# Degrees: Vasconcelos Contributions

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>and  Jooyoun Hong  Department of Mathematics,
Southern Connecticut State University, 501 Crescent Street, New Haven, CT
06515-1533, U.S.A<EMAIL_ADDRESS>
###### Abstract.

A degree of a module $M$ is a numerical measure of information carried by $M$.
We highlight some of Vasconcelos’ outstanding contributions to the theory of
degrees, bridging commutative algebra and computational algebra. We present
several degrees he introduced and developed, including arithmetic degree,
jdeg, homological degree, cohomological degrees, canonical degree and bi-
canonical degree. For the canonical and bi-canonical degrees we discuss recent
developments motivated by our joint works [25, 19, 9].

Key Words: multiplicity, arithmetic degree, jdeg, homological degree,
cohomological degree, canonical ideal, canonical degree, bi-canonical degree.
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.

In loving memory of Wolmer Vasconcelos. It has been an honor and a privilege
to be his friends and collaborators, and to witness the joy that mathematics
brought to him.

##  1\. Introduction

We discuss some of Vasconcelos’ outstanding results in the theory of degrees
and how they have inspired many researches in commutative algebra and beyond.
We present (in chronological order when appropriate) several degrees
Vasconcelos introduced and developed, such as arithmetic degree, jdeg,
homological degree, cohomological degrees, canonical degree and bi-canonical
degree. For more details and proofs on the statements of various degrees we
refer the reader to [64] and to the referenced literature. For canonical
degree and bi-canonical degree we focus on our joint work, informal
discussions, open questions that were of particular interest to Vasconcelos,
and recent developments. A more comprehensive survey is in [8] and a full
treatment is in our papers [25, 19, 9]. We refer to [11] for general
background and any unexplained terminology.

Let $(R,\mathfrak{m})$ be a Noetherian local ring (or a Noetherian graded
algebra) and let $\mathcal{M}(R)$ be the category of finitely generated
$R$-modules (or the appropriate category of graded modules). Broadly speaking,
a degree function is a numerical function $d:\mathcal{M}(R)\mapsto\mathbb{N}$.
Extensively studied degree functions include the multiplicity (classical
degree) and the Castelnuovo-Mumford regularity. The main point is that the
value $d(M)$ should capture several important elements of the structure of the
$R$-module $M$. A major theme of Vasconcelos’ prolific research is the
connection between commutative algebra and computational algebra (see for
instance [61]). Therefore we highlight some issues of particular interest when
we look for new degree functions.

Cohen-Macaulay structures can be considered as fundamental blocks (see [11]
for a beautiful treatment of the subject). Vasconcelos wrote in [64]: “ We
hold the view that Cohen-Macaulay structures are ubiquitous in algebra, are
central to commutative algebra and algebraic geometry and represent unique
computational efficiencies and that without an understanding of the ways it
appears in the structure of non Cohen-Macaulay structures the prospects for
successful large scale symbolic computations are rather dim.” Hence it is
desirable to seek functions that coincide with the classical multiplicity when
the $R$-module $M$ is Cohen-Macaulay and to be able to measure the deviation
of $M$ from being Cohen-Macaulay. It is also important to relate a new degree
function to other classical invariants of $M$ and to study is its behavior
under generic hyperplane sections.

A main topic of Vasconcelos’ research is the study of the complexity of
normalization processes of graded algebras. In this context, let $R$ be a
normal, unmixed integral domain and let $A$ be a semistandard graded
$R$-algebra (i.e., a finite extension of a standard graded $R$-algebra). Let
$\overline{A}$ be the integral closure of $A$. Estimating the number of steps
that general algorithms must take to build $\overline{A}$ can be viewed as an
invariant of A. For several interesting classes of algebras, including affine
graded algebras over fields and Rees algebras of ideals and modules,
multiplicity dependent bounds were derived for instance in [14, 41, 47, 54,
56]. One of Vasconcelos’ broad goals was to define new degree functions that
provide extensions of such bounds to general graded algebras and to find new
techniques to study the complexity.

Now we briefly describe how this manuscript is organized. The first half of
the paper focuses on arithmetic degree, jdeg, homological degree, and
cohomological degrees. The definitions, properties and developments of these
degrees are in the respective sections.

The arithmetic degree is addressed in Section 2. It is a refinement of the
classical multiplicity, but it is defined in a way that collects information
about the associated primes of $M$ in all codimensions. A major contribution
of Vasconcelos is a formula for the arithmetic degree that does not use a
primary decomposition (Theorem 2.2). Due to his result, software systems such
as Macaulay2 [34] or CoCoA [1] can be used to compute the arithmetic degree.

Section 3 is dedicated to $jdeg$, introduced by Pham and Vasconcelos as a
generalization of j-multiplicity. One important point is that the jdeg
provides estimates for the complexity of normalization (Theorem 3.3). It also
extends classical bounds.

In Section 4 we discuss the homological degree, which was introduced by
Vasconcelos as an extension of the usual notion of multiplicity, aimed to
provide a numerical signature for a module $M$ when $M$ is not necessarily
Cohen-Macaulay. The difference between the homological degree of $M$ and its
multiplicity is often called a Cohen-Macaulay deficiency in the literature.
For example, if $M$ is generalized Cohen-Macaulay, this difference is the
Stückrad-Vogel invariant of $M$. The homological degree is defined recursively
on the dimension of the module, which allows, in principle, for its
calculation by symbolic computer packages. It can be used to bound the Euler
characteristic of parameter ideals (Theorem 4.14). The homological degree is
used to define the homological torsion, which plays an important role in the
study of Hilbert coefficients. The homological torsion can be used to bound
the first Hilbert coefficient $\mathrm{e}_{1}(Q,M)$ of a module $M$ relative
to an ideal generated by a system of parameters $Q$ for $M$ (Theorem 4.17).
The coefficient $\mathrm{e}_{1}(Q,M)$ is often called the Chern number, and it
has been extensively studied by Vasconcelos.

In Section 5 we discuss cohomological degrees, introduced by Vasconcelos. The
notion of cohomological degree generalizes the homological degree and it is
the classical multiplicity if $M$ is Cohen-Macaulay. It provides an estimate
for the Castelnuovo-Mumford regularity of a standard graded algebra (Theorem
5.3). The cohomological degree can also be used to estimate the minimal number
of generators of an ideal, generalizing and improving many previously known
bounds (Theorem 5.6). We also discuss the bdeg, a cohomological degree
introduced by Vasconcelos’ former student Gunston.

The second half of the manuscript, Sections 6, 7, 8, 9, 10, 11, deals with
more recently introduced degrees, the canonical degree and the bi-canonical
degree [25, 19, 9]. We include topics and developments motivated by our
discussions with Vasconcelos.

The overarching goal here is the stratification of Cohen-Macaulay rings.
Several researchers have been interested in finding new significant classes of
rings which naturally include Gorenstein rings. We introduce invariants that
characterize some of these classes and measure the deviation from $R$ being
Gorenstein. Let $R$ be a Cohen-Macaulay local ring that has a canonical ideal
$\mathcal{C}$. A canonical ideal is an ideal isomorphic to a canonical module
of $R$. The condition that the canonical module is an ideal is equivalent to
$R$ being generically Gorenstein. We refer to [11, Section 3] for the
background on canonical modules. The ring $R$ is Gorenstein when $\mathcal{C}$
is isomorphic to $R$, therefore we look at the properties of $\mathcal{C}$ as
a way to refine our understanding of $R$.

In Section 6 we discuss the canonical degree of $R$, $\mbox{\rm cdeg}(R)$. It
is defined so that its vanishing characterizes the Gorenstein property of $R$
in codimension one. Furthermore, the minimal value of the canonical degree is
assumed on the class of almost Gorenstein rings (Proposition 6.9). Almost
Gorenstein rings local rings (AGL for short) were introduced in [5], and a
rich theory has been developed in [29] and [33]. Numerical semigroup rings, in
particular the three-generated ones, provide the set-up for many interesting
results and examples, which we summarize at the end of the section. In Section
7 we relate the canonical degree to the Sally module, introduced by
Vasconcelos. This connection inspired the definition of 2-AGL rings [12], as a
generalization of AGL rings. We propose a further generalization to 3-AGL
rings, we give a characterization in terms of canonical degree when the type
of the ring is small, we present supporting examples in three-generated
semigroup rings (computed with the aid of Macaulay2), and we conclude with
some open questions.

In Section 8 we discuss the bi-canonical degree of $R$, $\mbox{\rm bideg}(R)$.
Like the canonical degree, the vanishing of the bi-canonical degree
characterizes Gorensteinness in codimension one. However, the bi-canonical
degree seems more suitable for computations. We recall the important
properties and we explore rings with minimal value of bideg.

In Section 9 we discuss the Comparison Conjecture. In [19] we conjectured that
in general, $\mbox{\rm bideg}(R)\leq\mbox{\rm cdeg}(R)$. We prove that the
conjecture holds in three-generated numerical semigroup rings, using the
methods of [19], where we had shown a special case. There have been very
interesting recent developments for numerical semigroup rings. Using different
methods, Herzog and Kumashiro in [39] studied an inequality that turned out to
be equivalent to the Comparison Conjecture. Their results imply that the
conjecture is true when the type of $R$ is at most three, and they found a
counterexample when the type is five.

In Section 10 we summarize how the canonical degree and the bi-canonical
degree behave under change of rings, in particular augmented rings,
$(\mathfrak{m}:\mathfrak{m})$, and hyperplane sections.

Finally, in Section 11, we present the main results and open questions from
[9], our last paper with Vasconcelos. We extended the bi-canonical degree to
rings where the canonical module is not necessarily an ideal, and we discussed
generalizations to rings without canonical modules but admitting modules
sharing some of their properties.

##  2\. Arithmetic Degree

We begin this section by recalling the definition of multiplicity, which
motivated the definition of the arithmetic degree. Let $R$ be a Noetherian
local ring of dimension $d>0$ with maximal ideal $\mathfrak{m}$ and let $M$ be
a finitely generated $R$-module of dimension $s$. Let $I$ be an
$\mathfrak{m}$-primary ideal of $R$. We denote length by $\lambda$. We define
the Hilbert function of $M$ with respect to $I$ as

${\mathrm{H}}_{I,M}:n\mapsto{\lambda}(M/I^{n+1}M).$

There exists a polynomial ${\displaystyle\mbox{\rm
P}_{I,M}(X)\in\mathbb{Q}[X]}$ of degree $s$ such that
${\displaystyle{\mathrm{H}}_{I,M}(n)=\mbox{\rm P}_{I,M}(n)}$ for all
sufficiently large $n$. This polynomial ${\displaystyle\mbox{\rm P}_{I,M}(X)}$
is called the Hilbert polynomial of $M$ with respect to $I$ and can be written
as

$\mbox{\rm
P}_{I,M}(n)=\sum_{i=0}^{s}(-1)^{i}\mathrm{e}_{i}(I,M){{n+s-i}\choose{s-i}}.$

The integers $\mathrm{e}_{i}(I,M)$ are called the Hilbert coefficients of $M$
with respect to $I$. In particular, $\mathrm{e}_{0}(I,M)$ is called the
multiplicity (or degree) of the module $M$ with respect to $I$. When
$I=\mathfrak{m}$, we write
${\displaystyle\mathrm{e}_{0}(\mathfrak{m},M)=\mbox{\rm deg}(M)}$.

There is a chain of submodules ${\displaystyle 0=M_{0}\subset
M_{1}\subset\cdots\subset M_{r-1}\subset M_{r}=M}$ such that ${\displaystyle
M_{i}/M_{i-1}\simeq R/{\mathfrak{p}}_{i}}$ for some
${\displaystyle{\mathfrak{p}}_{i}\in\mbox{\rm Spec}(R)}$. In this case, we
obtain

$\mbox{\rm deg}(M)=\sum_{\tiny\dim(R/{\mathfrak{p}}_{i})=d}\mbox{\rm
deg}(R/{\mathfrak{p}}_{i}).$

Recall that ${\displaystyle\mbox{\rm
Ass}(M)\subseteq\\{{\mathfrak{p}}_{i}\\}_{i=1}^{n}}$. If
${\displaystyle{\mathfrak{p}}\in\mbox{\rm Ass}(M)}$ with
${\displaystyle\dim(R/{\mathfrak{p}})=d}$, then the length
${\displaystyle{\lambda}(M_{{\mathfrak{p}}})}$ is the number of times that
${\mathfrak{p}}$ occurs in the chain of submodules. That is, we have

$\sum_{\tiny\begin{array}[]{c}{\mathfrak{p}}\in\mbox{\rm Ass}(M)\\\
\dim(R/{\mathfrak{p}})=d\end{array}}{\lambda}(M_{{\mathfrak{p}}})\,\mbox{\rm
deg}(R/{\mathfrak{p}})\leq\mbox{\rm deg}(M).$

The arithmetic degree of $M$ was defined in a way that collects information
about the associated primes of $M$ in all codimensions.

######  Definition 2.1.

[59, Definition 1.6] Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$
a finitely generated $R$-module. The arithmetic degree of $M$ is the integer

$\mbox{\rm adeg}(M)=\sum_{\tiny{\mathfrak{p}}\in\mbox{\rm
Ass}(M)}{\lambda}(\Gamma_{{\mathfrak{p}}}(M_{{\mathfrak{p}}}))\,\mbox{\rm
deg}(R/{\mathfrak{p}}),$

where
${\displaystyle\Gamma_{{\mathfrak{p}}}(M_{{\mathfrak{p}}})=H^{0}_{{\mathfrak{p}}R_{{\mathfrak{p}}}}(M_{{\mathfrak{p}}})}$
is the finite length submodule of $M_{{\mathfrak{p}}}$ consisting of all
elements of ${\displaystyle M_{{\mathfrak{p}}}}$ with support in
${\displaystyle{\mathfrak{p}}R_{{\mathfrak{p}}}}$. (In some literatures, the
arithmetic degree is also denoted by ${\displaystyle\mbox{arith-deg}(M)}$.)

If all the associated primes of $M$ have the same codimension, then $\mbox{\rm
adeg}(M)$ is equal to $\mbox{\rm deg}(M)$. Both $\mbox{\rm deg}(M)$ and
$\mbox{\rm adeg}(M)$ can be defined in the same way as above when $M$ is a
(graded) finitely generated module over a standard graded algebra
${\displaystyle A=A_{0}[A_{1}]}$ with the Artinian local ring $A_{0}$. In this
case, the Hilbert function of $M$ is defined in terms of the homogeneous
maximal ideal of $A$. Bayer and Mumford [6] gave an extensive survey on
${\displaystyle\mbox{\rm adeg}(A)}$ and its significance in algebraic
geometry, where ${\displaystyle A=k[X_{1},\ldots,X_{n}]/I}$ is the quotient of
a polynomial ring over a field $k$.

Vasconcelos showed how to compute the arithmetic degree of a graded module
without using a primary decomposition. Due to his theorem, software systems
such as Macaulay2 or CoCoA can be used to compute the arithmetic degree.

######  Theorem 2.2.

([59, Proposition 1.11], [58, Proposition 5]) Let $R=k[X_{1},\ldots,X_{d}]$
and $M$ a graded $R$-module. Then

$\mbox{\rm adeg}(M)=\sum_{i=0}^{d}\mbox{\rm deg}\left(\mbox{\rm
Ext}^{i}_{R}\left(\mbox{\rm Ext}^{i}_{R}(M,R),R\right)\right).$

One of the main characteristics of a new degree function $\mbox{\rm Deg}(M)$
(see Section 5) that we wish to obtain is that it behaves well under generic
hyperplane sections. That is, if $A=A_{0}[A_{1}]$ is a standard graded algebra
with the Artinian local ring $A_{0}$ and if $h\in A_{1}$ is not contained in
any minimal prime of $A$, then ${\displaystyle\mbox{\rm Deg}(M)\geq\mbox{\rm
Deg}(M/hM)}$. Unfortunately, the arithmetic degree fails to obtain this
characteristic.

######  Theorem 2.3.

[59, Theorem 1.14] Let $A=A_{0}[A_{1}]$ be a standard graded algebra with the
Artinian local ring $A_{0}$ and $M$ a finitely generated graded $A$-module.
Let $h\in A_{1}$ be a regular element on $M$. Then

$\mbox{\rm adeg}(M)\leq\mbox{\rm adeg}(M/hM).$

In [58] Vasconcelos explored relations between the Castelnuovo-Mumford
regularity and the arithmetic degree of a standard algebra $A$. He also
studied comparisons between the arithmetic degree and the reduction number of
$A$. He extensively studied the reduction number as a measure of the
complexity of $A$. One interesting application of the arithmetic degree is
that it can be used to make predictions about the outcome of carrying out a
Noether normalization on a graded algebra.

######  Theorem 2.4.

([59, Theorem 2.11], [58, Theorem 7]) Let $A$ be an affine algebra over an
infinite field $k$ and let $k[{\mathbf{z}}]$ be a Noether normalization of
$A$. Let $M$ be a finitely generated, graded, and faithful $A$-module. Then
every element of $A$ satisfies a monic equation over $k[{\mathbf{z}}]$ of
degree at most $\mbox{\rm adeg}(M)$.

Miyazki, Vogel, and Yanagawa [42] presented a refined theory on the arithmetic
degree of a homogeneous ideal $I$ of a polynomial ring $S$ over a field. A key
point in their work is the description of the arithmetic degree in terms of
the Hilbert polynomial of an appropriate $\mbox{\rm Ext}_{S}^{i}(S/I,S)$,
based on Vasconcelos’ ideas [16, 58].

##  3\. jdeg

The notion of j-multiplicity was introduced and developed by Achilles and
Manaresi [2], Flenner, O’Carroll and Vogel [18], and evolved into a rich
extension of the ordinary multiplicity (see for instance [43, 53]). Let
$(R,\mathfrak{m})$ be a Noetherian local ring and let $A$ be a finitely
generated graded $R$-algebra. Let $M$ be a finitely generated graded
$A$-module. Then ${\mathrm{H}}^{0}_{\mathfrak{m}}(M)$ is a graded submodule of
$M$ and is annihilated by a sufficiently large power of $\mathfrak{m}$.
Therefore, ${\mathrm{H}}^{0}_{\mathfrak{m}}(M)$ can be considered as a graded
module over the Artinian ring ${\displaystyle A/\mathfrak{m}^{k}A}$ for some
sufficiently large $k$.

######  Definition 3.1.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $A$ be a finitely
generated graded $R$-algebra. Let $M$ be a finitely generated graded
$A$-module. Then the j-multiplicity of $M$ is

$\mbox{\rm
j}_{\mathfrak{m}}(M)=\left\\{\begin{array}[]{ll}{\displaystyle\mbox{\rm
deg}({\mathrm{H}}_{\mathfrak{m}}^{0}(M))}&\mbox{if}\;\;{\displaystyle\dim({\mathrm{H}}_{\mathfrak{m}}^{0}(M))=\dim_{R}(M)},\vspace{0.1
in}\\\ 0&\mbox{otherwise.}\end{array}\right.$

A special case of the j-multiplicity can be defined as follows: Let
$(R,\mathfrak{m})$ be a Noetherian local ring, $I$ an arbitrary $R$-ideal, and
$N$ a finite $R$-module of dimension $d$. Then the associated graded module
${M=\mbox{\rm gr}_{I}(N)=\bigoplus_{j=0}^{\infty}I^{j}N/I^{j+1}N}$ is a
finitely generated graded module over the associated graded ring ${A=\mbox{\rm
gr}_{I}(R)=\bigoplus_{j=0}^{\infty}I^{j}/I^{j+1}}$. In this case, the
j-multiplicity of ${\displaystyle\mbox{\rm gr}_{I}(N)}$ is sometimes called
the j-multiplicity of $I$ on $N$. Polini and Xie [48] established the depth of
the associated graded ring in terms of the j-multiplicity of an ideal.

Pham and Vasconcelos introduced the jdeg, extending the definition of
j-multiplicity by localization. We note that in contrast to other extensions
of the classical multiplicity, such as the arithmetic degree, that requires
$R$ to be a local ring, jdeg places no such restrictions on $R$.

######  Definition 3.2.

[46, Definition 1.1] Let $R$ be a Noetherian ring and $A$ a standard graded
$R$-algebra. Let $M$ be a finitely generated graded $A$-module. For each prime
ideal ${\mathfrak{p}}$ of $R$, we define

$\mbox{\rm
j}_{{\mathfrak{p}}}(M)=\left\\{\begin{array}[]{ll}{\displaystyle\mbox{\rm
deg}({\mathrm{H}}_{{\mathfrak{p}}R_{{\mathfrak{p}}}}^{0}(M_{{\mathfrak{p}}}))}&\mbox{if}\;\;{\displaystyle\dim({\mathrm{H}}_{{\mathfrak{p}}R_{{\mathfrak{p}}}}^{0}(M_{{\mathfrak{p}}}))=\dim_{R_{{\mathfrak{p}}}}(M_{{\mathfrak{p}}})},\vspace{0.1
in}\\\ 0&\mbox{otherwise.}\end{array}\right.$

Then jdeg of $M$ is the integer

$\mbox{\rm jdeg}(M)=\sum_{\tiny{\mathfrak{p}}\in\mbox{\rm Spec}(R)}\mbox{\rm
j}_{{\mathfrak{p}}}(M).$

If $(R,\mathfrak{m})$ is an Artinian local ring, then $\mbox{\rm jdeg}(M)$ is
equal to the traditional multiplicity $\mbox{\rm deg}(M)$. One important
application is that the jdeg provides estimates for the complexity of
normalization.

######  Theorem 3.3.

[45, Theorem 2] Let $R$ be a Noetherian domain and $A$ a semistandard graded
$R$-algebra of finite integral closure $\overline{A}$. Consider a sequence of
distinct integral graded extensions

$A\subseteq A_{0}\rightarrow A_{1}\rightarrow A_{2}\cdots\rightarrow
A_{n}=\overline{A},$

where the $A_{i}$ satisfy the $S_{2}$ condition of Serre. Then $n\leq\mbox{\rm
jdeg}(\overline{A}/A)$.

In [45] Pham and Vasconcelos also explored how the values of jdeg relate to
other well studied invariants of the algebra. For instance, let $A$ be a
semistandard graded domain over a field $k$. If $A$ satisfies $S_{2}$ and
$A\neq\overline{A}$, then $\mbox{\rm
jdeg}(\overline{A}/A)=\mathrm{e}_{1}(A)-\mathrm{e}_{1}(\overline{A})$. Among
other interesting results, using jdeg the authors generalize bounds of [47]
(see [45, Theorem 3]). In the sequel [46] the focus is on jdeg of blow-up
algebras. We denote the minimal number of generators of a module $M$ by
$\nu(M)$.

######  Theorem 3.4.

[46, Proposition 4.1] Let ${\displaystyle(R,\mathfrak{m})}$ be a Noetherian
local domain of positive dimension and let $M$ be a non-free module of rank r
that is free on the punctured spectrum. Let $S$ be the symmetric algebra of
$M$. Then

$\mbox{\rm jdeg}(S)=\left\\{\begin{array}[]{ll}1+\mbox{\rm
j}_{\mathfrak{m}}(S)&\mbox{if}\;\;{\displaystyle\nu(M)\geq\dim(R)+r,\vspace{0.1
in}}\\\ 1&\mbox{otherwise.}\end{array}\right.$

A more specific bound of jdeg of the symmetric algebra with additional
conditions on $R$ and $M$ can be found in [46, Theorem 4.2 and Proposition
4.3]. An important property of jdeg is its invariance property.

######  Theorem 3.5.

[44, Theorem 4.1.1] Let $R$ be a reduced universally catenary Noetherian ring
and $I$ an ideal containing a regular element. Let $A=R[It]$ be the Rees
algebra of $I$ and let $\overline{A}$ be the integral closure of $A$. Let $B$
be a finitely generated graded $A$-algebra defined by a decreasing filtration
such that ${\displaystyle A\subsetneq B\subset\overline{A}}$. Then

$\mbox{\rm j}_{{\mathfrak{p}}}(\mbox{\rm gr}(A))=\mbox{\rm
j}_{{\mathfrak{p}}}(\mbox{\rm gr}(B))\quad\mbox{for
every}\;\;{\mathfrak{p}}\in\mbox{\rm Spec}(R).$

Moreover, if $B$ satisfies the Serre’s condition $({\mathrm{S}}_{2})$, then
${\displaystyle\mbox{\rm jdeg}(\mbox{\rm gr}(A))=\mbox{\rm jdeg}(\mbox{\rm
gr}(B))}$.

##  4\. Homological Degree

The homological degree was introduced by Vasconcelos as an extension of the
usual notion of multiplicity, aimed to provide a numerical signature for a
module $M$ when $M$ is not necessarily Cohen-Macaulay. It is defined
recursively on the dimension of the module, which allows, in principle, for
its calculation by symbolic computer packages.

######  Definition 4.1.

[60, Definition 2.8] Let $(R,\mathfrak{m})$ be a Gorenstein local ring of
dimension $d$. Let $I$ be an $\mathfrak{m}$-primary ideal of $R$. Let $M$ be a
finitely generated $R$-module of dimension $s$. The homological degree of $M$
with respect to $I$ is defined as follows:

$\mbox{\rm
hdeg}_{I}(M)=\left\\{\begin{array}[]{ll}{\displaystyle{\lambda}(M)}&\mbox{if}\;\;{\displaystyle
s=0,\vspace{0.1 in}}\\\
{\displaystyle\mathrm{e}_{0}(I,M)+\sum_{i=d-s+1}^{d}{{s-1}\choose{i-d+s-1}}\mbox{\rm
hdeg}_{I}\left(\mbox{\rm Ext}^{i}_{R}(M,R)\right)}&\mbox{if}\;\;{\displaystyle
s>0.}\end{array}\right.$

When $I=\mathfrak{m}$, we simply write ${\displaystyle\mbox{\rm hdeg}(M)}$.

If $(R,\mathfrak{m})$ is a Gorenstein local ring, by the Local Duality Theorem
[11, Theorem 3.5.8], for all integers $i$, there exist natural isomorphisms

$\mbox{\rm Ext}^{i}_{R}(M,R)\simeq\mbox{\rm
Hom}_{R}(H^{d-i}_{\mathfrak{m}}(M),\,E),$

where $E=E(k)$ denotes the injective envelope of the residue field $k$.

Goto and Ozeki [31] introduced an alternative definition of a homological
degree.

######  Definition 4.2.

[31, Definition 1.1] Let $(R,\mathfrak{m})$ be a Gorenstein local ring with
infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal of $R$ and
$M$ a finitely generated $R$-module of dimension $s$. The homological degree
of $M$ with respect to $I$ is defined as follows:

$\mbox{\rm
hdeg}_{I}(M)=\left\\{\begin{array}[]{ll}{\displaystyle{\lambda}(M)}&\mbox{if}\;\;{\displaystyle
s=0\vspace{0.1 in}}\\\
{\displaystyle\mathrm{e}_{0}(I,M)+\sum_{i=0}^{s-1}{{s-1}\choose{i}}\mbox{\rm
hdeg}_{I}\left(\mbox{\rm
Hom}_{R}(H^{i}_{\mathfrak{m}}(M),E)\right)}&\mbox{if}\;\;{\displaystyle
s>0.}\end{array}\right.$

By [31, Fact 2.1], ${\displaystyle\dim\left(\mbox{\rm
Hom}_{R}(H^{i}_{\mathfrak{m}}(M),E)\right)\leq i}$ for each $i$. Therefore the
definition of the homological degree is a recursive definition. Moreover, if
$\dim(R)=\dim(M)$, then the definitions 4.1 and 4.2 coincide. Note that if $M$
is Cohen-Macaulay, then ${\displaystyle\mbox{\rm
hdeg}_{I}(M)=\mathrm{e}_{0}(I,M)}$.

In the original definition of the homological degree [60, Definition 2.8],
Vasconcelos supposed that $R$ is a graded algebra, $S$ a Gorenstein graded
algebra mapping onto $R$ and $M$ is a finitely generated graded module of $R$,
and he used ${\displaystyle\mbox{\rm Ext}_{S}^{i}(M,S)}$ in the definition. In
[31], the authors supposed that $R$ is $\mathfrak{m}$-adically complete. In
this case, $R$ is a homomorphic image of a Gorenstein local ring $S$. Thus, by
passing to $S$, without loss of generality, they assumed that $R$ is a
Gorenstein ring. Throughout this section we may assume that $R$ is a
Gorenstein local ring, not only for simplicity, but also because it enables us
to use both definitions ([60, Definition 2.8], [31, Definition 1.1]) due to
the Local Duality Theorem.

######  Remark 4.3.

The following are the homological degrees of a module $M$ of small dimension.

  1. (1)

Suppose $\dim(M)=1$. Then

$\mbox{\rm
hdeg}_{I}(M)=\mathrm{e}_{0}(I,M)+{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right).$

  2. (2)

Suppose $\dim(M)=2$. Let ${\displaystyle N=\mbox{\rm
Hom}(H^{1}_{\mathfrak{m}}(M),E)}$. Then

$\mbox{\rm
hdeg}_{I}(M)=\left\\{\begin{array}[]{ll}{\displaystyle\mathrm{e}_{0}(I,M)+{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)+{\lambda}\left(N\right)}&\mbox{if}\;\;\dim(N)=0,\vspace{0.1
in}\\\
{\displaystyle\mathrm{e}_{0}(I,M)+{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)+\mathrm{e}_{0}(I,N)+{\lambda}\left(H^{0}_{\mathfrak{m}}(N)\right)}&\mbox{if}\;\;\dim(N)=1.\end{array}\right.$

Also, by [59, Proposition 3.5], ${\displaystyle\mbox{\rm hdeg}(M)=\mbox{\rm
adeg}(M)+{\lambda}\left(\mbox{\rm Ext}^{2}_{R}\left(\mbox{\rm
Ext}^{1}_{R}(M,R),R\right)\right)}$.

A finitely generated module $M$ is said to be generalized Cohen-Macaulay if
$M_{{\mathfrak{p}}}$ is Cohen-Macaulay for every non-maximal prime ideal
${\mathfrak{p}}$. Recall that $M$ is generalized Cohen-Macaulay if and only if
$H^{i}_{\mathfrak{m}}(M)$ is finitely generated for all $i<\dim(M)$. By using
the Matlis Duality, we obtain the following.

######  Proposition 4.4.

Let $(R,\mathfrak{m})$ be a Gorenstein local ring and $I$ an
$\mathfrak{m}$-primary ideal. If $M$ is a generalized Cohen-Macaulay module
and $\dim(M)=s$, then

$\mbox{\rm
hdeg}_{I}(M)=\mathrm{e}_{0}(I,M)+\sum_{i=0}^{s-1}{{s-1}\choose{i}}\,{\lambda}\left(H^{i}_{\mathfrak{m}}(M)\right).$

The integer
${\displaystyle\sum_{i=0}^{s-1}{{s-1}\choose{i}}\,{\lambda}\left(H^{i}_{\mathfrak{m}}(M)\right)}$
is called the Stückrad-Vogel invariant of $M$ and is denoted by
${\displaystyle\mathbb{I}(M)}$. Thus, if $M$ is a generalized Cohen-Macaulay
module, then ${\displaystyle\mbox{\rm
hdeg}_{I}(M)=\mathrm{e}_{0}(I,M)+\mathbb{I}(M)}$. A family of elements
${\displaystyle{\mathbf{x}}=x_{1},\ldots,x_{s}}$ in the maximal ideal
$\mathfrak{m}$ is said to be a system of parameters for $M$ if
${\displaystyle\dim(M/({\mathbf{x}})M)=0}$, where $s=\dim(M)$. We call an
$R$-ideal $Q$ a parameter ideal for $M$ if there exists a system of parameters
${\displaystyle{\mathbf{x}}=x_{1},\ldots,x_{s}}$ for $M$ such that
$Q=({\mathbf{x}})$. A finitely generated module $M$ is said to be Buchsbaum if
${\displaystyle{\lambda}(M/QM)-\mathrm{e}_{0}(Q,M)=\mathbb{I}(M)}$ for every
parameter ideal $Q$ of $M$. It is well known that a Buchbaum module is a
generalized Cohen-Macaulay module. Thus, we obtain the following formula for
$\mbox{\rm hdeg}_{I}(M)$ in terms of multiplicities.

######  Proposition 4.5.

Let $(R,\mathfrak{m})$ be a Gorenstein local ring and $I$ an
$\mathfrak{m}$-primary ideal. If $M$ is a Buchsbaum module, then for every
parameter ideal $Q$ of $M$ we have

$\mbox{\rm
hdeg}_{I}(M)=\mathrm{e}_{0}(I,M)+{\lambda}(M/QM)-\mathrm{e}_{0}(Q,M).$

### Properties of homological degrees

Now we observe general properties of the homological degree. Particularly, we
are interested in bounds for hdeg, how hdeg works with short exact sequences,
and most importantly how hdeg behaves under hyperplane sections.

######  Proposition 4.6.

[63, Proposition 7.6] Let $(R,\mathfrak{m})$ be a Gorenstein local ring and
$I$ an $\mathfrak{m}$-primary ideal such that $\mathfrak{m}^{r}\subset I$. Let
$M$ be a $R$-module of dimension $s$. Then

$\mbox{\rm hdeg}_{I}(M)\leq r^{s}\,\mbox{\rm deg}(M)+r^{s-1}\,\left(\mbox{\rm
hdeg}(M)-\mbox{\rm deg}(M)\right).$

######  Proposition 4.7.

([59, Proposition 3.18], [62, Proposition 2.84], [31, Lemma 2.4])  Let
$(R,\mathfrak{m})$ be a Gorenstein local ring and $I$ an
$\mathfrak{m}$-primary ideal. Let ${\displaystyle 0\rightarrow L\rightarrow
M\rightarrow N\rightarrow 0}$ be an exact sequence of finitely generated
$R$-modules. Then

  1. (1)

If ${\displaystyle{\lambda}(L)<\infty}$, then ${\displaystyle\mbox{\rm
hdeg}_{I}(M)=\mbox{\rm hdeg}_{I}(L)+\mbox{\rm hdeg}_{I}(N)}$.

  2. (2)

If ${\displaystyle{\lambda}(N)<\infty}$, then ${\displaystyle\mbox{\rm
hdeg}_{I}(M)\leq\mbox{\rm hdeg}_{I}(L)+\mbox{\rm hdeg}_{I}(N)}$.

  3. (3)

If ${\displaystyle{\lambda}(N)<\infty}$ and ${\displaystyle\dim(M)=s}$, then
${\displaystyle\mbox{\rm hdeg}_{I}(L)\leq\mbox{\rm hdeg}_{I}(M)+(s-1)\mbox{\rm
hdeg}_{I}(N)}$.

If $R$ is a standard Gorenstein graded algebra and $M$ is a finitely generated
graded $R$-module of depth at least $1$, then Vasconcelos proved that
${\displaystyle\mbox{\rm hdeg}(M/hM)\leq\mbox{\rm hdeg}(M)}$ for any generic
hyperplane section $h$ on $M$ [60, Theorem 2.13]. Goto and Ozeki extended it
to a local ring case. We denote the Rees algebra of an ideal $I$ by
${\displaystyle\mathcal{R}(I)=\bigoplus_{n=0}^{\infty}I^{n}t^{n}}$. We write
${\displaystyle\mathcal{R}_{+}(I)=\bigoplus_{n=1}^{\infty}I^{n}t^{n}}$ and set

$\mbox{\rm
Proj}\left(\mathcal{R}(I)\right)=\left\\{{\mathfrak{p}}\mid{\mathfrak{p}}\;\mbox{is
a graded prime ideal of $\mathcal{R}(I)$ such that
$\mathcal{R}_{+}(I)\not\subseteq{\mathfrak{p}}$ }\right\\}.$

Let ${\displaystyle f:I\rightarrow\mathcal{R}(I)}$ be a map given by
${\displaystyle f(x)=xt}$.

######  Theorem 4.8.

[31, Lemma 2.6] Let $(R,\mathfrak{m})$ be a Gorenstein local ring and $I$ an
$\mathfrak{m}$-primary ideal. Let $M$ be a finitely generated $R$-module.
There exists a finite subset ${\displaystyle\mathcal{F}\subseteq\mbox{\rm
Proj}\left(\mathcal{R}(I)\right)}$ such that

  1. (1)

every element of
${H=I\setminus\bigcup_{{\mathfrak{p}}\in\mathcal{F}}\left(f^{-1}({\mathfrak{p}})+\mathfrak{m}I\right)}$
is superficial for $M$ with respect to $I$, and

  2. (2)

${\displaystyle\mbox{\rm hdeg}_{I}(M/aM)\leq\mbox{\rm hdeg}_{I}(M)}$ for each
$a\in H$.

######  Definition 4.9.

Let $(R,\mathfrak{m})$ be a Gorenstein local ring and $I$ an
$\mathfrak{m}$-primary ideal. Let $M$ be a finitely generated $R$-module. An
element $a\in R$ (or a sequence ${\mathbf{a}}=a_{1},\ldots,a_{s}$ of elements
of $R$) is said to be superficial relative to $\mbox{\rm hdeg}_{I}(M)$ if $a$
(or the sequence ${\mathbf{a}}$) is superficial for $M$ with respect to $I$
and ${\displaystyle\mbox{\rm hdeg}_{I}(M/aM)\leq\mbox{\rm hdeg}_{I}(M)}$ (or
${\displaystyle\mbox{\rm hdeg}_{I}(M/{\mathbf{a}}M)\leq\mbox{\rm
hdeg}_{I}(M)}$ respectively).

######  Corollary 4.10.

[63, Theorem 7.2] Let $(R,\mathfrak{m})$ be a Gorenstein local ring and $I$ an
$\mathfrak{m}$-primary ideal. Let $M$ be a finitely generated $R$-module of
$\dim(M)\geq 2$. Let ${\mathbf{a}}=a_{1},\ldots,a_{r}$ be a superficial
sequence relative to $\mbox{\rm hdeg}_{I}(M)$. If $r<\dim(M)$, then

${\lambda}\left(H^{0}_{\mathfrak{m}}\left(M/({\mathbf{a}})M\right)\right)\leq\mbox{\rm
hdeg}_{I}(M)-\mathrm{e}_{0}(I,M).$

Brennan and Ciupercă showed when the homological degrees are preserved under
modulo operation for modules of small dimension.

######  Theorem 4.11.

[7, Theorems 3.1 and 4.2] Let $R$ be a standard graded ring over a field and
$\mathfrak{m}$ the ideal generated by the elements of degree $1$. Let $M$ be a
finitely generated graded $R$-module.

  1. (1)

Suppose ${\displaystyle\dim(M)=1}$. Let $a$ be an element of
$\mathfrak{m}\setminus\mathfrak{m}^{2}$ such that
${\displaystyle\dim(M/aM)=0}$. Then ${\displaystyle\mbox{\rm
hdeg}(M)=\mbox{\rm hdeg}(M/aM)}$ if and only if
${\displaystyle(0:_{M}a)=H^{0}_{\mathfrak{m}}(M)}$.

  2. (2)

Suppose ${\displaystyle\dim(M)=2}$. Let $a$ be a homogeneous element of degree
$1$ regular on $M/H^{0}_{\mathfrak{m}}(M)$. Then ${\displaystyle\mbox{\rm
hdeg}(M)=\mbox{\rm hdeg}(M/aM)}$ if and only if
${\displaystyle(0:_{M}a)=H^{0}_{\mathfrak{m}}(M)}$ and ${\displaystyle
a\,H^{0}_{\mathfrak{m}}(\mbox{\rm Ext}^{1}_{S}(M,S))=0}$.

Homological torsion is a combinatorial expression of the terms of homological
degrees. It behaves well under hyperplane sections [31, Lemma 2.8] and plays a
very useful role in the study of Hilbert coefficients.

######  Definition 4.12.

[31, Definition 1.2] Let $(R,\mathfrak{m})$ be a Gorenstein local ring of
dimension $d>0$ with infinite residue field $k$. Let $E=E(k)$ be the injective
envelope of $k$. Let $I$ be an $\mathfrak{m}$-primary ideal of $R$ and $M$ a
finitely generated $R$-module of dimension $s\geq 2$. The $i$-th homological
torsion of $M$ with respect to $I$ is

$T_{I}^{i}(M)=\sum_{j=1}^{s-i}{{s-i-1}\choose{j-1}}\mbox{\rm
hdeg}_{I}\left(\mbox{\rm Hom}_{R}(H^{j}_{\mathfrak{m}}(M),E)\right).$

In particular, we write $T_{I}(M)=T^{1}_{I}(M)$ and call it the homological
torsion. That is,

$T_{I}(M)=\sum_{j=1}^{s-1}{{s-2}\choose{j-1}}\mbox{\rm
hdeg}_{I}\left(\mbox{\rm Hom}_{R}(H^{j}_{\mathfrak{m}}(M),E)\right).$

######  Proposition 4.13.

[63, Proposition 6.1] Let $(R,\mathfrak{m})$ be a Gorenstein local ring. Let
$M$ be a generalized Cohen-Macaulay $R$-module of dimension $s\geq 2$. Let
${\displaystyle Q=(x_{1},\ldots,x_{s})}$ be a parameter ideal for $M$. Then

${\lambda}\left(H^{0}_{\mathfrak{m}}(M/(x_{1},\ldots,x_{s-1})M)\right)\leq
T_{I}(M)$

for any $\mathfrak{m}$-primary ideal $I$. Moreover, if $M$ is Buchsbaum, then
the equality holds.

###### Proof.

The following is a modification of the proof given in [63] with added details.
We prove by induction on $s$. Suppose $s=2$. We may assume that $x=x_{1}$ is
$M$-regular. From the exact sequence ${\displaystyle 0\rightarrow
M\stackrel{{\scriptstyle x}}{{\longrightarrow}}M\rightarrow M/xM\rightarrow
0}$, we obtain

$H^{0}_{\mathfrak{m}}(M)=0\longrightarrow
H^{0}_{\mathfrak{m}}(M/xM)\longrightarrow
H^{1}_{\mathfrak{m}}(M)\longrightarrow\cdots.$

Thus,
${\displaystyle{\lambda}\left(H^{0}_{\mathfrak{m}}(M/xM)\right)\leq{\lambda}\left(H^{1}_{\mathfrak{m}}(M)\right)=T_{I}(M)}$.

Suppose $s\geq 3$ and the assertion is true for $s-1$. Let ${\displaystyle
N=M/x_{1}M}$, ${\displaystyle{\mathbf{x}}=x_{1},\ldots,x_{s-1}}$, and
${\displaystyle{\mathbf{x}}^{\prime}=x_{2},\ldots,x_{s-1}}$. Then by induction
hypothesis, we obtain

${\lambda}\left(H^{0}_{\mathfrak{m}}(M/{\mathbf{x}}M)\right)={\lambda}\left(H^{0}_{\mathfrak{m}}(N/{\mathbf{x}}^{\prime}N)\right)\leq
T_{I}(N).$

It is enough to show that ${\displaystyle T_{I}(N)\leq T_{I}(M)}$. From the
exact sequence ${\displaystyle 0\rightarrow M\stackrel{{\scriptstyle
x_{1}}}{{\longrightarrow}}M\rightarrow N\rightarrow 0}$, we obtain the long
exact sequence

$\cdots\longrightarrow H^{j}_{\mathfrak{m}}(M)\longrightarrow
H^{j}_{\mathfrak{m}}(M)\longrightarrow H^{j}_{\mathfrak{m}}(N)\longrightarrow
H^{j+1}_{\mathfrak{m}}(M)\longrightarrow\cdots.$

Therefore, for each $j=1,\ldots,s-2$, we have

${\lambda}\left(H^{j}_{\mathfrak{m}}(N)\right)\leq{\lambda}\left(H^{j}_{\mathfrak{m}}(M)\right)+{\lambda}\left(H^{j+1}_{\mathfrak{m}}(M)\right).$

Now we compute ${\displaystyle T_{I}(N)}$.

$\begin{array}[]{rcl}{\displaystyle
T_{I}(N)}&=&{\displaystyle\sum_{j=1}^{s-2}{{s-3}\choose{j-1}}{\lambda}\left(H^{j}_{\mathfrak{m}}(N)\right)\vspace{0.1
in}}\\\
&\leq&{\displaystyle\sum_{j=1}^{s-2}{{s-3}\choose{j-1}}{\lambda}\left(H^{j}_{\mathfrak{m}}(M)\right)+\sum_{j=1}^{s-2}{{s-3}\choose{j-1}}{\lambda}\left(H^{j+1}_{\mathfrak{m}}(M)\right)\vspace{0.1
in}}\\\
&=&{\displaystyle{\lambda}\left(H^{1}_{\mathfrak{m}}(M)\right)+\sum_{j=2}^{s-2}\left({{s-3}\choose{j-1}}+{{s-3}\choose{j-2}}\right){\lambda}\left(H^{j}_{\mathfrak{m}}(M)\right)+{\lambda}\left(H^{s-1}_{\mathfrak{m}}(M)\right)\vspace{0.1
in}}\\\ &=&{\displaystyle T_{I}(M).}\end{array}$

∎

### Euler Characteristic

Let $Q=(x_{1},\ldots,x_{s})$ be a parameter ideal for a module $M$. The $i$th
Kozul homology module generated by the system of parameters
$x_{1},\ldots,x_{s}$ with coefficients in $M$ is denoted by ${\displaystyle
H_{i}(Q,M)}$. The (first) Euler characteristic of $M$ relative to $Q$ is
defined as

$\chi_{1}(Q,M)=\sum_{i\geq 1}(-1)^{i-1}{\lambda}\left(H_{i}(Q,M)\right).$

Then by a classical result of Serre [4, 52], we have

$\chi_{1}(Q,M)={\lambda}(M/QM)-\mathrm{e}_{0}(Q,M).$

The Euler characteristic can be also bounded by the homological degree.

######  Theorem 4.14.

[22, Theorem 7.2] Let $(R,\mathfrak{m})$ be a Noetherian complete local ring
with infinite residue field and let $M$ be a finitely generated $R$-module
with ${\displaystyle\dim(M)=\dim(R)=d\geq 1}$. Then for every parameter ideal
$Q$ for $R$, we have

$\chi_{1}(Q,M)\leq\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M).$

Goto and Ozeki proved when the equality holds true. Recall that we denote the
Hilbert polynomial of $M$ with respect to $Q$ by ${\displaystyle\mbox{\rm
P}_{Q,M}(X)}$.

######  Theorem 4.15.

[31, Theorem 3.3] Let $(R,\mathfrak{m})$ be a Noetherian complete local ring
with infinite residue field and let $M$ be a finitely generated $R$-module
with ${\displaystyle\dim(M)=\dim(R)=d\geq 1}$. Let $Q$ be a parameter ideal
for $R$. Then

$\chi_{1}(Q,M)=\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M)$

if and only if

$(-1)^{i}\mathrm{e}_{i}(Q,M)=\left\\{\begin{array}[]{ll}{\displaystyle
T^{\,i}_{Q}(M)}&\mbox{if}\;\;1\leq i\leq d-1,\vspace{0.1 in}\\\
{\displaystyle{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)}&\mbox{if}\;\;i=d\end{array}\right.$

for all $1\leq i\leq d$, and
${\displaystyle{\lambda}\left(M/Q^{n+1}M\right)=\mbox{\rm P}_{Q,M}(n)}$ for
all $n\geq 0$.

######  Example 4.16.

[31, Example 3.8] Let $\ell\geq 2$ and $m\geq 1$ be integers. Let

$S=k[\\![X_{i},Y_{i},Z_{j}\mid 1\leq i\leq\ell,\;1\leq j\leq
m]\\!]\quad\mbox{and}\quad
R=S/(X_{1},\ldots,X_{\ell})\cap(Y_{1},\ldots,X_{\ell}),$

where $S$ is the formal power series ring over an infinite field $k$. Let
$x_{i},y_{i}$, and $z_{j}$ denote the images of $X_{i},Y_{i}$, and $Z_{j}$ in
$R$ respectively. Consider the following parameter ideal of $R$:

$Q=(x_{i}-y_{i}\mid 1\leq i\leq\ell)+(z_{j}\mid 1\leq j\leq m).$

Then we have

$\chi_{1}(Q,R)=\ell-1,\quad\mathrm{e}_{0}(Q,R)=2,\quad\mbox{and}\quad\mbox{\rm
hdeg}_{Q}(R)=2+{{\ell+m-1}\choose{m+1}}.$

In particular, if $\ell=2$, then ${\displaystyle\chi_{1}(Q,R)=\mbox{\rm
hdeg}_{Q}(R)-\mathrm{e}_{0}(Q,R)}$.

### Chern number

The homological torsion can be used as a bound for the first Hilbert
coefficient $\mathrm{e}_{1}(Q,M)$ of a module $M$ relative to an ideal
generated by a system of parameters $Q$ for $M$. This coefficient
$\mathrm{e}_{1}(Q,M)$ is often called the Chern number, and it has been
extensively studied by Vasconcelos and the authors [63, 26, 20, 21, 23, 22].

######  Theorem 4.17.

[22, Theorem 6.5] Let $(R,\mathfrak{m})$ be a Noetherian complete local ring
with infinite residue field and let $M$ be a finitely generated $R$-module
with ${\displaystyle\dim(R)=\dim(M)\geq 2}$. Let $Q$ be a parameter ideal for
$R$. Then ${\displaystyle-\mathrm{e}_{1}(Q,M)\leq T_{Q}(M)}$.

As a direct consequence of Theorem 4.15, we obtain that if
${\displaystyle\chi_{1}(Q,M)=\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M)}$,
then ${\displaystyle-\mathrm{e}_{1}(Q,M)=T_{Q}(M)}$. The converse is true if
$M$ is unmixed. We note that [31, Example 4.5] shows that the unmixedness is
necessary.

######  Theorem 4.18.

[31, Theorem 4.2] Suppose that ${\displaystyle\dim(R)=\dim(M)\geq 2}$ and that
$M$ is unmixed. Let $Q$ be a parameter ideal for $R$. Then

$-\mathrm{e}_{1}(Q,M)=T_{Q}(M)\;\;\mbox{if and only
if}\;\;\chi_{1}(Q,M)=\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M).$

Recall that if $M$ is Cohen-Macaulay, then $\mbox{\rm
hdeg}_{I}(M)=\mathrm{e}_{0}(I,M)$. Thus, sometimes the integer
${\displaystyle\mbox{\rm hdeg}_{I}(M)-\mathrm{e}_{0}(I,M)}$ is called a Cohen-
Macaulay deficiency. Vasconcelos conjectured that the negativity of
$\mathrm{e}_{1}(Q,R)$, where $Q$ is a parameter ideal of $R$, is an expression
of the lack of Cohen-Macaulayness of $R$ [63, Conjecture 3.1]. Ghezzi, Goto,
Hong, Ozeki, Phuong and Vasconcelos settled this conjecture affirmatively for
rings in [20] and for modules in [22]. Thus, $\mathrm{e}_{1}(Q,M)$ can be
considered also as a Cohen-Macaulay deficiency. The following proposition
shows the relation between these two Cohen-Macaulay deficiencies.

######  Proposition 4.19.

[63, Corollary 7.7] Let $(R,\mathfrak{m})$ be a Noetherian complete local ring
with infinite residue field and let $M$ be a finitely generated $R$-module
with ${\displaystyle\dim(M)=\dim(R)=d\geq 1}$. Let $Q=(a_{1},\ldots,a_{d})$ be
a parameter ideal for $R$ such that the sequence ${\displaystyle
a_{1},\ldots,a_{d}}$ is superficial relative to $\mbox{\rm hdeg}_{Q}(M)$. Then

$-\mathrm{e}_{1}(Q,M)\leq\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M).$

###### Proof.

We include a modification of the proof of [30, Lemma 2.4]. We prove by
induction on $d$. If $d=1$, let $Q=(a)$. Let $t$ be a sufficiently large
integer such that

$H^{0}_{\mathfrak{m}}(M)=(0:_{M}Q^{t}),\quad\mbox{and}\quad{\lambda}(M/Q^{t}M)=\mathrm{e}_{0}(Q,M)t-\mathrm{e}_{1}(Q,M).$

Then we obtain the following:

$\begin{array}[]{rcl}{\displaystyle-\mathrm{e}_{1}(Q,M)}&=&{\displaystyle{\lambda}(M/Q^{t}M)-\mathrm{e}_{0}(Q,M)t\vspace{0.1
in}}\\\
&=&{\displaystyle{\lambda}(M/Q^{t}M)-\mathrm{e}_{0}(Q^{t},M)\vspace{0.1
in}}\\\
&=&{\displaystyle{\lambda}(M/Q^{t}M)-\left({\lambda}(M/Q^{t}M)-\chi_{1}(Q^{t},M)\right)\vspace{0.1
in}}\\\ &=&{\displaystyle{\lambda}(H_{1}(Q^{t},M))\vspace{0.1 in}}\\\
&=&{\displaystyle{\lambda}((0:_{M}Q^{t}))\vspace{0.1 in}}\\\
&=&{\displaystyle{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right).}\end{array}$

Since $\dim(M)=1$, we have ${\displaystyle\mbox{\rm
hdeg}_{Q}(M)=\mathrm{e}_{0}(Q,M)+{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)}$.
Thus, the assertion follows.

Suppose $d\geq 2$. Let ${\mathbf{a}}=a_{1},\ldots,a_{d-1}$,
$S=R/({\mathbf{a}})$, $q=QS$, and $M^{\prime}=M/({\mathbf{a}})M$. Then as
proved in the $d=1$ case, we have
${\displaystyle-\mathrm{e}_{1}(q,M^{\prime})={\lambda}\left(H^{0}_{\mathfrak{m}}(M^{\prime})\right)}$.
Therefore,

$-\mathrm{e}_{1}(Q,M)\leq-\mathrm{e}_{1}(q,M^{\prime})={\lambda}\left(H^{0}_{\mathfrak{m}}(M^{\prime})\right)\leq\mbox{\rm
hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M),$

where the last inequality follows from Corollary 4.10. ∎

By Theorem 4.14 and Proposition 4.19, we obtain

$\max\\{\chi_{1}(Q,M),\;-\mathrm{e}_{1}(Q,M)\\}\leq\mbox{\rm
hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M).$

In [27], Goto, Hong, and Vasconcelos established the relationships among
$\mathrm{e}_{1}(Q)$ and various first Euler characteristics when $Q$ is a
parameter ideal of $R$. We state one of the main results of [27], which is
related to the inequality stated above.

######  Theorem 4.20.

[27, Theorem 4.2] Let $(R,\mathfrak{m})$ be a Noetherian local ring of
dimension $d\geq 2$ with ${\displaystyle\mbox{\rm depth}(R)\geq d-1}$ and
infinite residue field. Let $Q=({\mathbf{x}})$ be a parameter ideal of $R$.
Then ${\displaystyle\chi_{1}(Q,M)=-\mathrm{e}_{1}(Q,M)}$ if and only if
${\displaystyle{\mathbf{x}}}$ is a $d$-sequence.

The sectional genus ${\displaystyle g_{s}(I,M)}$ of $M$ with respect to an
$\mathfrak{m}$-primary ideal $I$ is defined as

$g_{s}(I,M)={\lambda}(M/IM)-\mathrm{e}_{0}(I,M)+\mathrm{e}_{1}(I,M).$

Suppose that ${\displaystyle\dim(R)=\dim(M)}$ and $Q$ is a parameter ideal for
$R$. Then due to Serre, we have
${\displaystyle\chi_{1}(Q,M)={\lambda}(M/QM)-\mathrm{e}_{0}(Q,M)}$. Thus, we
can rewrite

$g_{s}(Q,M)=\chi_{1}(Q,M)+\mathrm{e}_{1}(Q,M).$

As consequences of Theorems 4.14 and 4.17, both ${\displaystyle g_{s}(Q,M)}$
and ${\displaystyle\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M)-\,T_{Q}(M)}$ are
less than or equal to ${\displaystyle\mbox{\rm
hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M)+\mathrm{e}_{1}(Q,M)}$. Goto and Ozeki showed
a direct relation between ${\displaystyle g_{s}(Q,M)}$ and
${\displaystyle\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M)-\,T_{Q}(M)}$.

######  Theorem 4.21.

[32, Proposition 3.3 and Theorem 3.4] Suppose
${\displaystyle\dim(R)=\dim(M)\geq 2}$ and let $Q$ be a parameter ideal for
$R$. Then

$g_{s}(Q,M)\leq\mbox{\rm hdeg}_{Q}(M)-\mathrm{e}_{0}(Q,M)-T_{Q}(M).$

The equality holds true if and only if

$(-1)^{i}\mathrm{e}_{i}(Q,M)=\left\\{\begin{array}[]{ll}{\displaystyle
T^{i}_{Q}(M)}&\mbox{if}\;\;2\leq i\leq d-1,\vspace{0.1 in}\\\
{\displaystyle{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)}&\mbox{if}\;\;i=d,\end{array}\right.$

for all $2\leq i\leq d$ and

${\lambda}\left(M/QM\right)=\sum_{i=0}^{d}(-1)^{i}\mathrm{e}_{i}(Q,M).$

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an
$\mathfrak{m}$-primary ideal, and $Q$ a minimal reduction of $I$. Rossi proved
that, if ${\displaystyle\dim(R)\leq 2}$, then ${\displaystyle g_{s}(I,R)\geq
r_{Q}(I)-1}$, where $r_{Q}(I)$ is the reduction number of $I$ with respect to
$Q$ [49, Corollary 1.5]. Thus, the homological degree can be used as an upper
bound for a reduction number. Ghezzi, Goto, Hong, and Vasconcelos were able to
extend Rossi’s result to $2$-dimensional Buchsbaum rings [24, Theorem 4.3]. We
conclude this section with several open questions raised by Vasconcelos.

######  Question 4.22.

Let $R$ be a Cohen-Macaulay local ring which is a homomorphic image of a
Gorenstein local ring. Let $\mathcal{M}(R)$ be the category of all finitely
generated $R$-modules. Consider the following set of rational numbers:

$\left\\{\frac{\mbox{\rm hdeg}(M)-\mbox{\rm hdeg}(M/hM)}{\mbox{\rm
deg}(M)}\;\;\vline\;\;M\in\mathcal{M}(R),\;\mbox{and $h$ is a generic
hyperplane section}\right\\}.$

Is this set finite or bounded? Can it be expressed as an invariant of $R$?

######  Question 4.23.

Let $(R,\mathfrak{m})$ be a homomorphic image of a Gorenstein local ring and
let $Q=({\mathbf{x}})$ be a parameter ideal for $R$. If ${\mathbf{x}}$ is a
$d$-sequence, what is an estimation for $\mbox{\rm hdeg}_{Q}(R)$?

######  Question 4.24.

Let ${\displaystyle\varphi:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})}$ be a
flat homomorphisms of local rings which are homomorphic images of a Gorenstein
local rings. Can we relate ${\displaystyle\mbox{\rm hdeg}(R),\mbox{\rm
hdeg}(S)}$, and ${\displaystyle\mbox{\rm hdeg}(S/\mathfrak{m}S)}$?

##  5\. Cohomological Degree

Cohomological degrees were introduced by Vasconcelos and generalize the
homological degree.

######  Definition 5.1.

([59, Definition 3.1], [15]) Let $R$ be either a Noetherian local ring or a
standard graded algebra with the (irrelevant) maximal ideal $\mathfrak{m}$.
Let $\mathcal{M}(R)$ be the category of finitely generated $R$-modules. The
cohomological degree (or extended degree) on $\mathcal{M}(R)$ is a function
${\displaystyle\mbox{\rm Deg}:\mathcal{M}(R)\rightarrow\mathbb{R}}$ satisfying
the following three conditions:

  1. (i)

${\displaystyle\mbox{\rm Deg}(M)=\mbox{\rm
Deg}\left(M/H^{0}_{\mathfrak{m}}(M)\right)+{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)}$.

  2. (ii)

Suppose that $R$ has positive depth and that $h\in R$ is a regular, generic
hyperplane section on $M$. Then ${\displaystyle\mbox{\rm
Deg}(M/hM)\leq\mbox{\rm Deg}(M)}$.

  3. (iii)

If $M$ is a Cohen-Macaulay module, then ${\displaystyle\mbox{\rm
Deg}(M)=\mbox{\rm deg}(M)}$.

If $(R,\mathfrak{m})$ is a homomorphic image of a Gorenstein local ring and
$I$ an $\mathfrak{m}$-primary ideal, then the homological degree $\mbox{\rm
hdeg}_{I}$ with respect to $I$ is a cohomological degree of $R$ (Proposition
4.7, Corollary 4.10).

######  Remark 5.2.

There is just one cohomological degree in dimension at most 1.

  1. (1)

If ${\displaystyle\dim(M)=0}$, then ${\displaystyle\mbox{\rm
Deg}(M)={\lambda}(M)}$.

  2. (2)

If ${\displaystyle\dim(M)=1}$, then ${\displaystyle\mbox{\rm Deg}(M)=\mbox{\rm
deg}(M)+{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)=\mbox{\rm
adeg}(M)=\mbox{\rm hdeg}(M)}$.

An important property of the cohomological degree is that it provides an
estimate for the Castelnuovo-Mumford regularity $\mbox{\rm reg}(A)$ of a
standard graded algebra $A$.

######  Theorem 5.3.

[15, Theorem 2.4] Let $A$ be a standard graded algebra over an infinite field
$k$. Then $\mbox{\rm reg}(A)<\mbox{\rm Deg}(A)$.

Another important property of the cohomological degree is that it bounds the
minimal number of generators of a module.

######  Proposition 5.4.

[15, Proposition 2.1] Let $M$ be a finitely generated $R$-module.

  1. (1)

${\displaystyle\mbox{\rm Deg}(M)\geq\mbox{\rm deg}(M)}$ and the equality holds
if and only if $M$ is Cohen-Macaulay.

  2. (2)

${\displaystyle\nu(M)\leq\mbox{\rm Deg}(M)}$.

  3. (3)

Let $L$ be a submodule of $M$. If ${\displaystyle{\lambda}(L)<\infty}$, then
${\displaystyle\mbox{\rm Deg}(M)=\mbox{\rm Deg}(M/L)+{\lambda}(L)}$.

######  Proposition 5.5.

[59, Proposition 3.9] Let $(R,\mathfrak{m})$ be a Noetherian local ring and
$M$ a finitely generated $R$-module. Then

$\mbox{\rm deg}(M)\leq\mbox{\rm adeg}(M)\leq\mbox{\rm Deg}(M).$

###### Proof.

We collect the associated primes of $M$ by their dimensions:
${\displaystyle\dim(M)=s_{1}>s_{2}>\ldots>s_{n}}$. Then

$\mbox{\rm adeg}(M)=a_{s_{1}}(M)+a_{s_{2}}(M)+\cdots+a_{s_{n}}(M),$

where $a_{s_{i}}(M)$ is the contribution of all primes in $\mbox{\rm Ass}(M)$
of dimension $s_{i}$. Also, ${\displaystyle a_{s_{1}}(M)=\mbox{\rm deg}(M)}$,
which proves the first inequality. Since both adeg and Deg are additive on the
exact sequence ${\displaystyle 0\rightarrow H^{0}_{\mathfrak{m}}(M)\rightarrow
M\rightarrow M/H^{0}_{\mathfrak{m}}(M)\rightarrow 0}$, we may assume that
$H^{0}_{\mathfrak{m}}(M)=0$. Let $h$ be a regular hyperplane section on $M$.
Then by Theorem 2.3, the induction hypothesis on $\dim(M)$, and Definition
5.1, we obtain

$\mbox{\rm adeg}(M)\leq\mbox{\rm adeg}(M/hM)\leq\mbox{\rm
Deg}(M/hM)\leq\mbox{\rm Deg}(M).\qed$

Cohomological degrees can be used to estimate the minimal number of generators
of an ideal in an arbitrary local ring. The following theorem generalizes a
bound that holds when $R$ is Cohen-Macaulay [15, Theorem 4.1], and it improves
previously known bounds (see [15, Section 3]).

######  Theorem 5.6.

[15, Theorem 4.6] Let $(R,\mathfrak{m})$ be a local ring of dimension $d\geq
1$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal.
Suppose that ${\displaystyle\mathfrak{m}^{s}\subseteq I}$. Then, for any
cohomological degree Deg on $\mathcal{M}(R)$, we have

$\nu(I)\leq\mbox{\rm Deg}(R)\,{{s+d-2}\choose{d-1}}+{{s+d-2}\choose{d-2}}.$

Gunston introduced the following cohomological degree in his Ph.D. thesis.

######  Definition 5.7.

[35, Definition 3.1.1] Let $\mathcal{D}(R)$ denote the set of all
cohomological degree functions on $\mathcal{M}(R)$. For a finitely generated
$R$-module, we define

$\mbox{\rm bdeg}(M)=\min\\{\mbox{\rm Deg}(M)\mid\mbox{\rm
Deg}\in\mathcal{D}(R)\\}.$

The inductive approach for bdeg works.

######  Theorem 5.8.

[35, Theorem 3.1.2] Let $M$ be a finitely generated $R$-module of positive
depth. Then for a generic hyperplane $h$, we have

$\mbox{\rm bdeg}(M)=\mbox{\rm bdeg}(M/hM).$

######  Corollary 5.9.

Let $M$ be a finitely generated $R$-module of dimension $s$. There exist a
generic superficial sequence ${\displaystyle x_{1},\ldots,x_{s}}$ for $M$ such
that

$\begin{array}[]{rcl}{\displaystyle\mbox{\rm
bdeg}(M)}&\leq&{\displaystyle{\lambda}\left(H^{0}_{\mathfrak{m}}(M)\right)+\sum_{i=1}^{s}{\lambda}\left(H^{0}_{\mathfrak{m}}(M/(x_{1},\ldots,x_{i})M)\right)\vspace{0.1
in}}\\\ &\leq&{\displaystyle(s+1)\mbox{\rm hdeg}(M)-s\cdot\mbox{\rm
deg}(M)}\end{array}$

###### Proof.

The first inequality is a consequence of Theorem 5.8 and the second follows
from Corollary 4.10. ∎

One of the difficulties of computing cohomological degrees lies on their
behavior on short exact sequences. It turns out bdeg has more amenable
properties.

######  Proposition 5.10.

([35, Proposition 3.2.2], [13, Propositions 3.2 and 3.3])

Let ${\displaystyle 0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0}$ be
an exact sequence of finitely generated $R$-modules.

  1. (1)

If ${\displaystyle{\lambda}(N)<\infty}$, then ${\displaystyle\mbox{\rm
bdeg}(M)\leq\mbox{\rm bdeg}(L)+{\lambda}(N)}$.

  2. (2)

${\displaystyle\mbox{\rm bdeg}(M)\leq\mbox{\rm bdeg}(L)+\mbox{\rm bdeg}(N)}$.

  3. (3)

${\displaystyle\mbox{\rm bdeg}(L)\leq\mbox{\rm
bdeg}(M)+\left(\dim(L)-1\right)\mbox{\rm bdeg}(N)}$.

We denote ${\displaystyle M^{*}=\mbox{\rm Hom}_{R}(M,R)}$ for an $R$-module
$M$.

######  Theorem 5.11.

[13, Theorems 5.2 and 5.3] Let $(R,\mathfrak{m})$ be a Gorenstein local ring
of dimension $d$ and $M$ a finitely generated $R$-module. Then

$\nu_{R}(M^{*})\leq\mbox{\rm bdeg}_{R}(M^{*})\leq\mbox{\rm
hdeg}(M)\left(\mbox{\rm deg}(R)+\frac{d(d-1)}{2}\right).$

##  6\. The canonical degree

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring that has a canonical
ideal $\mathcal{C}$. The canonical degree was introduced in [25] with the
overall goal to refine our understanding of $R$ and to measure the deviation
from $R$ being Gorenstein. The canonical degree requires knowledge of the
multiplicity of $\mathfrak{m}$-primary ideals.

We begin with the definition of canonical degree in $1$-dimensional Cohen-
Macaulay local rings, which is our main setting. We denote length by
$\lambda$, and the multiplicity associated with the $\mathfrak{m}$-adic
filtration by $\mbox{\rm deg}(\mbox{\rule{8.53581pt}{0.56905pt}$\;\\!$})$.

######  Definition 6.1.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring with a canonical ideal
$\mathcal{C}$. The canonical degree of $R$ is the integer $\mbox{\rm
cdeg}(R)=\mathrm{e}_{0}(\mathcal{C})-{\lambda}(R/\mathcal{C})$. In particular,
if $(a)$ is a minimal reduction of $\mathcal{C}$, then $\mbox{\rm
cdeg}(R)={\lambda}(\mathcal{C}/(a))\geq 0$.

We show in [25, Proposition 2.1] that $\mbox{\rm cdeg}(R)$ is well defined;
that is, the integer $\mathrm{e}_{0}(\mathcal{C})-{\lambda}(R/\mathcal{C})$ is
independent of the canonical ideal $\mathcal{C}$.

######  Remark 6.2.

In $1$-dimensional Cohen-Macaulay local rings the vanishing of $\mbox{\rm
cdeg}(R)$ is equivalent to $R$ being Gorenstein, since we have $\mbox{\rm
cdeg}(R)=0$ if and only if $\mathcal{C}$ is a principal ideal.

We define $\mbox{\rm cdeg}(R)$ in full generality as follows.

######  Definition 6.3.

Let $R$ be a Cohen-Macaulay local ring of dimension $d\geq 1$ that has a
canonical ideal $\mathcal{C}$. The canonical degree of $R$ is the integer

$\mbox{\rm cdeg}(R)=\sum_{\tiny\mbox{\rm ht}({\mathfrak{p}})=1}\mbox{\rm
cdeg}(R_{{\mathfrak{p}}})\mbox{\rm deg}(R/{\mathfrak{p}})=\sum_{\tiny\mbox{\rm
ht}({\mathfrak{p}})=1}[\mathrm{e}_{0}(\mathcal{C}_{{\mathfrak{p}}})-{\lambda}((R/\mathcal{C})_{{\mathfrak{p}}})]\mbox{\rm
deg}(R/{\mathfrak{p}}).$

We show in [25, Theorem 2.2] that $\mbox{\rm cdeg}(R)$ is a well-defined
finite sum independent of the canonical ideal $\mathcal{C}$. The definition
yields a nice formula in terms of multiplicity when $\mathcal{C}$ is
equimultiple, and several results in dimension one extend to higher dimension
when $\mathcal{C}$ is equimultiple.

######  Remark 6.4.

If the canonical ideal $\mathcal{C}$ is equimultiple with a minimal reduction
$(a)$, then by the associativity formula [11, Corollary 4.7.8] we have

$\mbox{\rm cdeg}(R)=\mbox{\rm
deg}(\mathcal{C}/(a))=\mathrm{e}_{0}(\mathfrak{m},\mathcal{C}/(a)).$

In general, the vanishing of $\mbox{\rm cdeg}(R)$ characterizes Gorensteinness
in codimension one.

######  Remark 6.5.

$\mbox{\rm cdeg}(R)\geq 0$ and vanishes if and only if $R$ is Gorenstein in
codimension $1$.

######  Remark 6.6.

If $R$ is Gorenstein, it is clear that $\mbox{\rm cdeg}(R)=0$. However, the
converse does not hold in general. For example, if $R$ is a normal domain we
have that $\mbox{\rm cdeg}(R)=0$, since $R_{{\mathfrak{p}}}$ is Gorenstein for
every prime ${\mathfrak{p}}$ of $R$ with $\mbox{\rm ht}({\mathfrak{p}})\leq
1$. Therefore if $R$ is a non-Gorenstein normal domain then $\mbox{\rm
cdeg}(R)=0$.

Generalizing the one-dimensional case, we observe that when $\mathcal{C}$ is
equimultiple the vanishing of $\mbox{\rm cdeg}(R)$ is equivalent to $R$ being
Gorenstein. We denote the type of $R$ by $r(R)$.

######  Corollary 6.7.

[25, Corollary 2.5] Suppose that the canonical ideal of $R$ is equimultiple.
Then we have the following.

  1. (1)

$\mbox{\rm cdeg}(R)\geq r(R)-1$.

  2. (2)

$\mbox{\rm cdeg}(R)=0$ if and only if $R$ is Gorenstein.

###### Proof.

Let $(a)$ be a minimal reduction of the canonical ideal $\mathcal{C}$. Then

$\mbox{\rm
cdeg}(R)=\mathrm{e}_{0}(\mathfrak{m},\mathcal{C}/(a))\geq\nu(\mathcal{C}/(a))=r(R)-1.$

If $\mbox{\rm cdeg}(R)=0$ then $r(R)=1$, which proves that $R$ is Gorenstein.
∎

In light of the above corollary, we are interested in examining extremal
values of the canonical degree. We see that the minimal value for the
canonical degree is assumed on a new class of Cohen-Macaulay rings, called
almost Gorenstein rings, introduced in [5]. A rich theory has been developed
in dimension one by Goto, Matsuoka and Phuong [29], and in higher dimension by
Goto, Takahashi and Taniguchi [33]. We recall the definition.

######  Definition 6.8.

[33, Definition 3.3] A Cohen-Macaulay local ring $R$ with a canonical module
$\omega$ is said to be an almost Gorenstein local ring (AGL for short) if
there exists an exact sequence of $R$-modules ${\displaystyle 0\rightarrow
R\rightarrow\omega\rightarrow X\rightarrow 0}$ such that
$\nu(X)=\mathrm{e}_{0}(X)$, where $\nu(X)$ is the minimal number of generators
of $X$. In particular, if $R$ is Gorenstein, then $X=0$, so $R$ is almost
Gorenstein.

The following result characterizes AGL rings in terms of minimal canonical
degree. We include the proof for completeness.

######  Proposition 6.9.

[25, Proposition 3.3] Let $(R,\mathfrak{m})$ be a $1$-dimensional Cohen-
Macaulay local ring with a canonical ideal $\mathcal{C}$. Then $\mbox{\rm
cdeg}(R)=r(R)-1$ if and only if $R$ is an almost Gorenstein ring.

###### Proof.

We may assume that $R$ is not a Gorenstein ring. Suppose that $\mbox{\rm
cdeg}(R)=r(R)-1$. Let $(a)$ be a minimal reduction of $\mathcal{C}$. Consider
the exact sequence of $R$-modules

$0\to R\overset{\varphi}{\to}\mathcal{C}\to X\to
0,\;\;\mbox{where}\;\;\varphi(1)=a.$

Then $\nu(X)=r(R)-1=\mbox{\rm cdeg}(R)=\mathrm{e}_{0}(X)$. Thus, $R$ is an
almost Gorenstein ring.

For the converse, assume that $R$ is almost Gorenstein. There exists an exact
sequence of $R$-modules

$0\to R\overset{\varphi}{\to}\mathcal{C}\to X\to 0\;\;\mbox{such
that}\;\;\nu(X)=\mathrm{e}_{0}(X).$

Since $\dim(X)=0$ by [33, Lemma 3.1], we have
$\lambda(X)=\mathrm{e}_{0}(X)=\nu(X)=\lambda(X/\mathfrak{m}X)$, so that
$\mathfrak{m}X=(0)$. Let $a=\varphi(1)\in\mathcal{C}$ and set $Q=(a)$. Then
$\mathfrak{m}Q\subseteq\mathfrak{m}\mathcal{C}\subseteq Q$. Therefore, since
$R$ is not a discrete valuation ring and ${\lambda}(Q/\mathfrak{m}Q)=1$, we
get $\mathfrak{m}\mathcal{C}=\mathfrak{m}Q$, so that $Q$ is a minimal
reduction of $\mathcal{C}$. It follows that $\mbox{\rm
cdeg}(R)=\mathrm{e}_{0}(X)=\nu(X)=r(R)-1$. ∎

######  Remark 6.10.

Assume that $\mathcal{C}$ is equimultiple. The proof of Proposition 6.9 shows
that if $\mbox{\rm cdeg}(R)=r(R)-1$, then $R$ is an almost Gorenstein ring
[25, Proposition 3.3]. In particular, if $\mbox{\rm cdeg}(R)=1$, then by
Corollary 6.7, $R$ is not Gorenstein and $r(R)=2$. Therefore $R$ is almost
Gorenstein (not Gorenstein).

Under a more technical assumption we can generalize Proposition 6.9 to higher
dimension. We refer the reader to [25, Theorem 3.5].

### Numerical semigroup rings

Several interesting results and examples can be found in numerical semigroup
rings. We briefly recall the definitions and basic properties that we will use
throughout the manuscript. Our focus will be on 3-generated numerical
semigroup rings. For reference we use [66, Section 8.7].

Given positive integers $a_{1},\ldots,a_{n}$, let
$H=<a_{1},\ldots,a_{n}>=\\{c_{1}a_{1}+\cdots+c_{n}a_{n}\mid
c_{i}\in\mathbb{Z}_{\geq 0}\\}$ be the numerical semigroup generated by
$a_{1},\ldots,a_{n}$. We assume that $\gcd(a_{1},\ldots,a_{n})=1$. Let $k$ be
a field and let $k[\\![t]\\!]$ be the formal power series ring over $k$. We
denote the subring of $k[\\![t]\\!]$ generated by the monomials $t^{a_{i}}$ by
$k[\\![H]\\!]$ and we call it the semigroup ring of $H$ over $k$. The ring
$k[\\![H]\\!]$ is a one-dimensional Cohen-Macaulay local domain.

There are several integers playing roles in deriving properties of
$k[\\![H]\\!]$. There is an integer $s$ such that $t^{n}\in k[\\![H]\\!]$ for
all $n\geq s$. The smallest such $s$ is called the conductor of $H$ or of
$k[\\![H]\\!]$, which we denote by $c$, and $t^{c}k[\\![t]\\!]$ is the largest
ideal of $k[\\![t]\\!]$ contained in $k[\\![H]\\!]$. Then $c-1$ is called the
Frobenius number of $k[\\![H]\\!]$ and expresses its multiplicity,
$c-1=\mbox{\rm deg}(k[\\![H]\\!])$. For the computation of the canonical
module of $k[\\![H]\\!]$ see for instance [12, Section 6].

#### 3-generated numerical semigroup rings

Let $k$ be a field, let $a,b,c$ be positive integers with $\gcd(a,b,c)=1$,
$H=<a,b,c>$ and $R=k[\\![t^{a},t^{b},t^{c}]\\!]$. Assume $R$ is not Gorenstein
(i.e., $H$ is not symmetric). Then $r(R)=2$ (see [29, Section 4]). There
exists a surjective homomorphism $\Gamma:S=k[\\![X,Y,Z]\\!]\rightarrow R$
given by $\Gamma(X)=t^{a},\Gamma(Y)=t^{b},\Gamma(Z)=t^{c}$. Let $P=\mbox{\rm
ker}(\Gamma)$. Then there exists a matrix

$\left(\begin{matrix}X^{a_{1}}&Y^{b_{1}}&Z^{c_{1}}\\\
Y^{b_{2}}&Z^{c_{2}}&X^{a_{2}}\\\ \end{matrix}\right)$

such that $P=I_{2}(\varphi)$, the ideal generated by $2\times 2$ minors of
$\varphi$. That is,

$P=(f_{1},f_{2},f_{3})=(X^{a_{1}+a_{2}}-Y^{b_{2}}Z^{c_{1}},\;Y^{b_{1}+b_{2}}-X^{a_{1}}Z^{c_{2}},\;Z^{c_{1}+c_{2}}-X^{a_{2}}Y^{b_{1}}).$

We call this matrix the Herzog matrix of the semigroup $H$ [36]. The
computation of the Herzog matrix is also summarized in [38, Section 3].

We recall a way to compute the canonical ideal in this case. See also [62,
Theorem 6.25].

######  Remark 6.11.

Let $x,y,z$ be the images of $X,Y,Z$ in $R\simeq S/P$ respectively. Let
$L=(f_{1},f_{2})$. Since $f_{1},f_{2}$ form a regular sequence, the canonical
module of $R$ is

$\mathcal{C}=\mbox{\rm Ext}^{2}_{S}(R,S)\simeq\mbox{\rm
Ext}^{2}_{S}(S/P,S)\simeq\mbox{\rm
Hom}_{S/L}(S/P,S/L)\simeq(L:P)/L=(L:f_{3})/L\simeq(x^{a_{1}},y^{b_{2}}).$

Similarly,
$\mathcal{C}\simeq(y^{b_{1}},z^{c_{2}})\simeq(x^{a_{2}},z^{c_{1}})$.

Goto, Matsuoka and Phuong proved a nice formula for the canonical degree [29,
Theorem 4.1].

######  Theorem 6.12.

With notation as above we have

  1. (1)

$\mbox{\rm cdeg}(R)=a_{1}b_{1}c_{1}$, if $bb_{2}-aa_{1}>0$;

  2. (2)

$\mbox{\rm cdeg}(R)=a_{2}b_{2}c_{2}$, if $bb_{2}-aa_{1}<0$.

######  Corollary 6.13.

[29, Corollary 4.2] The ring $k[\\![t^{a},t^{b},t^{c}]\\!]$ is an almost
Gorenstein ring if and only if the Herzog matrix is either
${\displaystyle\left(\begin{matrix}X&Y&Z\\\ Y^{b_{2}}&Z^{c_{2}}&X^{a_{2}}\\\
\end{matrix}\right)}$ or
${\displaystyle\left(\begin{matrix}X^{a_{1}}&Y^{b_{1}}&Z^{c_{1}}\\\ Y&Z&X\
\end{matrix}\right)}$.

######  Example 6.14.

Let $R=k[\\![t^{3},t^{4},t^{5}]\\!]$. The Herzog matrix is
${\displaystyle\left(\begin{matrix}X&Y&Z\\\ Y&Z&X^{2}\ \end{matrix}\right)}$,
so $\mbox{\rm cdeg}(R)=1$. $R$ is almost Gorenstein.

######  Example 6.15.

Let $R=k[\\![t^{5},t^{7},t^{9}]\\!]$. The Herzog matrix is
$\left(\begin{matrix}X&Y&Z^{2}\\\ Y&Z&X^{4}\ \end{matrix}\right)$, so
$\mbox{\rm cdeg}(R)=2$. $R$ is not almost Gorenstein.

##  7\. Canonical index and Sally module

In view of Corollary 6.7 and Proposition 6.9, the next case to consider is
$\mbox{\rm cdeg}(R)=r(R)$. It is open in general. Some examples are provided
in [25, Example 4.12]. In the smallest case, $\mbox{\rm cdeg}(R)=r(R)=2$, we
obtained nice relationships among several invariants of $R$ in [25, Section
4]. Inspired by [25] and [65], Chau, Goto, Kumashiro and Matsuoka defined the
notion of 2-AGL rings as a generalization of AGL rings of dimension one [12].
The theory of these rings has then been developed by Goto, Isobe, and
Taniguchi [28]. Both [25] and [12] use the Sally module, introduced by
Vasconcelos in [57]. In order to discuss these results, we recall the
necessary background.

We showed in [25, Proposition 4.1] that the reduction number of the canonical
ideal of $R$ is an invariant of the ring. Therefore we have the following
definition.

######  Definition 7.1.

[25, Definition 4.2] Let $R$ be a Cohen-Macaulay local ring of dimension
$d\geq 1$ with a canonical ideal $\mathcal{C}$. The canonical index of $R$ is
the reduction number of the canonical ideal $\mathcal{C}$ of $R$ and is
denoted by $\rho(R)$.

######  Remark 7.2.

Assume $d=1$ and that $R$ is not Gorenstein. Let $(a)$ be a minimal reduction
of $\mathcal{C}$. Notice that we can not have $\rho(R)=1$, otherwise
$\mathcal{C}^{2}=a\mathcal{C}$. Then $\mathcal{C}a^{-1}\subset\mbox{\rm
Hom}(\mathcal{C},\mathcal{C})=R$ so that $\mathcal{C}=(a)$, a contradiction.
Furthermore, if $R$ is almost Gorenstein (i.e., $\mbox{\rm cdeg}(R)=r(R)-1$),
then $\rho(R)=2$, by [29, Theorem 3.16]. We note that the converse is not
true, as shown in the following example.

######  Example 7.3.

[25, Example 4.9] Let $S=k[\\![X,Y,Z]\\!]$, let
$I=(X^{4}-Y^{2}Z^{2},Y^{4}-X^{2}Z^{2},Z^{4}-X^{2}Y^{2})$ and $R=S/I$. Let
$x,y,z$ be the images of $X,Y,Z$ in $R$. Then $\mathcal{C}=(x^{2},z^{2})$ is a
canonical ideal of $R$ with a minimal reduction $(x^{2})$. We have that
$\rho(R)=2$, $\mathrm{e}_{1}(\mathcal{C})=16$ and $\mbox{\rm cdeg}(R)=8$.

We briefly recall the Sally module associated to the canonical ideal
$\mathcal{C}$ in rings of dimension $1$. Let $Q=(a)$ be a minimal reduction of
$\mathcal{C}$ and consider the exact sequence of finitely generated
$R[QT]$-modules

$0\rightarrow\mathcal{C}R[QT]\rightarrow\mathcal{C}R[\mathcal{C}T]\rightarrow
S_{Q}(\mathcal{C})\rightarrow 0.$

Then the Sally module $S=S_{Q}(\mathcal{C})=\bigoplus_{n\geq
1}\mathcal{C}^{n+1}/\mathcal{C}Q^{n}$ of $\mathcal{C}$ relative to $Q$ is
Cohen-Macaulay and, by [24, Theorem 2.1], we have

(1) $\mathrm{e}_{1}(\mathcal{C})=\mbox{\rm
cdeg}(R)+\sum_{j=1}^{\rho(R)-1}\lambda(\mathcal{C}^{j+1}/a\mathcal{C}^{j})=\sum_{j=0}^{\rho(R)-1}\lambda(\mathcal{C}^{j+1}/a\mathcal{C}^{j}).$

The following property of Cohen-Macaulay rings of type $2$ is a useful
calculation that we use to characterize rings with minimal canonical index
$2$.

######  Proposition 7.4.

[25, Proposition 4.5] Let $R$ be a $1$-dimensional Cohen-Macaulay local ring
with a canonical ideal $\mathcal{C}$. Let $(a)$ be a minimal reduction of
$\mathcal{C}$. If $\nu(\mathcal{C})=2$, then
${\lambda}(\mathcal{C}^{2}/a\mathcal{C})={\lambda}(\mathcal{C}/(a))$.

From Equation (1) and Proposition 7.4 we obtain the following (see also
Example 7.3).

######  Theorem 7.5.

[25, Theorem 4.6] Let $R$ be a $1$-dimensional Cohen-Macaulay local ring with
a canonical ideal $\mathcal{C}$. Suppose that the type of $R$ is $2$. Then
$\rho(R)=2$ if and only if
${\displaystyle\mathrm{e}_{1}(\mathcal{C})=2\,\mbox{\rm cdeg}(R)}$.

######  Remark 7.6.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring with a canonical ideal
$\mathcal{C}$. Then the multiplicity of the Sally module
$s_{0}(S)=\mathrm{e}_{1}(\mathcal{C})-\mathrm{e}_{0}(\mathcal{C})+\lambda(R/\mathcal{C})=\mathrm{e}_{1}(\mathcal{C})-\mbox{\rm
cdeg}(R)$ is an invariant of $R$, by [29, Corollary 2.8] and [25, Proposition
2.1].

######  Remark 7.7.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring with a canonical ideal
$\mathcal{C}$. Suppose that the type of $R$ is $2$. Then by Equation (1),
Proposition 7.4 and Remark 7.6 we have that $s_{0}(S)=\mbox{\rm cdeg}(R)$, if
$\rho(R)=2$, and $s_{0}(S)=\mbox{\rm
cdeg}(R)+\sum_{j=2}^{\rho(R)-1}\lambda(\mathcal{C}^{j+1}/a\mathcal{C}^{j})$ if
$\rho(R)\geq 3$.

It is shown in [29, Theorem 3.16] that $R$ is an almost Gorenstein (but not
Gorenstein) ring if and only if $s_{0}(S)=1$. Therefore it is natural to
generalize as follows.

######  Definition 7.8.

[12, Definition 1.3] Let $R$ be a $1$-dimensional Cohen-Macaulay local ring
with a canonical ideal $\mathcal{C}$. Then $R$ is a $2$-almost Gorenstein
local ring ($2$-AGL for short), if $s_{0}(S)=2$; that is,
$\mathrm{e}_{1}(\mathcal{C})=\mathrm{e}_{0}(\mathcal{C})-\lambda(R/\mathcal{C})+2=\mbox{\rm
cdeg}(R)+2$.

######  Remark 7.9.

We recall the notation of [12, Section 2], and we refer the reader there for
details. Let $R$ be a Cohen-Macaulay local ring of dimension one with
canonical ideal $\mathcal{C}$. Let $(a)$ be a minimal reduction of
$\mathcal{C}$. Set ${\displaystyle
K=\dfrac{\mathcal{C}}{a}=\left\\{\dfrac{x}{a}\;\vline\;x\in\mathcal{C}\right\\}}$
in the total ring of fractions of $R$. Then $K/R\simeq\mathcal{C}/(a)$.
Therefore ${\lambda}(K/R)$ that appears in [12] is $\mbox{\rm cdeg}(R)$.

Among many interesting results in [12], Proposition 3.8 characterizes 2-AGL
rings of type two. They are rings with minimal canonical degree and minimal
canonical index. We give here an alternative short proof using our set-up.

######  Proposition 7.10.

[12, Proposition 3.8] Let $R$ be a $1$-dimensional Cohen-Macaulay local ring
with a canonical ideal. Suppose that the type of $R$ is $2$. The following
conditions are equivalent :

  1. (i)

$R$ is a $2$-AGL ring ;

  2. (ii)

$\mbox{\rm cdeg}(R)=2$ and $\rho(R)=2$.

###### Proof.

We may assume that $\mbox{\rm cdeg}(R)\geq 2$, since $R$ is not almost
Gorenstein. The conclusion follows from Remark 7.7. ∎

In [12, Section 6] the authors studied 2-AGL numerical semigroup rings. In
particular, for 3 generated semigroup rings they obtained the following
characterization, in the spirit of Corollary 6.13.

######  Theorem 7.11.

[12, Theorem 6.4] The ring $k[\\![t^{a},t^{b},t^{c}]\\!]$ is $2$-AGL if and
only if after a suitable permutation of $a,b,c$ the Herzog matrix is
${\displaystyle\left(\begin{matrix}X^{2}&Y&Z\\\
Y^{b_{2}}&Z^{c_{2}}&X^{a_{2}}\\\ \end{matrix}\right)}$ with $a_{2}\geq 2$ and
$b_{2},c_{2}>0$.

######  Example 7.12.

We consider some examples of rings of type 2.

  1. (1)

Three generated semigroup rings with small multiplicity that are 2-AGL are
characterized in [12, Corollary 6.7]. One example is
$R=k[\\![t^{3},t^{7},t^{8}]\\!]$.

  2. (2)

The ring $R=k[\\![t^{5},t^{7},t^{9}]\\!]$ of Example 6.15 has $\mbox{\rm
cdeg}(R)=2$. After a permutation the Herzog matrix is
${\displaystyle\left(\begin{matrix}X^{2}&Y&Z\\\ Y^{4}&Z&X\\\
\end{matrix}\right)}$. The ring is not 2-AGL, according to Theorem 7.11 or
[12, Corollary 6.7]. We have that $\mathcal{C}=(x^{2},y^{4})$ is a canonical
ideal of $R$ with minimal reduction $(x^{2})$ and $\rho(R)=4$.

  3. (3)

The ring $R$ of Example 7.3 has $\rho(R)=2$, but it is not 2-AGL, since
$\mbox{\rm cdeg}(R)=8$.

To conclude this section, we would like to propose a possible further
generalization.

######  Definition 7.13.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring with a canonical ideal
$\mathcal{C}$. Then $R$ is a $n$-almost Gorenstein local ring ($n$-AGL for
short), if $s_{0}(S)=n$; that is,
$\mathrm{e}_{1}(\mathcal{C})=\mathrm{e}_{0}(\mathcal{C})-\lambda(R/\mathcal{C})+n=\mbox{\rm
cdeg}(R)+n$.

### 3-AGL rings

It would be interesting to develop a theory of $3$-AGL rings, in particular
for $3$-generated semigroup rings. We begin with the following
characterization of rings of type 2, in the spirit of Proposition 7.10. We
note that 3-AGL does not imply that $\mbox{\rm cdeg}(R)=3$.

######  Proposition 7.14.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring of type $2$ with a
canonical ideal $\mathcal{C}$. Let $(a)$ be a minimal reduction of
$\mathcal{C}$. Then $R$ is a $3$-AGL ring if and only if one of the following
holds :

  1. (1)

$\mbox{\rm cdeg}(R)=3$ and $\rho(R)=2$ ;

  2. (2)

$\mbox{\rm cdeg}(R)=2$, $\rho(R)=3$, and
${\lambda}(\mathcal{C}^{3}/a\mathcal{C}^{2})=1$.

###### Proof.

We may assume that $\mbox{\rm cdeg}(R)\geq 2$, since $R$ is not almost
Gorenstein. The conclusion follows from Remark 7.7. ∎

######  Example 7.15.

After a permutation $R=k[\\![t^{7},t^{12},t^{15}]\\!]$ has Herzog matrix
${\displaystyle\left(\begin{matrix}X^{3}&Y&Z\\\ Y^{2}&Z^{2}&X^{3}\\\
\end{matrix}\right)}$. Then $\mathcal{C}=(x^{3},y^{2})$ is a canonical ideal
of $R$ with minimal reduction $(x^{3})$ and $\rho(R)=2$. We have $\mbox{\rm
cdeg}(R)=3$ by Theorem 6.12. Therefore $R$ is $3$-AGL by Proposition 7.14.

######  Example 7.16.

After a permutation $R=k[\\![t^{5},t^{9},t^{11}]\\!]$ has Herzog matrix
${\displaystyle\left(\begin{matrix}X^{2}&Y&Z\\\ Y^{2}&Z^{3}&X\\\
\end{matrix}\right)}$. Then $\mathcal{C}=(z,x)$ is a canonical ideal of $R$
with minimal reduction $(z)$. We have $\rho(R)=3$,
${\lambda}(\mathcal{C}^{3}/z\mathcal{C}^{2})=1$. We also have $\mbox{\rm
cdeg}(R)=2$ by Theorem 6.12. Therefore $R$ is $3$-AGL by Proposition 7.14.

######  Example 7.17.

The ring $R=k[\\![t^{5},t^{7},t^{9}]\\!]$ of Example 7.12 has $\mbox{\rm
cdeg}(R)=2$, $\rho(R)=4$. Therefore $R$ is not $3$-AGL by Proposition 7.14.

In accordance with Theorem 7.11 and the above examples we ask the following.

######  Question 7.18.

Suppose that after a suitable permutation of $a,b,c$, the ring
$R=k[\\![t^{a},t^{b},t^{c}]\\!]$ has Herzog matrix
${\displaystyle\left(\begin{matrix}X^{3}&Y&Z\\\
Y^{b_{2}}&Z^{c_{2}}&X^{a_{2}}\\\ \end{matrix}\right)}$ with $a_{2}\geq 3$ and
$b_{2},c_{2}>0$. Is $R$ a $3$-AGL ring?

######  Question 7.19.

Suppose that after a suitable permutation of $a,b,c$, the ring
$R=k[\\![t^{a},t^{b},t^{c}]\\!]$ has Herzog matrix
${\displaystyle\left(\begin{matrix}X^{2}&Y&Z\\\ Y^{b_{2}}&Z^{c_{2}}&X\\\
\end{matrix}\right)}$ with $b_{2},c_{2}>0$. When is $R$ a $3$-AGL ring?

##  8\. bi-canonical degree

The canonical degree is dependent on finding minimal reductions, which in
general is a particularly hard task. We consider in [19] a degree that seems
more amenable to computation.

######  Definition 8.1.

Let $(R,\mathfrak{m})$ be a $1$-dimensional Cohen-Macaulay local ring with a
canonical ideal $\mathcal{C}$. The bi-canonical degree of $R$ is the integer
$\mbox{\rm bideg}(R)=\lambda(\mathcal{C}^{**}/\mathcal{C})$, where
${\mathcal{C}}^{**}=\mbox{\rm Hom}_{R}(\mbox{\rm Hom}_{R}(\mathcal{C},R),R)$
is the bidual of $\mathcal{C}$.

######  Remark 8.2.

We note that the bi-canonical degree is an invariant of the ring. We have
$\mbox{\rm bideg}(R)=0$ if and only if $\mathcal{C}$ is reflexive. Therefore,
by [40, Corollary 7.29] in $1$-dimensional Cohen-Macaulay local rings the
vanishing of $\mbox{\rm bideg}(R)$ is equivalent to $R$ being Gorenstein.

We define $\mbox{\rm bideg}(R)$ in full generality as follows. We recall that
$\mbox{\rm deg}(\mbox{\rule{8.53581pt}{0.56905pt}$\;\\!$})$ is the
multiplicity defined by the $\mathfrak{m}$-adic filtration.

######  Definition 8.3.

[19, Theorem 3.1] Let $R$ be a Cohen-Macaulay local ring of dimension $d\geq
1$ that has a canonical ideal $\mathcal{C}$. Then the bi-canonical degree of
$R$ is the integer

$\mbox{\rm bideg}(R)=\mbox{\rm
deg}(\mathcal{C}^{**}/\mathcal{C})=\sum_{\tiny\mbox{\rm
ht}({\mathfrak{p}})=1}\mbox{\rm bideg}(R_{{\mathfrak{p}}})\mbox{\rm
deg}(R/{\mathfrak{p}})=\sum_{\tiny\mbox{\rm
ht}({\mathfrak{p}})=1}[\lambda(R_{{\mathfrak{p}}}/\mathcal{C}_{{\mathfrak{p}}})-\lambda(R_{{\mathfrak{p}}}/\mathcal{C}^{**}_{{\mathfrak{p}}})]\mbox{\rm
deg}(R/{\mathfrak{p}}).$

We show in [19, Theorem 3.1] that $\mbox{\rm bideg}(R)$ is a well-defined
finite sum independent of the canonical ideal $\mathcal{C}$.

######  Remark 8.4.

$\mbox{\rm bideg}(R)\geq 0$ and vanishes if and only if $R$ is Gorenstein in
codimension $1$.

######  Remark 8.5.

Let $R$ be a Cohen-Macaulay local ring with a canonical ideal $\mathcal{C}$.
The residue of $R$ was introduced by Herzog, Hibi, and Stamate in [37, Section
2]. A standard metric is ${\rm res}(R)=\mbox{\rm deg}(R/\mbox{\rm
tr}(\mathcal{C}))$, where $\mbox{\rm
tr}(\mathcal{C})=\mathcal{C}\cdot\mathcal{C}^{*}=\\{f(x):f\in\mathcal{C}^{*},x\in\mathcal{C}\\}$
is the trace ideal of $\mathcal{C}$. In dimension $1$, we have $\mbox{\rm
bideg}(R)=\lambda(R/\mbox{\rm tr}(\mathcal{C}))={\rm res}(R)$ [19, Proposition
2.3]. They may differ in dimension $d>1$.

######  Remark 8.6.

In dimension $1$, $R$ is Gorenstein if and only if one of $\mbox{\rm
cdeg}(R)$, $\mbox{\rm bideg}(R)$ or ${\rm res}(R)$ vanishes, in which case all
three vanish. If $R$ is not Gorenstein, $\mbox{\rm bideg}(R)\geq 1$, ${\rm
res}(R)\geq 1$. The cases when the minimal values are reached have the
following designations.

  1. (1)

$\mbox{\rm bideg}(R)=1$ if and only if $R$ is a Goto ring.

  2. (2)

${\rm res}(R)=1$ if and only if $R$ is a nearly Gorenstein ring.

We now recall very useful formulas for the computation of duals and biduals.
Let $\mathcal{Q}$ be the total ring of fractions of a Noetherian local ring
$R$. If $I$ is an ideal containing a non zero divisor, then
${\displaystyle\mbox{\rm Hom}_{R}(I,R)=(R:_{\mathcal{Q}}I)}$. A difficulty is
that computer systems such as Macaulay2 [34] are set to calculate quotients of
the form $(A:_{R}B)$ for ideals $A,B\subset R$, which is done with
calculations of syzygies. But it is possible to compute quotients in the total
rings of fractions as follows.

######  Proposition 8.7.

[19, Proposition 3.3] Let $I,J$ be $R$-ideals and let $a\in I$ be a regular
element.

  1. (1)

$\mbox{\rm Hom}_{R}(I,J)=a^{-1}(aJ:_{R}I)$. In particular, $I^{*}=\mbox{\rm
Hom}_{R}(I,R)=a^{-1}((a):_{R}I)$.

  2. (2)

$I^{**}=\mbox{\rm Hom}_{R}(\mbox{\rm Hom}_{R}(I,R),R)=((a):_{R}((a):_{R}I))$.

When $R$ is $1$-dimensional we discuss relationships between almost Gorenstein
rings (i.e., rings with minimal value $\mbox{\rm cdeg}(R)=r(R)-1$) and Goto
rings.

######  Theorem 8.8.

([19, Theorem 4.2], [37, Proposition 6.1]) Let $(R,\mathfrak{m})$ be a Cohen-
Macaulay local ring of dimension $1$ with a canonical ideal $\mathcal{C}$. If
$R$ is almost Gorenstein $($not Gorenstein$)$, then $R$ is a Goto ring.

###### Proof.

From the proof of Proposition 6.9 there exists an exact sequence of
$R$-modules

$0\rightarrow(a)\rightarrow\mathcal{C}\rightarrow X\rightarrow 0$

such that $(a)$ is a minimal reduction of $\mathcal{C}$ and
$X\simeq(R/\mathfrak{m})^{r(R)-1}$. By applying $\mbox{\rm
Hom}_{R}(\cdot,(a))$ to the exact sequence above, we obtain the exact sequence

$0\rightarrow\mbox{\rm Hom}_{R}(\mathcal{C},(a))\rightarrow
R\rightarrow\mbox{\rm Ext}_{R}^{1}(X,(a)).$

Since $R$ is not Gorenstein, we have $\mathcal{C}\neq(a)$. Therefore the image
of $\mbox{\rm Hom}_{R}(\mathcal{C},(a))=(a):_{R}\mathcal{C}$ in $R$ is a
proper ideal of $R$. Since $\mathfrak{m}\mathcal{C}\subset(a)$ it follows that
$\mathfrak{m}=(a):_{R}\mathcal{C}$.

By Proposition 8.7 we have that
$\mathcal{C}^{**}=((a):_{R}((a):_{R}\mathcal{C}))=(a):_{R}\mathfrak{m}$.
Therefore $\mathcal{C}^{**}/(a)$ is isomorphic to the socle of $R/(a)$ in $R$
and we obtain

$\mbox{\rm
bideg}(R)=\lambda(\mathcal{C}^{**}/(a))-\lambda(\mathcal{C}/(a))=r(R)-(r(R)-1)=1.\qed$

######  Corollary 8.9.

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $1$ with a
canonical ideal $\mathcal{C}$. If $\mbox{\rm cdeg}(R)=1$, then $\mbox{\rm
bideg}(R)=1$.

###### Proof.

Since $\mbox{\rm cdeg}(R)=1$, by Remark 6.10 $R$ is almost Gorenstein (not
Gorenstein). Therefore $\mbox{\rm bideg}(R)=1$ by Theorem 8.8. ∎

The example below shows that the converse of Theorem 8.8 (and of Corollary
8.9) does not hold true. Therefore, in dimension one, Goto rings are a larger
class than almost Gorenstein rings.

######  Example 8.10.

Let $R=k[\\![t^{5},t^{7},t^{9}]\\!]$ as in Example 6.15. We have that
$\mbox{\rm cdeg}(R)=2$, and $\mbox{\rm bideg}(R)=1$, by Proposition 9.3, or by
a direct calculation (see [19, Example 4.3]).

##  9\. Comparison of canonical and bi-canonical degrees

In this section we discuss the Comparison Conjecture, proposed in [19].

######  Conjecture 9.1.

(Comparison Conjecture) Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring
that has a canonical ideal $\mathcal{C}$. Then

$\mbox{\rm bideg}(R)\leq\mbox{\rm cdeg}(R).$

######  Remark 9.2.

The Comparison Conjecture holds in the following cases:

  1. (1)

$R$ is Gorenstein in codimension one, since $\mbox{\rm cdeg}(R)=0$ and
$\mbox{\rm bideg}(R)=0$.

  2. (2)

$R$ is a Goto ring, since $\mbox{\rm bideg}(R)=1$ implies $\mbox{\rm
cdeg}(R)\geq 1$, otherwise $R$ would be Gorenstein in codimension one.

  3. (3)

$R$ is an almost Gorenstein ring of dimension one, by Theorem 8.8.

### The Comparison Conjecture for 3-generated numerical semigroup rings

We prove the conjecture for 3-generated numerical semigroup rings. Notation is
as in Section 6. In [19, Proposition 6.1] we calculated $\mbox{\rm bideg}(R)$
when $a_{1}\leq a_{2}$, $b_{2}\leq b_{1}$, and $c_{1}\leq c_{2}$. We obtained
that $\mbox{\rm bideg}(R)=a_{1}b_{2}c_{1}$, which supported the Comparison
Conjecture, since $\mbox{\rm cdeg}(R)$ is either $a_{1}b_{1}c_{1}$ or
$a_{2}b_{2}c_{2}$, by Theorem 6.12. Here we complete the calculation to obtain
$\mbox{\rm bideg}(R)$ in general.

######  Proposition 9.3.

Let $R=k[\\![t^{a},t^{b},t^{c}]\\!]$. Let
$\varphi=\left(\begin{matrix}X^{a_{1}}&Y^{b_{1}}&Z^{c_{1}}\\\
Y^{b_{2}}&Z^{c_{2}}&X^{a_{2}}\\\ \end{matrix}\right)$ be the Herzog matrix
such that $R\simeq k[\\![X,Y,Z]\\!]/I_{2}(\varphi)$. Then

${\displaystyle\mbox{\rm
bideg}(R)=\min\\{a_{1},a_{2}\\}*\min\\{b_{1},b_{2}\\}*\min\\{c_{1},c_{2}\\}}.$

###### Proof.

We note that there are essentially two (possibly overlapping) cases, after a
suitable change of variables.

  1. (1)

Two among $\min\\{a_{1},a_{2}\\},\min\\{b_{1},b_{2}\\},\min\\{c_{1},c_{2}\\}$
occur as exponents on the same column of the matrix.

  2. (2)

All $\min\\{a_{1},a_{2}\\},\min\\{b_{1},b_{2}\\},\min\\{c_{1},c_{2}\\}$ occur
as exponents on the same row of the matrix.

Let $x,y,z$ be the images of $X,Y,Z$ in $R$ respectively. Then we have

$x^{a_{1}+a_{2}}=y^{b_{2}}z^{c_{1}},\quad
y^{b_{1}+b_{2}}=x^{a_{1}}z^{c_{2}},\quad z^{c_{1}+c_{2}}=x^{a_{2}}y^{b_{1}}.$

The ideal ${\displaystyle\mathcal{C}=(x^{a_{1}},y^{b_{2}})}$ is the canonical
ideal of $R$ and
${\displaystyle(x^{a_{1}}):\mathcal{C}=(x^{a_{1}},y^{b_{1}},z^{c_{1}})}$,
since

$\begin{array}[]{ll}{\displaystyle
y^{b_{1}}y^{b_{2}}=x^{a_{1}}z^{c_{2}}\in(x^{a_{1}}),}&{\displaystyle
z^{c_{1}}y^{b_{2}}=x^{a_{1}+a_{2}}\in(x^{a_{1}}).}\end{array}$

Then by Proposition 8.7, we have

$\mathcal{C}^{**}=(x^{a_{1}}):((x^{a_{1}}):\mathcal{C})=(x^{a_{1}}):(x^{a_{1}},y^{b_{1}},z^{c_{1}}).$

For case (1) we may assume that $a_{1}\leq a_{2}$, $b_{2}\leq b_{1}$, and
$c_{1}\leq c_{2}$. Then we have

$\begin{array}[]{ll}{\displaystyle
z^{c_{2}}y^{b_{1}}=z^{c_{1}}y^{b_{2}}z^{c_{2}-c_{1}}y^{b_{1}-b_{2}}=x^{a_{1}+a_{2}}z^{c_{2}-c_{1}}y^{b_{1}-b_{2}}\in(x^{a_{1}}),}&{\displaystyle
z^{c_{2}}z^{c_{1}}=x^{a_{2}}y^{b_{1}}\in(x^{a_{1}}).}\end{array}$

It follows that
${\displaystyle\mathcal{C}^{**}=(x^{a_{1}},y^{b_{2}},z^{c_{2}})}$. Moreover,
since $z^{c_{1}+c_{2}}\in(x^{a_{1}})$, we have that
$\mathcal{C}=(x^{a_{1}},y^{b_{2}})=(x^{a_{1}},y^{b_{2}},z^{c_{1}+c_{2}})$.
Therefore,

$\begin{array}[]{rcl}{\displaystyle\mbox{\rm
bideg}(R)}&=&{\displaystyle\lambda(R/\mathcal{C})-\lambda(R/\mathcal{C}^{**})\vspace{0.1
in}}\\\
&=&{\displaystyle\lambda(R/(x^{a_{1}},y^{b_{2}},z^{c_{1}+c_{2}}))-\lambda(R/(x^{a_{1}},y^{b_{2}},z^{c_{2}}))\vspace{0.1
in}}\\\ &=&{\displaystyle a_{1}b_{2}(c_{1}+c_{2})-a_{1}b_{2}c_{2}\vspace{0.1
in}}\\\ &=&{\displaystyle{a_{1}}{b_{2}}{c_{1}}.}\end{array}$

For case (2) we may assume that that $a_{1}\leq a_{2}$, $b_{1}\leq b_{2}$, and
$c_{1}\leq c_{2}$. We have that

$\mathcal{C}^{**}=(x^{a_{1}}):((x^{a_{1}}):\mathcal{C})=(x^{a_{1}}):(x^{a_{1}},y^{b_{1}},z^{c_{1}})=(x^{a_{1}},y^{b_{2}},z^{c_{2}}y^{b_{2}-b_{1}}),$

since

$\begin{array}[]{l}z^{c_{2}}y^{b_{2}-b_{1}}y^{b_{1}}=z^{c_{2}}y^{b_{2}}=z^{c_{2}-c_{1}}y^{b_{2}}z^{c_{1}}=z^{c_{2}-c_{1}}x^{a_{1}+a_{2}}\in(x^{a_{1}}),\\\
\\\
z^{c_{2}}y^{b_{2}-b_{1}}z^{c_{1}}=z^{c_{1}+c_{2}}y^{b_{2}-b_{1}}=x^{a_{2}}y^{b_{1}}y^{b_{2}-b_{1}}=x^{a_{2}}y^{b_{2}}\in(x^{a_{1}}).\end{array}$

Since $z^{c_{1}+c_{2}}=x^{a_{2}}y^{b_{1}}\in(x^{a_{1}})$, we have
$\mathcal{C}=(x^{a_{1}},y^{b_{2}},z^{c_{1}+c_{2}})$,
$\mathcal{C}^{**}=(x^{a_{1}},y^{b_{2}},z^{c_{1}+c_{2}},z^{c_{2}}y^{b_{2}-b_{1}})$.
Because there are $a_{1}b_{1}c_{1}$ monomials of the form $x^{i}y^{j}z^{k}$,
$0\leq i\leq a_{1}-1$, $b_{2}-b_{1}\leq j\leq b_{2}-1$, $c_{2}\leq i\leq
c_{1}+c_{2}-1$, it follows that

$\begin{array}[]{rcl}{\displaystyle\mbox{\rm
bideg}(R)}&=&{\displaystyle\lambda(R/(x^{a_{1}},y^{b_{2}},z^{c_{1}+c_{2}}))-\lambda(R/(x^{a_{1}},y^{b_{2}},z^{c_{1}+c_{2}},z^{c_{2}}y^{b_{2}-b_{1}})\vspace{0.1
in}}\\\ &=&{\displaystyle
a_{1}b_{2}(c_{1}+c_{2})-\big{(}a_{1}b_{2}(c_{1}+c_{2})-a_{1}b_{1}c_{1}\big{)}\vspace{0.1
in}}\\\ &=&{\displaystyle{a_{1}}{b_{1}}{c_{1}}.}\end{array}$

∎

From Theorem 6.12 and Proposition 9.3 we have the following.

######  Corollary 9.4.

Let $R=k[\\![t^{a},t^{b},t^{c}]\\!]$ be a $3$-generated semigroup ring. Then
$\mbox{\rm bideg}(R)\leq\mbox{\rm cdeg}(R)$; that is, the Comparison
Conjecture holds.

######  Remark 9.5.

For $R=k[\\![t^{a},t^{b},t^{c}]\\!]$, the value of $\mbox{\rm bideg}(R)$ was
also independently calculated in [38, Proposition 3.1], in accordance with the
fact that in dimension one $\mbox{\rm bideg}(R)$ is the residue of $R$ (Remark
8.5).

### Recent developments on the Comparison Conjecture for numerical semigroup
rings

In the recent paper [39], Herzog and Kumashiro studied upper bounds on the
colength of the trace of the canonical module in numerical semigroup rings,
i.e., upper bounds on the bi-canonical degree. Let $R=K[\\![H]\\!]$. They
study when the inequality ${\lambda}(R/\mbox{\rm tr}(\mathcal{C}))\leq
g(H)-n(H)$ holds. In their set-up $g(H)=|\mathbb{N}\setminus H|$ and
$n(H)=|\\{h\in H:h<F(H)\\}|$ denote the number of gaps and the number of non-
gaps, respectively, and $F(H)$ denotes the Frobenius number of $H$; that is,
the largest integer in $\mathbb{N}\setminus H$. We refer to [39] for the
background. It follows from [39, Lemma 2.2] and Remark 7.9 that
$g(H)-n(H)=\mbox{\rm cdeg}(R)$. Therefore the above inequality is equivalent
to the Comparison Conjecture. We summarize their results. Recall that $r(R)$
denotes the type of $R$.

  1. (1)

Suppose that $r(R)\leq 3$. Then the Comparison Conjecture holds [39, Theorem
2.4]. See also Proposition 3.2 in the earlier paper [38] for the case of
3-generated semigroups.

  2. (2)

There is a counterexample to the Comparison Conjecture when $r(R)=5$ [39,
Example 2.5]. Let $H=<13,14,15,16,17,18,21,23>$. Then $\mbox{\rm
bideg}(R)=9>\mbox{\rm cdeg}(R)=8$. The authors note that this example arises
from far-flung Gorenstein rings, studied in [39, Section 4]. It is shown in
[38, Corollary 2.2] that the inequality ${\lambda}(R/\mbox{\rm
tr}(\mathcal{C}))\leq n(H)$ always holds. $R$ is far-flung Gorenstein if and
only if ${\lambda}(R/\mbox{\rm tr}(\mathcal{C}))=n(H)$. Therefore, far-flung
Gorenstein rings are good candidates for counterexamples, since the bi-
canonical degree is as big as possible.

  3. (3)

The case $r(R)=4$ is open [39, Question 2.6]. The examples we tried so far
support the conjecture.

We conclude this section with the following question.

######  Question 9.6.

Can we find other classes of rings for which the Comparison Conjecture holds?
What about 2-AGL rings?

##  10\. Canonical degrees and change of rings

Degree formulas are often statements about change of rings. In [25] and [19]
we examined several cases for cdeg and bi-deg respectively. In particular we
considered polynomial extensions, power series and completions in [25, Section
6]. In this section we highlight and summarize the cases of augmented rings,
$(\mathfrak{m}:\mathfrak{m})$, and hyperplane sections.

### Augmented rings

Let $(R,\mathfrak{m})$ be a $1$-dimensional Cohen-Macaulay local ring with a
canonical ideal $\mathcal{C}$. Consider the augmented ring
$R\ltimes\mathfrak{m}$. That is,
$R\ltimes\mathfrak{m}=R\oplus\mathfrak{m}\epsilon$, $\epsilon^{2}=0$. (Just to
keep the components apart in computations we use $\epsilon$ as a place
holder.) From the results in [25, Section 6] we have that
$(\mathcal{C}:_{R}\mathfrak{m})\times\mathcal{C}$ is a canonical ideal of
$R\ltimes\mathfrak{m}$. Motivated by [29, Theorem 6.5] we obtained the
following. Recall that $r(R)$ denotes the type of $R$.

######  Theorem 10.1.

([25, Theorem 6.8, Corollary 6.9], [19, Proposition 5.1]) Let
$(R,\mathfrak{m})$ be a $1$-dimensional Cohen-Macaulay local ring with a
canonical ideal.

  1. (1)

Suppose that $R$ is not a discrete valuation ring. Then

$\mbox{\rm cdeg}(R\ltimes\mathfrak{m})=2\,\mbox{\rm cdeg}(R)+2\quad\mbox{\rm
and}\quad r(R\ltimes\mathfrak{m})=2\,r(R)+1.$

  2. (2)

$R$ is an almost Gorenstein ring if and only if $R\ltimes\mathfrak{m}$ is an
almost Gorenstein ring.

  3. (3)

Suppose that $R$ is not Gorenstein. Then

$\mbox{\rm bideg}(R\ltimes\mathfrak{m})=2\,\mbox{\rm bideg}(R)-1.$

In particular if $R$ is a Goto ring, then $R\ltimes\mathfrak{m}$ is also a
Goto ring.

Augmented rings in the 2-AGL context are discussed in [12, Section 4].

### Calculations in $(\mathfrak{m}:\mathfrak{m})$

Let $(R,\mathfrak{m})$ be a $1$-dimensional Cohen-Macaulay local ring with a
canonical ideal $\mathcal{C}$. Let $\mathcal{Q}$ be the total ring of
fractions of $R$ and $A=(\mathfrak{m}:_{\mathcal{Q}}\mathfrak{m})$. The
Gorenstein property of $A$ was studied in [29]. In [19, Section 8] we found a
formula for $\mbox{\rm cdeg}(A)$ in relation to the invariants of $R$. We have
that the ideal $\mathfrak{m}\mathcal{C}$ is the canonical ideal of $A$.

######  Theorem 10.2.

[19, Theorem 8.3] Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring of
dimension $1$ with a canonical ideal. Let $\mathcal{Q}$ be the total ring of
fractions of $R$ and $A=(\mathfrak{m}:_{\mathcal{Q}}\mathfrak{m})$. Suppose
that $R$ is not a discrete valuation ring and that $(A,\mathfrak{M})$ is a
local ring. Let $e=[A/\mathfrak{M}:R/\mathfrak{m}]$. Then

$\mbox{\rm cdeg}(A)=e^{-1}(\mbox{\rm
cdeg}(R)+\mathrm{e}_{0}(\mathfrak{m})-2r(R)).$

######  Question 10.3.

Under the same assumptions of Theorem 10.2 can we find a formula for
$\mbox{\rm bideg}(A)$?

######  Example 10.4.

[19, Example 8.8] Let $R=k[\\![t^{5},t^{7},t^{9}]\\!]$ as in Example 6.15. We
have $\mbox{\rm cdeg}(R)=2$. Moreover we have
$\mathrm{e}_{0}(\mathfrak{m})=5$, $r(R)=2$ and $e=1$. Therefore $\mbox{\rm
cdeg}(A)=3$. Let $\mathcal{D}$ be the canonical ideal of $A$. With a direct
calculation we get $\lambda(A/\mathcal{D})=11$ and
$\lambda(A/\mathcal{D}^{**})=9$. Therefore $\mbox{\rm bideg}(A)=2$.

The structure of $(\mathfrak{m}:\mathfrak{m})$ in connection with the 2-AGL
property is studied in [12, Section 5].

### Hyperplane sections

A change of rings issue is the comparison $\mbox{\rm cdeg}(R)$ to $\mbox{\rm
cdeg}(R/(x))$ and $\mbox{\rm bideg}(R)$ to $\mbox{\rm bideg}(R/(x))$ for an
appropriate regular element $x$. We know that if $\mathcal{C}$ is a canonical
module for $R$ then $\mathcal{C}/x\mathcal{C}$ is a canonical module for
$R/(x)$ with the same number of generators, so type is preserved under
specialization. However $\mathcal{C}/x\mathcal{C}$ may not be isomorphic to an
ideal of $R/(x)$. Here is a case of good behavior. Suppose $x$ is regular
modulo $\mathcal{C}$. Then for the sequence

$0\rightarrow\mathcal{C}\longrightarrow R\longrightarrow
R/\mathcal{C}\rightarrow 0,$

we get the exact sequence

$0\rightarrow\mathcal{C}/x\mathcal{C}\longrightarrow R/(x)\longrightarrow
R/(\mathcal{C},x)\rightarrow 0,$

so the canonical module $\mathcal{C}/x\mathcal{C}$ embeds in $R/(x)$. Note
that this leads to $\rho(\mathcal{C})\geq\rho(\mathcal{C}/x\mathcal{C})$.

######  Proposition 10.5.

[25, Proposition 6.11] Suppose $(R,\mathfrak{m})$ is a Cohen-Macaulay local
ring of dimension $d\geq 2$ that has a canonical ideal $\mathcal{C}$. Suppose
that $\mathcal{C}$ is equimultiple and $x$ is regular modulo $\mathcal{C}$.
Then

$\mbox{\rm cdeg}(R)\leq\mbox{\rm cdeg}(R/(x)).$

######  Question 10.6.

An interesting question is when there exists $x$ such that $\mbox{\rm
cdeg}(R)=\mbox{\rm cdeg}(R/(x))$ in $2$ cases: (i) $\mathcal{C}$ is
equimultiple and (ii) $\mathcal{C}$ is not necessarily equimultiple. It would
be a Bertini type theorem.

For bideg$(R)$, in accordance with the Comparison Conjecture we propose the
following.

######  Conjecture 10.7.

[19, Conjecture 5.5] If $\mathcal{C}$ is equimultiple and $x$ is regular mod
$\mathcal{C}$, then $\mbox{\rm bideg}(R)\geq\mbox{\rm bideg}(R/(x))$.

##  11\. Generalization of bi-canonical degrees

In our last paper with Vasconcelos [9] we defined the bi-canonical degree for
rings where the canonical module is not necessarily an ideal. We also
discussed generalizations to rings without canonical modules but admitting
modules sharing some of their properties. In this section we summarize the
main results and open questions.

### bi-canonical degree of rings with a canonical module

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ that
has a canonical module $\omega$. For an $R$-module $E$, we denote the dual of
$E$ by ${\displaystyle E^{*}=\mbox{\rm Hom}_{R}(E,R)}$. By dualizing the
finite free presentation of $\omega$

$F_{1}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}F_{0}\rightarrow\omega\rightarrow
0,$

we obtain the exact sequence

$0\rightarrow\omega^{*}\rightarrow
F_{0}^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}F_{1}^{*}\rightarrow
D(\omega)\rightarrow 0,$

where ${\displaystyle D(\omega)=\mbox{\rm coker}(\varphi^{*})}$ is the
Auslander dual of $\omega$. The module $D(\omega)$ depends on the chosen
presentation but the values of ${\displaystyle\mbox{\rm
Ext}_{R}^{i}(D(\omega),R)}$, for $i\geq 1$, are independent of the
presentation. By [3, Proposition 2.6], there exists an exact sequence

$0\longrightarrow\mbox{\rm
Ext}_{R}^{1}(D(\omega),R)\longrightarrow\omega\longrightarrow\omega^{**}\longrightarrow\mbox{\rm
Ext}_{R}^{2}(D(\omega),R)\longrightarrow 0.$

######  Definition 11.1.

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with a canonical module
$\omega$. The bi-canonical degree of $R$ is

$\mbox{\rm bideg}(R)=\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{1}(D(\omega),R))+\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{2}(D(\omega),R)),$

where ${\displaystyle D(\omega)}$ is the Auslander dual of $\omega$ and
$\mbox{\rm deg}(\mbox{\rule{8.53581pt}{0.56905pt}$\;\\!$})$ is the
multiplicity associated with the $\mathfrak{m}$-adic filtration.

An $R$-module $E$ is said to be torsionless if the natural homomorphism
${\displaystyle\sigma:E\rightarrow E^{**}}$ is injective. We show that if $R$
has a canonical ideal, then Definition 8.3 and Definition 11.1 coincide.

######  Proposition 11.2.

[9, Proposition 3.2] Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring
with a canonical module $\omega$. Then $\omega$ is torsionless if and only if
$\omega$ is isomorphic to an ideal of $R$.

Notice that by Definition 11.1, $\mbox{\rm bideg}(R)=0$ if and only if
$\omega$ is reflexive. By [40, Korollar 7.29], this is equivalent to
$R_{{\mathfrak{p}}}$ is Gorenstein for all primes ${\mathfrak{p}}$ with height
$1$. Therefore we can generalize Remark 8.4.

######  Proposition 11.3.

[9, Proposition 3.3] Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring
with a canonical module $\omega$. Then $\mbox{\rm bideg}(R)=0$ if and only if
$R$ is Gorenstein in codimension $1$.

In accordance with Theorem 8.8, we ask when the minimal value of bideg is
attained.

######  Question 11.4.

Let $R$ be a Cohen-Macaulay local ring of dimension $1$ with a canonical
module $\omega$. When is $\mbox{\rm bideg}(R)=1$?

######  Remark 11.5.

By Definition 11.1, $\mbox{\rm bideg}(R)=1$ if and only if one of the
following holds:

  1. (1)

$\mbox{\rm Ext}_{R}^{1}(D(\omega),R)=0$ and $\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{2}(D(\omega),R))=1$;

  2. (2)

$\mbox{\rm Ext}_{R}^{2}(D(\omega),R)=0$ and $\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{1}(D(\omega),R))=1$.

By Proposition 11.2 we have $\mbox{\rm Ext}_{R}^{1}(D(\omega),R)=0$ if and
only if $\omega$ is isomorphic to an ideal $\mathcal{C}$ of $R$. Therefore
case (1) is equivalent to $\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{2}(D(\omega),R))={\lambda}(\mathcal{C}^{**}/\mathcal{C})=1$, i.e.,
$R$ is a Goto ring. It would be interesting to find an explicit example of
case (2).

The following proposition, which follows from classical results in [11], is
helpful in computing examples with Macaulay2, since it provides an explicit
presentation of the Auslander dual as well as a criterion for the canonical
module to be an ideal in terms of a matrix.

######  Proposition 11.6.

[9, Proposition 3.4] Let $(S,\mathfrak{n})$ be a regular local ring. Let $I$
be an $S$-ideal of height $g$ such that $R=S/I$ is Cohen-Macaulay. Let

$0\rightarrow
G_{g}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}G_{g-1}\rightarrow\cdots\rightarrow
G_{1}\rightarrow G_{0}=S\rightarrow 0$

be a minimal free $S$-resolution of $R$. Then:

  1. (1)

${\displaystyle\omega_{R}\simeq\mbox{\rm coker}(\sigma^{T}\otimes R)}$.

  2. (2)

${\displaystyle D(\omega_{R})\simeq\mbox{\rm coker}(\sigma\otimes R)}$.

  3. (3)

Let ${\displaystyle\varphi=\sigma^{T}\otimes R}$ and ${\displaystyle
G_{g}=S^{p}}$. Then $\omega_{R}$ is an ideal if and only if
${\displaystyle\mbox{\rm ht}(I_{p-1}(\varphi))\geq 1}$ and $I_{p}(\varphi)=0$.

Using Proposition 11.6 we compute $\mbox{\rm bideg}(R)$ in a ring when the
canonical module is not an ideal.

######  Example 11.7.

[9, Example 3.5] Let $S$ be a regular local ring with the maximal ideal
$(X,Y,Z)$. Let $R=S/I$ be given by the minimal resolution

$0\longrightarrow
S^{2}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}S^{3}\longrightarrow
S\longrightarrow R\longrightarrow
0,\;\mbox{where}\;\sigma=\left(\begin{matrix}X^{2}&Y^{2}\\\ YZ&ZX\\\
XZ&XY\end{matrix}\right).$

Let $x,y,z$ be the images of $X,Y,Z$ in $R=S/I$ respectively. We obtain the
free presentation of the canonical module $\omega_{R}$ of $R$:

$R^{3}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}R^{2}\rightarrow\omega_{R}\rightarrow
0,$

where ${\displaystyle\varphi=\sigma^{T}\otimes
R=\left(\begin{matrix}x^{2}&yz&xz\\\ y^{2}&zx&xy\end{matrix}\right)}$. Since
${\displaystyle\mbox{\rm ht}(I_{1}(\varphi))=0}$, the canonical module
$\omega_{R}$ is not an ideal. Moreover, we have

$0\rightarrow\omega_{R}^{*}\rightarrow
R^{2}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}R^{3}\rightarrow
D(\omega_{R})\rightarrow 0,$

where ${\displaystyle\varphi^{*}=\sigma\otimes
R=\left(\begin{matrix}x^{2}&y^{2}\\\ yz&zx\\\ xz&xy\end{matrix}\right)}$. By
using Macaulay2, we compute the bi-canonical degree:

$\mbox{\rm bideg}(R)=\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{1}(D(\omega_{R}),R))+\mbox{\rm deg}(\mbox{\rm
Ext}_{R}^{2}(D(\omega_{R}),R))=1+6=7.$

### Precanonical ideals

We consider a Cohen-Macaulay local ring $R$ that does not necessarily have a
canonical module. We look for rings that have ideals with properties similar
to those of a canonical ideal. Ideals with similar set of conditions have
appeared in the literature, notably among them closed ideals, studied by
Brennan and Vasconcelos in [10], and spherical modules, introduced by
Vasconcelos in [55]. We recall the necessary background.

An ideal $I$ is said to be closed if $\mbox{\rm Hom}_{R}(I,I)=R$. An
$R$-module $E$ is said to be spherical if ${\displaystyle\mbox{\rm
Hom}_{R}(E,E)=R}$ and ${\displaystyle\mbox{\rm Ext}^{i}_{R}(E,E)=0}$ for all
$i\geq 1$ [55, Section 4.3]. Spherical modules are also called semidualizing
and have been studied extensively by Sather-Wagstaff. We refer to [50] for a
survey.

We note that the canonical module is semidualizing. If $R$ is not Gorenstein,
principal ideals generated by a non-zero divisor are semidualizing. Examples
of rings with non trivial semidualizing ideals are constructed in [51].

We note an early result of Vasconcelos in this context.

######  Proposition 11.8.

[55, Proposition 4.9]. Let $(R,\mathfrak{m})$ be an Artinian local ring with
$\mathfrak{m}^{2}=0$. Let $E$ denote the the injective hull of
$R/\mathfrak{m}$. Let $P$ be a finitely generated $R$-module that satisfies
${\displaystyle\mbox{\rm Hom}_{R}(P,P)=R}$ and ${\displaystyle\mbox{\rm
Ext}^{1}_{R}(P,P)=0}$. Then $P\simeq R$, or $P\simeq E$.

In the set-up of the above proposition, an example of a module $P$ that is
neither $R$ nor $E$ is in [55, Section 4.3]. We now define a class of ideals
in this setting.

######  Definition 11.9.

Let $R$ be a Cohen-Macaulay local ring. An $R$-ideal $\mathcal{P}$ is called a
precanonical ideal if $\mbox{\rm Hom}_{R}(\mathcal{P},\mathcal{P})=R$ and
$\mbox{\rm Ext}_{R}^{1}(\mathcal{P},\mathcal{P})=0$.

In [9, Theorem 4.2] we prove conditions under which a precanonical ideal is
canonical. It would be interesting to find ideals that are precanonical, but
not canonical. Next, we propose the following definition.

######  Definition 11.10.

Let $R$ be a Cohen-Macaulay local ring with a precanonical ideal
$\mathcal{P}$. The bi-canonical degree of $R$ relative to $\mathcal{P}$ is

$\mbox{\rm bideg}_{\mathcal{P}}(R)=\mbox{\rm
deg}(\mathcal{P}^{**}/\mathcal{P}).$

######  Question 11.11.

[9, Question 4.3] Which are the properties we can obtain from the bi-canonical
degree of $R$ relative to $\mathcal{P}$ to understand the structures of the
ideal and the ring?

Recall that if $\mathcal{C}$ is a canonical ideal of a $1$-dimensional Cohen-
Macaulay local ring, then $\mbox{\rm
bideg}(R)=\lambda(\mathcal{C}^{**}/\mathcal{C})=0$ (i.e., $\mathcal{C}$ is
reflexive) if and only if $R$ is Gorenstein (i.e., $\mathcal{C}$ is isomorphic
to $R$). In [9, Proposition 4.5] we generalize as follows.

######  Proposition 11.12.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring and $I$ an ideal that
contains a non-zero-divisor. If $I$ is closed and reflexive, then $I$ is a
principal ideal.

######  Corollary 11.13.

Let $R$ be a $1$-dimensional Cohen-Macaulay local ring with a precanonical
ideal $\mathcal{P}$ that contains a non-zero-divisor. Then $\mbox{\rm
bideg}_{\mathcal{P}}(R)=0$ if and only if $\mathcal{P}$ is a principal ideal.

###### Proof.

If $\mbox{\rm bideg}_{\mathcal{P}}(R)=0$, then $\mathcal{P}$ is reflexive and
so it is principal by Proposition 11.12. If $\mathcal{P}$ is a principal ideal
the conclusion follows by Proposition 8.7. ∎

######  Question 11.14.

Can we remove the hypothesis that $I$ contains a non-zero-divisor in
Proposition 11.12?

######  Question 11.15.

How does $\mbox{\rm Ext}_{R}^{1}(I,I)=0$ affect the structure of $I$?

We conclude this manuscript by mentioning that the questions on torsionless
and reflexivity that came up while the authors worked on [9] inspired the
paper [17] that studies torsionless and reflexivity in the more general
context of $q$-torsionfree modules and for rings that are not necessarily
Cohen-Macaulay. In particular, [17, Proposition 2.3] generalizes Proposition
11.2.

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