(3)$

$I/I^{2}$ is a free $A/I$-module ;

  4. $(4)$

$I$ is a reflexive $A$-module.

###### Proof.

By Proposition 5.1, it suffices to show $(2)\Rightarrow(1)$. Enlarging the
residue class field $A/\mathfrak{m}$ of $A$ if necessary, we may assume that
$A/\mathfrak{m}$ is infinite. Let $I=(x_{1},x_{2},\ldots,x_{n})$ ($n>0$) so
that each $(x_{i})$ is a reduction of $I$. We set $K_{i}=x_{i}^{-1}I$ and
choose a non-zerodivisor $b$ of $A$ so that $bK_{i}^{2}\subseteq A$ for all
$1\leq i\leq n$. Let $J=bI$ and $y_{i}=bx_{i}$ for $1\leq i\leq n$. Then
$(y_{i})$ is a reduction of $J$. Notice that $A/I$ and $A/J$ are both
Gorenstein rings, since $I,J\cong\mathrm{K}_{A}$ as $A$-modules.

######  Claim 1.

$J^{2}:_{A}J=J$.

###### Proof of Claim 1.

Suppose that $J^{2}:_{A}J\supsetneq J$. Then, $J:_{A}\mathfrak{m}\subseteq
J^{2}:_{A}J$. Since $A/J$ is a Gorenstein ring, we have
$J:_{A}\mathfrak{m}=J+A\varphi$ for some
$\varphi\in(J:_{A}\mathfrak{m})\setminus J$. Hence,
$\frac{\varphi}{b}\in\mathrm{Q}(A)$ and
$\mathfrak{m}{\cdot}\frac{\varphi}{b}\subseteq I$, so that
$\frac{\varphi}{b}\in I:\mathfrak{m}$. Because $I\subsetneq
I:_{A}\mathfrak{m}\subseteq I:\mathfrak{m}$ and
$\ell_{A}\left((I:\mathfrak{m})/I\right)=1$ (since $A/I$ is a Gorenstein
ring), we get $\frac{\varphi}{b}\in I:_{A}\mathfrak{m}$, so that
$\frac{\varphi}{b}\in A$. On the other hand,
$\frac{\varphi}{b}{\cdot}I\subseteq I^{2}$, since $\varphi{\cdot}bI=\varphi
J\subseteq J^{2}=b^{2}I^{2}$. Consequently, $\frac{\varphi}{b}\in
I^{2}:_{A}I=I$, whence $\varphi\in bI=J$, which is impossible. Thus
$J^{2}:_{A}J=J$. ∎

Let $\overline{y_{i}}$ denote the image of $y_{i}$ in $J/J^{2}$. We then have
$J/J^{2}=\sum_{i=1}^{n}(A/J){\cdot}\overline{y_{i}}$, and therefore,
$(0):_{A/J}\overline{y_{i}}=(0)$ for some $i$, since $A/J$ is a Gorenstein
ring and $(0):_{A/J}J/J^{2}=(0)$ by Claim 1. Without loss of generality,
assume $i=1$. Then, $J^{2}:_{A}y_{1}=J$. On the other hand, since
$bK_{1}^{2}\subseteq A$ and $K_{1}=y_{1}^{-1}J$, we get
$b{\cdot}(y_{1}^{-1}J)^{2}\subseteq A$ , whence $bJ^{2}\subseteq(bx_{1})^{2}$.
Therefore, $J^{2}\subseteq(bx_{1}^{2})\subseteq(bx_{1})=(y_{1})$. Hence,
$J^{2}=y_{1}{\cdot}(J^{2}:_{A}y_{1})=y_{1}J$. Thus $A$ is a Gorenstein ring
(see [12, Theorem 3.7]). ∎

##  6\. $q$-torsionfree canonical modules

The purpose of this section is to give a generalization of Proposition 2.3 and
Theorem 3.6, which characterize local rings with $q$-torsionfree canonical
modules for $q=1,2$.

Let $R$ be a Noetherian (not necessarily local) ring and $q$ an integer. Let
$M$ be a finitely generated $R$-module with a finite projective presentation
${\displaystyle
P_{1}\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}P_{0}\rightarrow
M\rightarrow 0}$. By applying the $R$-dual functor
$(-)^{*}=\operatorname{Hom}_{R}(-,R)$, we obtain the exact sequence

$0\longrightarrow M^{*}\longrightarrow
P_{0}^{*}\stackrel{{\scriptstyle\sigma^{*}}}{{\longrightarrow}}P_{1}^{*}\longrightarrow\mathrm{D}_{\sigma}(M)\longrightarrow
0$

of $R$-modules. We set ${\displaystyle\mathrm{D}(M)=\mathrm{D}_{\sigma}(M)}$
and call it the Auslander transpose of $M$. Note that $\mathrm{D}(M)$ is
uniquely determined up to projective equivalence.

######  Definition 6.1 ([1, Definition 2.15]).

A finitely generated $R$-module $M$ is said to be $q$-torsionfree if
$\mbox{\rm Ext}_{R}^{i}(\mathrm{D}(M),R)=0$ for all $i=1,2,\ldots,q$.

By [1, Proposition 2.6], there exists an exact sequence

$0\longrightarrow\mbox{\rm Ext}^{1}_{R}(\mathrm{D}(M),R)\longrightarrow
M\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}M^{**}\longrightarrow\mbox{\rm
Ext}^{2}_{R}(\mathrm{D}(M),R)\rightarrow 0$

of $R$-modules, where $\varphi$ is the canonical homomorphism, and
furthermore, we have

$\mbox{\rm Ext}_{R}^{i+2}(\mathrm{D}(M),R)\cong\mbox{\rm
Ext}_{R}^{i}(M^{*},R)\ \ \ \text{for all}\ \ i>0.$

This shows $M$ is torsionless (resp. reflexive) if and only if $M$ is
$1$-torsionfree ($2$-torsionfree). When $q\geq 3$, the $R$-module $M$ is
$q$-torsionfree if and only if $M$ is reflexive and $\mbox{\rm
Ext}_{R}^{i}(M^{*},R)=(0)$ for all $i=1,2,\ldots,q-2$.

######  Example 6.2.

Let $S=k[\\![X,Y,Z]\\!]$ be the formal power series ring over a field $k$ and
set $A=S/[(X)\cap(Y,Z)]$. Let $x,y,z$ denote the images of $X,Y,Z$ in $A$,
respectively. Then we have
${\displaystyle\operatorname{Assh}A=\left\\{(x)\right\\}}$, ${\displaystyle
U=(x)}$, and ${\displaystyle\mathrm{K}_{A}=A/U}$, where $U$ denotes the
unmixed component of $(0)$ in $A$. By dualizing the exact sequence

$A\stackrel{{\scriptstyle\cdot x}}{{\longrightarrow}}A\rightarrow
A/(x)=\mathrm{K}_{A}\rightarrow 0,$

we obtain

$0\rightarrow\mathrm{K}_{A}^{*}\rightarrow A\stackrel{{\scriptstyle\cdot
x}}{{\longrightarrow}}A\rightarrow A/(x)\rightarrow 0.$

Thus, ${\displaystyle\mathrm{D}(\mathrm{K}_{A})=\mathrm{K}_{A}}$. Consider the
free resolution of $\mathrm{D}(\mathrm{K}_{A})=D$:

$A^{5}\stackrel{{\scriptstyle\tau_{4}}}{{\longrightarrow}}A^{3}\stackrel{{\scriptstyle\tau_{3}}}{{\longrightarrow}}A^{2}\stackrel{{\scriptstyle\tau_{2}}}{{\longrightarrow}}A\stackrel{{\scriptstyle\cdot
x}}{{\longrightarrow}}A\longrightarrow D\longrightarrow 0,$

where
${\displaystyle\tau_{2}=[y\;z],\;\tau_{3}=\left[\begin{array}[]{rrr}x&0&z\\\
0&x&-y\end{array}\right],\;\text{and}\;\tau_{4}=\left[\begin{array}[]{ccccc}y&z&0&0&0\\\
0&0&y&z&0\\\ 0&0&0&0&x\end{array}\right].}$ Dualize this free resolution to
obtain

$0\longrightarrow D^{*}\longrightarrow A\stackrel{{\scriptstyle\cdot
x}}{{\longrightarrow}}A\stackrel{{\scriptstyle\sigma_{2}}}{{\longrightarrow}}A^{2}\stackrel{{\scriptstyle\sigma_{3}}}{{\longrightarrow}}A^{3}\stackrel{{\scriptstyle\sigma_{4}}}{{\longrightarrow}}A^{5},$

where ${\displaystyle\sigma_{2},\sigma_{3},\sigma_{4}}$ are the transposes of
${\displaystyle\tau_{2},\tau_{3},\tau_{4}}$, respectively. Let ${\displaystyle
a\in\operatorname{Ker}\sigma_{2}}$. Then

$a\in\big{[}(0):_{A}y\big{]}\cap\big{[}(0):_{A}z\big{]}=(x)\cap(x)=(x)=\operatorname{Im}(\cdot
x).$

Thus, ${\displaystyle\mbox{\rm Ext}^{1}_{A}(D,A)=(0)}$. Let
${\displaystyle\left(\begin{smallmatrix}a_{1}\\\ a_{2}\\\
\end{smallmatrix}\right)\in\operatorname{Ker}\sigma_{3}}$. Then
${\displaystyle a_{1},a_{2}\in(0):_{A}x=(y,z)}$ and ${\displaystyle
a_{1}z-a_{2}y=0}$. Hence ${\displaystyle\left(\begin{smallmatrix}-a_{2}\\\
a_{1}\\\
\end{smallmatrix}\right)\in\operatorname{Ker}\tau_{2}=\operatorname{Im}(\tau_{3})}$.
Let ${\displaystyle-a_{2}=c_{1}x-c_{3}z}$, and ${\displaystyle
a_{1}=c_{2}x+c_{3}y}$ for some ${\displaystyle c_{1},c_{2},c_{3}\in A}$. Then
${\displaystyle c_{1}x=-a_{2}+c_{3}z\in(y,z)}$. Thus, $c_{1}=0$ and
$a_{2}=c_{3}z$. Similarly, ${\displaystyle a_{1}=c_{3}y}$. We obtain
${\displaystyle\left(\begin{smallmatrix}a_{1}\\\ a_{2}\\\
\end{smallmatrix}\right)\in\operatorname{Im}\sigma_{2}}$. Then
${\displaystyle\mbox{\rm Ext}^{2}_{A}(D,A)=(0)}$. Hence $\mathrm{K}_{A}$ is
$2$-torsionfree. Note that ${\displaystyle\operatorname{Ker}\sigma_{4}}$ is
generated by $\left(\begin{smallmatrix}x\\\ 0\\\ 0\end{smallmatrix}\right)$,
$\left(\begin{smallmatrix}0\\\ x\\\ 0\\\ \end{smallmatrix}\right)$,
$\left(\begin{smallmatrix}0\\\ 0\\\ y\end{smallmatrix}\right)$,
$\left(\begin{smallmatrix}0\\\ 0\\\ z\end{smallmatrix}\right)$. Then
${\displaystyle\mbox{\rm Ext}^{3}_{A}(D,A)\neq(0)}$. Thus, $\mathrm{K}_{A}$ is
not $3$-torsionfree.

######  Definition 6.3 ([1, Definition 2.15]).

A finitely generated $R$-module $M$ is called $q$-syzygy, if there exist
finite free $R$-modules $F_{1},F_{2},\ldots,F_{q}$ and an exact sequence
${\displaystyle 0\to M\to F_{1}\to F_{2}\to\cdots\to F_{q}}$ of $R$-modules.

Note that $(a)$ $M$ is torsionless if and only if $M$ is $1$-syzygy, (b) every
$q$-torsionfree $R$-module is $q$-syzygy, and (c) if $M$ is $q$-syzygy and $x$
is an $R$-regular element, then $M/xM$ is $(q-1)$-syzygy as an $R/xR$-module.

Although the following theorem has been proved by Foxby in a more general
setting involving Gorenstein modules, we restate it and give its proof in our
context for the sake of completeness. Recall that $R$ is $q$-Gorenstein if
$R_{\mathfrak{p}}$ is Gorenstein for every prime $\mathfrak{p}$ with
${\displaystyle\operatorname{depth}R_{\mathfrak{p}}<q}$.

######  Theorem 6.4 ([10, Proposition 3.2]).

Let $A$ be a Cohen-Macaulay local ring admitting the canonical module
$\mathrm{K}_{A}$. Then the following conditions are equivalent :

  1. (1)

$A$ is $q$-Gorenstein ;

  2. (2)

$\mathrm{K}_{A}$ is $q$-torsionfree ;

  3. (3)

$\mathrm{K}_{A}$ is $q$-syzygy.

###### Proof.

Since $A$ is Cohen-Macaulay, we have
${\displaystyle\operatorname{Spec}A=\operatorname{Supp}_{A}\mathrm{K}_{A}}$
and
${\displaystyle[\mathrm{K}_{A}]_{\mathfrak{p}}=\mathrm{K}_{A_{\mathfrak{p}}}}$
for every $\mathfrak{p}\in\operatorname{Spec}A$. Notice that
${\displaystyle\mathrm{K}_{A_{\mathfrak{p}}}}$ is maximal Cohen-Macaulay as an
$A_{\mathfrak{p}}$-module.

$(1)\Rightarrow(2)$ Since every $A$-regular sequence is
$\mathrm{K}_{A}$-regular, the $A$-module $\mathrm{K}_{A}$ is $q$-torsionfree
by [11, Proposition 2.3].

$(2)\Rightarrow(3)$ This follows from [11, Proposition 2.1].

$(3)\Rightarrow(1)$ Let $\mathfrak{p}\in\operatorname{Spec}A$ with
$\operatorname{depth}A_{\mathfrak{p}}<q$. Set $n=\dim A_{\mathfrak{p}}$. When
$n=0$, the ring $A_{\mathfrak{p}}$ is Gorenstein. Assume $n>0$ and choose a
system $f_{1},f_{2},\ldots,f_{n}$ of parameters of $A_{\mathfrak{p}}$. Then it
is an $A_{\mathfrak{p}}$-regular sequence, so that
$\mathrm{K}_{A_{\mathfrak{p}}}/(f_{1},f_{2},\ldots,f_{n})\mathrm{K}_{A_{\mathfrak{p}}}$
is $1$-syzygy because $n<q$. Since
$\mathrm{K}_{A_{\mathfrak{p}}}/(f_{1},f_{2},\ldots,f_{n})\mathrm{K}_{A_{\mathfrak{p}}}\cong\mathrm{K}_{A_{\mathfrak{p}}/(f_{1},f_{2},\ldots,f_{n})A_{\mathfrak{p}}}$,
we conclude that $A_{\mathfrak{p}}/(f_{1},f_{2},\ldots,f_{n})A_{\mathfrak{p}}$
is Gorenstein by Proposition 2.3, whence so is the ring $A_{\mathfrak{p}}$.
This completes the proof. ∎

As a direct consequence of Theorem 6.4, we have the following.

######  Corollary 6.5.

Let $A$ be a Cohen-Macaulay local ring with $d=\dim A$ admitting the canonical
module $\mathrm{K}_{A}$. Then $A$ is a Gorenstein ring if and only if
$\mathrm{K}_{A}$ is $(d+1)$-torsionfree.

Theorem 6.4 and Corollary 6.5 lead us to the question of the structure of a
local ring $A$ with $q$-torsionfree canonical module without the assumption
that $A$ is Cohen-Macaulay.

For a subset $\Phi$ of prime ideals in a Noetherian ring $R$, we say that

  * •

$R$ satisfies $(\mathrm{S}_{n})$ on $\Phi$ if
${\displaystyle\operatorname{depth}R_{\mathfrak{p}}\geq\min\left\\{n,\,\dim
R_{\mathfrak{p}}\right\\}}$ for every $\mathfrak{p}\in\Phi$,

  * •

$R$ satisfies $(\mathrm{G}_{n})$ on $\Phi$ if $R_{\mathfrak{p}}$ is Gorenstein
for every $\mathfrak{p}\in\Phi$ with ${\displaystyle\dim R_{\mathfrak{p}}\leq
n}$.

The main result of this section gives an answer to the above question.

######  Theorem 6.6.

Let $A$ be a Noetherian local ring admitting the canonical module
$\mathrm{K}_{A}$. Suppose that $\mathrm{K}_{A}$ satisfies $(\mathrm{S}_{q})$.
Then the following conditions are equivalent :

  1. (1)

$A$ satisfies both $(\mathrm{G}_{q-1})$ and $(\mathrm{S}_{q-1})$ on
$\operatorname{Supp}_{A}\mathrm{K}_{A}$ ;

  2. (2)

$\mathrm{K}_{A}$ is $q$-torsionfree ;

  3. (3)

$\mathrm{K}_{A}$ is $q$-syzygy.

To show this, we need some auxiliaries. The following plays an important role
in our argument.

######  Lemma 6.7 ([7, Lemma 4.9]).

Let $A$ be a Noetherian local ring and $M$ a nonzero finitely generated
$A$-module. Assume that $q\geq\operatorname{depth}A+2$ and $M$ is $q$-syzygy.
Then $\operatorname{depth}_{A}M=\operatorname{depth}A$.

We apply Lemma 6.7 to get the following.

######  Theorem 6.8.

Let $R$ be a Noetherian ring and $M$ a finitely generated $R$-module. If $M$
is $(q+1)$-syzygy, then one has

$\operatorname{depth}R_{\mathfrak{p}}\geq\min\\{q,\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\\}\
\ \ \text{for all}\ \ \mathfrak{p}\in\operatorname{Supp}_{R}M.$

In particular, if $M$ satisfies $(\mathrm{S}_{q})$, then $R$ satisfies
$(\mathrm{S}_{q})$ on $\operatorname{Supp}_{R}M$.

###### Proof.

By localizing at $\mathfrak{p}\in\operatorname{Supp}_{R}M$, it suffices to
show $\operatorname{depth}R\geq\min\\{q,\operatorname{depth}_{R}M\\}$. If
$\operatorname{depth}R\geq q$, the assertion is obvious. Otherwise, if
$\operatorname{depth}R<q$, the assertion follows from Lemma 6.7. ∎

As consequences of Theorems 6.4, 6.8, we get the following.

######  Corollary 6.9.

Let $A$ be a Noetherian local ring with $d=\dim A$ admitting the canonical
module $\mathrm{K}_{A}$. Then the following conditions are equivalent :

  1. (1)

$A$ is Gorenstein ;

  2. (2)

$\mathrm{K}_{A}$ is a $(d+1)$-torsionfree maximal Cohen-Macaulay $A$-module ;

  3. (3)

$\mathrm{K}_{A}$ is a $(d+1)$-syzygy maximal Cohen-Macaulay $A$-module.

###### Proof.

We only need to show (3) $\Rightarrow$ (1). By Theorem 6.8 we have that
$\operatorname{depth}A=d$, so that $A$ is Cohen-Macaulay. Hence the assertion
follows from Theorem 6.4. ∎

######  Corollary 6.10.

Let $(A,\mathfrak{m})$ be a Noetherian local ring with $d=\dim A$ admitting
the canonical module $\mathrm{K}_{A}$. Furthermore, we assume one of the
following conditions (i) and (ii).

  1. (i)

$\mathrm{H}^{i}_{\mathfrak{m}}(A)=(0)$ for every integer $i\not=0,1,d$.

  2. (ii)

$d\leq 2$.

Then the following conditions are equivalent :

  1. (1)

$A$ is Gorenstein ;

  2. (2)

$\mathrm{K}_{A}$ is $(d+1)$-torsionfree ;

  3. (3)

$\mathrm{K}_{A}$ is $(d+1)$-syzygy.

###### Proof.

(i) By passing to the $\mathfrak{m}$-adic completion, we may assume $A$ is
$\mathfrak{m}$-adically complete. In view of [17, (2.3) Satz], it follows that
$\mathrm{K}_{A}$ is maximal Cohen-Macaulay. Therefore the assertion follows
from Corollary 6.9.

(ii) Since $\mathrm{K}_{A}$ satisfies $(\mathrm{S}_{2})$, the assertion
follows from Corollary 6.9. ∎

######  Remark 6.11.

Let $A$ be a Noetherian local ring admitting the canonical module
$\mathrm{K}_{A}$. We say that $A$ is quasi-Gorenstein if $\mathrm{K}_{A}\cong
A$ as an $A$-module. When $d\geq 3$, there exist non-Gorenstein quasi-
Gorenstein local rings of dimension $d$ (see e.g., [2, Theorem 2.11]). Notice
that, in such a ring $A$, $\mathrm{K}_{A}$ is $q$-torsionfree for all $q\geq
1$. So, Corollary 6.10 fails without the condition (i) or (ii).

Based on the above observation, it is natural to raise the following question.

######  Question 6.12.

Let $A$ be a Noetherian local ring with $d=\dim A\geq 3$ admitting the
canonical module $\mathrm{K}_{A}$. When are the following conditions
equivalent?

  1. (i)

$A$ is a quasi-Gorenstein ring, i.e., $\mathrm{K}_{A}\cong A$.

  2. (ii)

$\mathrm{K}_{A}$ is a $(d+1)$-torsionfree $A$-module.

In what follows, let $R$ be a Noetherian ring and $M$ a finitely generated
$R$-module. The equivalence of $(1)$ and $(2)$ in the next theorem was
essentially proved by Auslander and Bridger [1]. Notice that this is a
$q^{\mbox{\tiny th}}$ version of [6, Proposition 1.4.1].

######  Theorem 6.13.

The following conditions are equivalent :

  1. (1)

$M$ is $q$-torsionfree ;

  2. (2)

$M$ satisfies the conditions below :

    1. (i)

$M_{\mathfrak{p}}$ is $q$-torsionfree for every
$\mathfrak{p}\in\operatorname{Supp}_{R}M$ with
$\operatorname{depth}R_{\mathfrak{p}}<q$ ;

    2. (ii)

$M$ satisfies $(\widetilde{\mathrm{S}}_{q})$ ;

  3. (3)

$M$ satisfies the conditions below :

    1. (i)

$M_{\mathfrak{p}}$ is $q$-torsionfree for every
$\mathfrak{p}\in\operatorname{Supp}_{R}M$ with
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}<q$ ;

    2. (ii)

$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}=\operatorname{depth}R_{\mathfrak{p}}$
for every $\mathfrak{p}\in\operatorname{Supp}_{R}M$ with
$\operatorname{depth}R_{\mathfrak{p}}<q-1$.

###### Proof.

Without loss of generality, we may assume $q\geq 1$.

$(1)\Rightarrow(2)$ This follows from [11, Proposition 2.1].

$(2)\Rightarrow(3)$ (i) Let $\mathfrak{p}\in\operatorname{Supp}_{R}M$ with
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}<q$. Since $M$
satisfies $(\widetilde{\mathrm{S}}_{q})$, we have
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq\operatorname{depth}R_{\mathfrak{p}}$.
This implies $\operatorname{depth}R_{\mathfrak{p}}<q$, and hence
$M_{\mathfrak{p}}$ is $q$-torsionfree. (ii) Let
$\mathfrak{p}\in\operatorname{Supp}_{R}M$ with
$\operatorname{depth}R_{\mathfrak{p}}<q-1$. Then $M_{\mathfrak{p}}$ is
$q$-torsionfree, so that
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}=\operatorname{depth}R_{\mathfrak{p}}$
by Lemma 6.7.

$(3)\Rightarrow(1)$ For each $i\in\\{1,2,\dots,q\\}$, we set $E^{i}=\mbox{\rm
Ext}^{i}_{R}(\mathrm{D}(M),R)$. Suppose $E^{q}\not=0$ and seek a
contradiction. Take $\mathfrak{p}\in\operatorname{Ass}_{R}E^{q}$. Since
$\mathfrak{p}\in\operatorname{Supp}_{R}M$, by (i) we have
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq q$. Then by (ii),
$\operatorname{depth}R_{\mathfrak{p}}\geq q-1$. By passing to the localization
$R_{\mathfrak{p}}$ at $\mathfrak{p}$, we may assume $R$ is a local ring,
$\operatorname{depth}R\geq q-1$, $\operatorname{depth}_{R}M\geq q$, and
$\operatorname{depth}_{R}E^{q}=0$.

We proceed by induction on $q$. First, assume that $q=1$. Since $E^{1}$ is
isomorphic to a submodule of $M$, it follows that
$\operatorname{depth}_{R}M=0$, a contradiction. Thus $E^{1}=(0)$. Next, we
assume $q=2$. Applying the depth lemma to the exact sequence $0\to M\to
M^{**}\to E^{2}\to 0$ of $R$-modules, we get $\operatorname{depth}_{R}M=1$, as
$\operatorname{depth}_{R}M^{**}\geq 1$. This is impossible, whence
$E^{2}=(0)$. Suppose $q\geq 3$ and the assertion holds for $q-1$, i.e., $M$ is
$(q-1)$-torsionfree. Hence $E^{1}=\cdots=E^{q-1}=(0)$. Consider a free
resolution $(F_{i},\partial_{i})$ of $M^{*}$. Applying the $R$-dual functor
$(-)^{*}$, we get the exact sequence

$0\to M^{**}\to F_{0}^{*}\xrightarrow[]{\partial_{1}^{*}}F_{1}^{*}\to\cdots\to
F_{q-3}^{*}\xrightarrow[]{\partial_{q-2}^{*}}F_{q-2}^{*}$

of $R$-modules because $E^{3}=\cdots=E^{q-1}=(0)$. Let $C$ be the cokernel of
$\partial_{q-2}^{*}$. Since $M$ is reflexive as an $R$-module, we obtain the
exact sequence of the form

$0\to M\to F_{0}^{*}\to\cdots\to F_{q-2}^{*}\to C\to 0.$

Since $E^{q}=\mbox{\rm Ext}^{q-2}_{R}(M^{*},R)$ may be regarded as a submodule
of $C$, we see that $\operatorname{depth}_{R}C=0$. Hence
$\operatorname{depth}_{R}M=q-1$. This gives a contradiction. Hence we conclude
that $E^{q}=(0)$, which shows $M$ is $q$-torsionfree. ∎

######  Corollary 6.14.

Suppose that the following conditions are satisfied :

  1. (a)

$M_{\mathfrak{p}}$ is $q$-torsionfree for every
$\mathfrak{p}\in\operatorname{Supp}_{R}M$ with $\dim R_{\mathfrak{p}}<q$ ;

  2. (b)

$M$ satisfies $(\mathrm{S}_{q})$ ;

  3. (c)

$\operatorname{depth}R_{\mathfrak{p}}\geq\min\\{q-1,\dim
R_{\mathfrak{p}}-1\\}$ for every $\mathfrak{p}\in\operatorname{Supp}_{R}M$.

Then $M$ is $q$-torsionfree.

###### Proof.

We will check condition (3) in Theorem 6.13. Let
$\mathfrak{p}\in\operatorname{Supp}_{R}M$. (i) Assume that
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}<q$. By (b),
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}=\dim
R_{\mathfrak{p}}$, so that $\dim R_{\mathfrak{p}}<q$. Therefore
$M_{\mathfrak{p}}$ is $q$-torsionfree by (a). (ii) Assume that
$\operatorname{depth}R_{\mathfrak{p}}<q-1$. By (c),
$\operatorname{depth}R_{\mathfrak{p}}\geq\dim R_{\mathfrak{p}}-1$, so that
$\dim R_{\mathfrak{p}}<q$. Therefore $M_{\mathfrak{p}}$ is $q$-torsionfree by
(a), whence
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}=\operatorname{depth}R_{\mathfrak{p}}$
by Lemma 6.7. ∎

######  Corollary 6.15.

Suppose that the following conditions are satisfied :

  1. (a)

$M$ satisfies $(\mathrm{S}_{q})$ ;

  2. (b)

$R$ satisfies both $(\mathrm{G}_{q-1})$ and $(\mathrm{S}_{q-1})$ on
$\operatorname{Supp}_{R}M$.

Then $M$ is $q$-torsionfree.

###### Proof.

By Corollary 6.14, it suffices to show that $M_{\mathfrak{p}}$ is
$q$-torsionfree on $\\{\mathfrak{p}\in\operatorname{Supp}_{R}M\mid\dim
R_{\mathfrak{p}}<q\\}$. Let $\mathfrak{p}\in\operatorname{Supp}_{R}M$ with
$\dim R_{\mathfrak{p}}<q$. Then by (b), $R_{\mathfrak{p}}$ is Gorenstein. As
$M$ satisfies $(\mathrm{S}_{q})$,
$\operatorname{depth}_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq\dim
R_{\mathfrak{p}}$. Hence $M_{\mathfrak{p}}$ is maximal Cohen-Macaulay as an
$R_{\mathfrak{p}}$-module. In particular, $M_{\mathfrak{p}}$ is
$q$-torsionfree. ∎

We are now ready to prove Theorem 6.6.

###### Proof of Theorem 6.6.

$(1)\Rightarrow(2)$ This follows from Corollary 6.15.

$(2)\Rightarrow(3)$ This follows from [11, Proposition 2.1].

$(3)\Rightarrow(1)$ By Theorem 6.8, the ring $A$ satisfies
$(\mathrm{S}_{q-1})$ on $\operatorname{Supp}_{A}\mathrm{K}_{A}$. Let
$\mathfrak{p}\in\operatorname{Supp}_{A}\mathrm{K}_{A}$ with $\dim
A_{\mathfrak{p}}\leq q-1$. Then $A_{\mathfrak{p}}$ is a Cohen-Macaulay ring of
dimension at most $q-1$. Hence Theorem 6.4 implies that $A_{\mathfrak{p}}$ is
a Gorenstein ring. ∎

As consequences of Theorem 6.6, we get the following corollaries.

######  Corollary 6.16.

Let $A$ be a Noetherian local ring admitting the canonical module
$\mathrm{K}_{A}$. Suppose that $q\geq 2$ and $\mathrm{K}_{A}$ satisfies
$(\mathrm{S}_{q})$. Consider the following conditions :

  1. (1)

$\mathrm{K}_{A}$ is $(q+1)$-torsionfree ;

  2. (2)

$\mathrm{K}_{A}$ is $(q+1)$-syzygy ;

  3. (3)

$A$ satisfies both $(\mathrm{S}_{q})$ and $(\mathrm{G}_{q-1})$ on
$\operatorname{Supp}_{A}\mathrm{K}_{A}$ ;

  4. (4)

$A$ satisfies both $(\mathrm{S}_{q})$ and $(\mathrm{G}_{q-1})$, that is, $A$
is $q$-Gorenstein ;

  5. (5)

$A$ satisfies $(\mathrm{S}_{q})$ and $\mathrm{K}_{A}$ is $q$-torsionfree ;

  6. (6)

$A$ satisfies $(\mathrm{S}_{q})$ and $\mathrm{K}_{A}$ is $q$-syzygy.

Then the implications
(1)$\Leftrightarrow$(2)$\Rightarrow$(3)$\Leftrightarrow$(4)$\Leftrightarrow$(5)$\Leftrightarrow$(6)
hold true.

###### Proof.

The implications $(1)\Rightarrow(2)$ and $(4)\Rightarrow(3)$ are clear. The
equivalence of $(3)$, $(5)$, and $(6)$ immediately follows from Theorem 6.6.
Thus it suffices to check the implications $(2)\Rightarrow(3)$,
$(3)\Rightarrow(4)$, and $(2)\Rightarrow(1)$.

$(2)\Rightarrow(3)$ Theorem 6.6 shows the ring $A$ satisfies
$(\mathrm{G}_{q-1})$ on $\operatorname{Supp}_{A}\mathrm{K}_{A}$. On the other
hand, by Theorem 6.8, we deduce that $A$ satisfies $(\mathrm{S}_{q})$ on
$\operatorname{Supp}_{A}\mathrm{K}_{A}$.

$(3)\Rightarrow(4)$ Since $q\geq 2$, by [3, Lemma 1.1] we have
$\operatorname{Supp}_{A}\mathrm{K}_{A}=\operatorname{Spec}A$.

$(2)\Rightarrow(1)$ The implication $(2)\Rightarrow(4)$ guarantees that $A$ is
$q$-Gorenstein. Hence $\mathrm{K}_{A}$ is $q$-torsionfree by [1, Proposition
4.21]. ∎

Since $\mathrm{K}_{A}$ satisfies $(\mathrm{S}_{2})$, from Corollary 6.16 we
have the following.

######  Corollary 6.17.

Let $A$ be a Noetherian local ring admitting the canonical module
$\mathrm{K}_{A}$. Consider the following conditions :

  1. (1)

$\mathrm{K}_{A}$ is $3$-torsionfree ;

  2. (2)

$\mathrm{K}_{A}$ is $3$-syzygy ;

  3. (3)

$A$ satisfies both $(\mathrm{S}_{2})$ and $(\mathrm{G}_{1})$, that is, $A$ is
quasi-normal ;

  4. (4)

$A$ satisfies $(\mathrm{S}_{2})$ and $\mathrm{K}_{A}$ is $2$-torsionfree ;

  5. (5)

$A$ satisfies $(\mathrm{S}_{2})$ and $\mathrm{K}_{A}$ is $2$-syzygy.

Then the implications
(1)$\Leftrightarrow$(2)$\Rightarrow$(3)$\Leftrightarrow$(4)$\Leftrightarrow$(5)
hold true.

######  Corollary 6.18.

Let $A$ be a Noetherian local ring admitting the canonical module
$\mathrm{K}_{A}$. Suppose that $q\geq 2$ and $\mathrm{K}_{A}$ satisfies
$(\mathrm{S}_{q+1})$. Then the following conditions are equivalent :

  1. (1)

$\mathrm{K}_{A}$ is $(q+1)$-torsionfree ;

  2. (2)

$\mathrm{K}_{A}$ is $(q+1)$-syzygy ;

  3. (3)

$A$ satisfies both $(\mathrm{S}_{q})$ and $(\mathrm{G}_{q})$ on
$\operatorname{Supp}_{A}\mathrm{K}_{A}$ ;

  4. (4)

$A$ satisfies both $(\mathrm{S}_{q})$ and $(\mathrm{G}_{q})$.

###### Proof.

This follows from Theorem 6.6 and the fact that
$\operatorname{Supp}_{A}\mathrm{K}_{A}=\operatorname{Spec}A$ ([3, Lemma 1.1]).
∎

######  Corollary 6.19.

Let $A$ be a Noetherian local ring with $d=\dim A$ which is a homomorphic
image of a Gorenstein ring. Suppose that $q\geq\frac{d}{2}+1$ and
$\mathrm{K}_{A}$ is $(q+1)$-syzygy satisfying $(\mathrm{S}_{q})$. Then $A$ is
a Cohen-Macaulay ring.

###### Proof.

By Theorem 6.8, we see that $A$ satisfies $(\mathrm{S}_{q})$. We may assume
$d>0$. Then $q\geq 2$, so that $A$ is equidimensional by [3, Lemma 1.1].
Furthermore, either $A$ is Cohen-Macaulay or $\operatorname{depth}A\geq q$. We
assume $\operatorname{depth}A\geq q$. Since $\mathrm{K}_{A}$ satisfies
$(\mathrm{S}_{q})$, every $A$-regular sequence of length at most $q$ is
$\mathrm{K}_{A}$-regular ([11, Proposition 2.1]). The assertion follows from
[9, Corollary (2.6)] (see also [10, Proposition 4.2]). ∎

##  7\. Examples of $q$-Gorenstein rings

Closing this paper, in order to illustrate our theorems, we provide additional
examples of Cohen-Macaulay and $q$-Gorenstein rings, i.e., rings with
$(\mathrm{S}_{q})$ and $(\mathrm{G}_{q-1})$ conditions, or equivalently, rings
with $(\widetilde{\mathrm{G}}_{q-1})$ condition.

######  Theorem 7.1.

Let $A$ be a Gorenstein local ring with $d=\dim A\geq 3$ and let
$a_{1},a_{2},\ldots,a_{d}$ be a system of parameters of $A$. Let
$\mathfrak{a}=(a_{1},a_{2},\ldots,a_{\ell})$ $(3\leq\ell\leq d)$ and let

$\mathcal{R}=A[a_{1}t,a_{2}t,\ldots,a_{\ell}t]\ \subseteq\ A[t]$

be the Rees algebra of $\mathfrak{a}$, where $t$ denotes an indeterminate.
Then, $\mathcal{R}$ is not a Gorenstein ring, but it is a Cohen-Macaulay
$(\ell+1)$-Gorenstein ring of dimension $d+1$.

###### Proof.

Recall that $\mathcal{R}$ is a Cohen-Macaulay ring of dimension $d+1$. Let
$S=A[X_{1},X_{2},\ldots,X_{\ell}]$ be the polynomial ring over $A$ and let
$\varphi:S\to\mathcal{R}$ denote the surjective homomorphism of $A$-algebras
defined by $\varphi(X_{i})=a_{i}t$ for each $1\leq i\leq\ell$. The
homomorphism $\varphi$ preserves the grading and ${\rm
Ker}(\varphi)=\mathbf{I}_{2}\left(\begin{smallmatrix}X_{1}&X_{2}&\ldots&X_{\ell}\\\
a_{1}&a_{2}&\ldots&a_{\ell}\\\ \end{smallmatrix}\right)$ is the perfect ideal
of $S$ of grade $\ell-1$ generated by the $2\times 2$ minors of the matrix
$\left(\begin{smallmatrix}X_{1}&X_{2}&\ldots&X_{\ell}\\\
a_{1}&a_{2}&\ldots&a_{\ell}\\\ \end{smallmatrix}\right)$. We set $I={\rm
Ker}(\varphi)$. We then have the following.

######  Claim 2.

Let $P\in\operatorname{Spec}S$ such that $I\subseteq P$ but
$(X_{1},X_{2},\ldots,X_{\ell})+(a_{1},a_{2},\ldots,a_{\ell})\not\subseteq P$.
Then, $S_{P}/IS_{P}$ is a Gorenstein ring.

###### Proof of Claim 2.

We may assume that $X_{1}\not\in P$. Let $\widetilde{S}=S[\frac{1}{X_{1}}]$,
$\widetilde{A}=A[X_{1},\frac{1}{X_{1}}]$, and $Y_{i}=\frac{X_{i}}{X_{1}}$ for
$2\leq i\leq\ell$. Then,
$\widetilde{S}=\widetilde{A}[Y_{2},Y_{3},\ldots,Y_{\ell}]$ and
$I\widetilde{S}=(a_{i}-a_{1}Y_{i}\mid 2\leq i\leq\ell)\widetilde{S}$. Because
$a_{1}\widetilde{S}+(a_{i}-a_{1}Y_{i}\mid 2\leq
i\leq\ell)\widetilde{S}=(a_{i}\mid 1\leq i\leq\ell)\widetilde{S}$ and
$a_{1},a_{2},\ldots,a_{\ell}$ is an $\widetilde{S}$-regular sequence, the
sequence $a_{2}-a_{1}Y_{2},a_{3}-a_{1}Y_{3},\ldots,a_{\ell}-a_{1}Y_{\ell}$ is
$\widetilde{S}_{P}$-regular, so that $S_{P}/IS_{P}$ is a Gorenstein ring. ∎

Let $P\in\operatorname{Spec}S$ and suppose that $I\subseteq P$. We set
$\mathfrak{p}=\varphi(P)\in\operatorname{Spec}\mathcal{R}$. Then,
$(X_{1},X_{2},\ldots,X_{\ell})+(a_{1},a_{2},\ldots,a_{\ell})\not\subseteq P$
if $\operatorname{ht}_{S}P<2\ell$, while

$\operatorname{ht}_{\mathcal{R}}\mathfrak{p}=\operatorname{ht}_{S/I}P/I=\operatorname{ht}_{S}P-(\ell-1).$

Therefore, if $\operatorname{ht}_{\mathcal{R}}\mathfrak{p}<\ell+1$, then
$\operatorname{ht}_{S}P-(\ell-1)<\ell+1$, that is
$\operatorname{ht}_{S}P<2\ell$, so that
$(X_{1},X_{2},\ldots,X_{\ell})+(a_{1},a_{2},\ldots,a_{\ell})\not\subseteq P$,
whence $\mathcal{R}_{\mathfrak{p}}=S_{P}/IS_{P}$ is a Gorenstein ring by Claim
2. Thus, $\mathcal{R}$ is an $(\ell+1)$-Gorenstein ring. ∎

Since the proofs of the following assertions are standard, we left them to the
interested readers.

######  Lemma 7.2.

Let $\varphi:A\to B$ be a flat local homomorphism of Noetherian local rings
and $q\geq 1$ be an integer. Then the following conditions are equivalent :

  1. $(1)$

$B$ is a $q$-Gorenstein ring ;

  2. $(2)$

$A$ is a $q$-Gorenstein ring and $B_{P}/\mathfrak{p}B_{P}$ is a Gorenstein
ring for every $P\in\operatorname{Spec}B$ with $\operatorname{depth}B_{P}<q$,
where $\mathfrak{p}=\varphi^{-1}(P)$.

######  Proposition 7.3.

Let $R$ be a Noetherian ring. Then the following assertions hold true.

  1. $(1)$

Let $q\geq 1$ be an integer. Then $R[t]$ is a $q$-Gorenstein ring if and only
if $R$ is a $q$-Gorenstein ring, where $t$ is an indeterminate.

  2. $(2)$

Let $H$ be a symmetric numerical semigroup. If $R$ is a $q$-Gorenstein ring,
then the semigroup ring $R[H]$ of $H$ over $R$ is a $q$-Gorenstein ring.

  3. $(3)$

Let $X=\\{X_{ij}\\}_{1\leq i\leq\ell,1\leq j\leq m}$ be indeterminates where
$\ell,m\geq 2$, and set $T=R[X]$. Let $t$ be an integer such that $2\leq
t\leq\min~{}\\{\ell,m\\}$ and let $I=\mathbf{I}_{t}(X)$ denote the ideal of
$S$ generated by the $t\times t$ minors of the matrix $X$. We set $S=T/I$.

    1. $(\rm{a})$

Let $\ell=m$. If $R$ is a $q$-Gorenstein ring, then $S$ is a $q$-Gorenstein
ring.

    2. $(\rm{b})$

Suppose that $R$ is a field and let $t=2$. Then $S$ is a $d$-Gorenstein ring,
where $d=\ell+m-1$.

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