# Powers of componentwise linear ideals: The Herzog–Hibi–Ohsugi Conjecture and
related problems

Huy Tài Hà  Department of Mathematics, Tulane University, 6823 St. Charles
Ave., New Orleans, LA 70118<EMAIL_ADDRESS>http://www.math.tulane.edu/ tai
and  Adam Van Tuyl  Department of Mathematics and Statistics, McMaster
University, Hamilton, ON, L8S 4L8, Canada<EMAIL_ADDRESS>https://ms.mcmaster.ca/ vantuyl/ Dedicated to Jürgen Herzog on the occasion of
his $80^{th}$ birthday

###### Abstract.

In 1999 Herzog and Hibi introduced componentwise linear ideals. A homogeneous
ideal $I$ is componentwise linear if for all non-negative integers $d$, the
ideal generated by the homogeneous elements of degree $d$ in $I$ has a linear
resolution. For square-free monomial ideals, componentwise linearity is
related via Alexander duality to the property of being sequentially Cohen-
Macaulay for the corresponding simplicial complexes. In general, the property
of being componentwise linear is not preserved by taking powers. In 2011,
Herzog, Hibi, and Ohsugi conjectured that if $I$ is the cover ideal of a
chordal graph, then $I^{s}$ is componentwise linear for all $s\geq 1$. We
survey some of the basic properties of componentwise linear ideals, and then
specialize to the progress on the Herzog-Hibi-Ohsugi conjecture during the
last decade. We also survey the related problem of determining when the
symbolic powers of a cover ideal are componentwise linear.

###### Key words and phrases:

componentwise linear ideals, linear quotients, powers of ideals, symbolic
powers, cover ideals, edge ideals, simplicial complexes

###### 2000 Mathematics Subject Classification:

Primary: 13D02 Secondary: 05E40, 13F20

##  1\. Introduction

Let $R$ be a polynomial ring over a field $k$ and let $I\subseteq R$ be a
homogeneous ideal. The ideal $I$ is often considered computationally simple if
it has a _linear resolution_. In 1999, Herzog and Hibi [32] defined the “next
best” class of ideals, called the componentwise linear ideals. Specifically,
by letting $\langle I_{d}\rangle$ be the ideal generated by homogeneous
polynomials of degree $d$ in $I$, the ideal $I$ is called a _componentwise
linear_ ideal if $\langle I_{d}\rangle$ has a linear resolution for all $d\geq
0$.

Componentwise linear ideals and their properties have become ubiquitous in
commutative algebra, especially in combinatorial commutative algebra. Indeed,
as shown in Herzog and Hibi’s original paper, the square-free monomial ideals
with the componentwise linear property provide an algebraic characterization
of sequentially Cohen-Macaulay simplicial complexes. Componentwise linear
ideals are also closely related to another class of ideals that have been
extensively examined. A finitely generated $R$-module $M$ is called a _Koszul_
module if its associated graded module
$\operatorname{gr}_{\mathfrak{m}}(M)=\bigoplus_{i\geq
0}{\mathfrak{m}}^{i}M/{\mathfrak{m}}^{i+1}M$, with respect to the maximal
homogeneous ideal ${\mathfrak{m}}$ of $R$, has a linear resolution over the
associated graded ring $\operatorname{gr}_{\mathfrak{m}}(R)=\bigoplus_{i\geq
0}{\mathfrak{m}}^{i}/{\mathfrak{m}}^{i+1}$. Koszul algebras (first introduced
by Priddy [47]) and Koszul modules (first considered by Şega [50] and by
Herzog and Iyengar [38]) have enjoyed much attention from various areas of
mathematics. Over a polynomial ring, a homogeneous ideal $I$ is Koszul if and
only if it is componentwise linear (see [49, Theorem 3.2.8] and [62,
Proposition 4.9]), thus cementing the importance of componentwise linearity.

In the two decades since their introduction, one research theme in commutative
algebra is to find new classes of componentwise linear ideals. At the same
time, a broader research theme is to understand how properties of an ideal are
preserved when taking powers. It is natural to ask the following question: if
$I$ is an ideal that is componentwise linear, does the ideal $I^{s}$ also have
this property? The answer to this question will be, in general, no. In fact,
there are examples of ideals such that $I$ has a linear resolution (and so is
componentwise linear), but $I^{2}$ does not have a linear resolution. For a
specific example, see Example 4.2.

One is then led to ask what extra hypotheses are required on $I$ to ensure
that $I^{s}$ is also componentwise linear. The first investigation into this
question was carried out by Herzog, Hibi, and Ohsugi [36]. While we will
define necessary terminology in later sections, for now, it is enough to know
that the ideal $I$ in the statement below is constructed from the properties
of a finite simple graph. Based upon their results and experiments, Herzog,
Hibi and Ohsugi posited the following conjecture:

######  Conjecture 1.1 (Herzog-Hibi-Ohsugi).

Let $I$ be the cover ideal of a chordal graph. Then $I^{s}$ is componentwise
linear for all $s\geq 1$.

One of the main goals of this paper is to provide an up-to-date survey on what
is known about Conjecture 1.1, and what is known about the more general
question of powers of ideals that are componentwise linear. Besides describing
which families of chordal graphs satisfy the above conjecture, we will also
sketch out the broad strategies that have been used to verify the conjecture.

We will also survey a variation of Conjecture 1.1, which was initiated by
Seyed Fakhari [53]. In this variation, it is asked whether or not $I^{(s)}$,
the $s$-th symbolic power of the cover ideal, is componentwise linear. The two
problems dovetail when the regular powers of a cover ideal are the same as its
symbolic powers. This approach is typified by Kumar and Kumar’s recent proof
that Conjecture 1.1 holds for all trees (see [41]); in the case of trees,
regular powers and symbolic powers of cover ideals agree, which allow Kumar
and Kumar to exploit the properties of symbolic powers of ideals.

The majority of the results in this paper have appeared in the literature. We
have, however, included one new result about edge ideals. In particular we
show that the edge ideals of complete $m$-partite graphs have the property
that all of their symbolic powers are componentwise linear (see Corollary
5.22). To encourage further work on Conjecture 1.1, we have included some
research questions in the final section.

We end this introduction with some final comments on our intended audience.
When writing this survey, we assumed that our readers are familiar with
minimal graded free resolutions, and possibly the Stanley-Reisner
correspondence between simplicial complexes and square-free monomial ideals.
Our goal in Sections 2 and 3 is to provide a quick summary of some of the
basic facts surrounding componentwise linear ideals, plus pointers to the
literature for proofs of these facts. We expect graduate students and
researchers new to componentwise linear ideals will appreciate this approach.
In the second half of the paper, we have been attempted to be more
encyclopedic in our approach to the Herzog-Hibi-Ohsugi Conjecture, and the
related questions concerning the symbolic powers. We hope experts will
appreciate this snapshot of the current state-of-the-art regarding these
problems.

Acknowledgements. The authors would like to thank Aldo Conca, Nursel Erey,
Ajay Kumar, Hop D. Nguyen, and Seyed Amin Seyed Fakhari for their suggestions
and feedback. Hà acknowledges supports from the Louisiana Board of Regents and
the Simons Foundation. Van Tuyl’s research is supported by NSERC Discovery
Grant 2019-05412.

##  2\. Basics of Componentwise Linear Ideals

In this section, we introduce componentwise linear ideals, and describe some
of their basic properties. We emphasize techniques to determine if an ideal is
componentwise linear (a theme that will be stressed throughout this survey).
Let $k$ be a field and let $R=k[x_{1},\dots,x_{n}]$ denote a polynomial ring
over $k$.

We begin by recalling the notion of a minimal graded free resolution. Given a
homogeneous ideal $I$ of $R$ (an ideal generated by homogeneous elements of
$R$), we can associate to $I$ a minimal graded free resolution, that is, a
long exact sequence of length $p\leq n$ of graded $R$-modules of the form

$0\rightarrow\bigoplus_{j\in\mathbb{N}}R(-j)^{\beta_{p,j}(I)}\rightarrow\bigoplus_{j\in\mathbb{N}}R(-j)^{\beta_{p-1,j}(I)}\rightarrow\cdots\rightarrow\bigoplus_{j\in\mathbb{N}}R(-j)^{\beta_{0,j}(I)}\rightarrow
I\rightarrow 0.$

Here, $R(-j)$ denotes the twisted graded $R$-module formed by setting
$R(-j)_{d}=R_{d-j}$. The invariants $\beta_{i,j}(I)$ are the $(i,j)$-th graded
Betti numbers of $I$, and they count the number of degree $j$ generators of
the $i$-th syzygy module of $I$.

We write $I=\langle f_{1},\ldots,f_{t}\rangle$ if $I$ is generated by
$\\{f_{1},\ldots,f_{t}\\}$. A homogeneous ideal $I=\langle
f_{1},\ldots,f_{t}\rangle$ has a linear resolution if $\deg f_{1}=\cdots=\deg
f_{t}=d$ for some integer $d\geq 0$ and, for all $i\geq 1$,

$\beta_{i,i+j}(I)=0~{}~{}\mbox{for all $j\neq d$.}$

This is equivalent to the fact that the minimal graded free resolution of $I$
has the form

$0\rightarrow R(-d-p)^{\beta_{p,p+d}(I)}\rightarrow\cdots\rightarrow
R(-d-1)^{\beta_{1,1+d}(I)}\rightarrow R(-d)^{\beta_{0,d}(I)}\rightarrow
I\rightarrow 0.$

######  Example 2.1.

Let $I=\langle x_{1}x_{2},x_{2}x_{3},x_{3}x_{4}\rangle$ in
$R=k[x_{1},x_{2},x_{3},x_{4}]$. Then the minimal graded free resolution of $I$
has the form

$0\rightarrow R(-3)^{2}\rightarrow R(-2)^{3}\rightarrow I\rightarrow 0.$

So, the ideal $I$ has a linear resolution.

On the other hand, consider the ideal $J=\langle
x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle$ in
$R=k[x_{1},\ldots,x_{5}]$. Note that $J$ is not generated by forms of the same
degree, so it cannot have a linear resolution. In particular, the graded free
resolution of $J$ has the form

$0\rightarrow R(-3)^{1}\oplus R(-5)^{1}\rightarrow R(-2)^{2}\oplus
R(-4)^{1}\rightarrow J\rightarrow 0.$

We will return to these ideals below.

We are now in a position to define the main objects of this survey. For any
homogeneous ideal $I\subseteq R$ and integer $d\geq 0$, let

$\langle I_{d}\rangle=\langle\\{F\in I~{}|~{}\mbox{$F$ is homogeneous of
degree $d$}\\}\rangle$

denote the ideal generated by all the homogeneous elements of degree $d$ in
$I$. The following definition is then due to Herzog and Hibi [32]:

######  Definition 2.2 (Herzog–Hibi).

A homogeneous ideal $I$ of $R$ is componentwise linear if for all integers
$d\geq 0$, the ideal $\langle I_{d}\rangle$ has a linear resolution.

Based only upon the above definition, verifying whether or not a homogeneous
ideal $I$ is componentwise linear would involve checking an infinite number of
conditions. Fortunately, it is possible to verify if $I$ is componentwise
linear by only checking a finite number of $d\in\mathbb{N}$ by using the
Castelnuovo-Mumford regularity of $I$.

The (Castelnuovo-Mumford) regularity of $I$ is defined to be

${\rm reg}(I)=\max\\{j-i~{}|~{}\beta_{i,j}(I)\neq 0\\}.$

Roughly speaking, ${\rm reg}(I)$ measures the largest degree of a generator of
a syzygy of $I$. This invariant can also be viewed as a measure of the
complexity of $I$. We then require the following well-known fact (see, for
example, [22, Proposition 1.1]).

######  Theorem 2.3.

Let $I$ be a homogeneous ideal with $r={\rm reg}(I)$. Then for all $d\geq r$,
the ideal $\langle I_{d}\rangle$ has a linear resolution.

Consequently, determining whether $I$ is componentwise linear can be reduced
to checking a finite number of cases.

######  Corollary 2.4.

Let $I$ be a homogeneous ideal with $r={\rm reg}(I)$. Then $I$ is
componentwise linear if and only if $\langle I_{d}\rangle$ has a linear
resolution for all $0\leq d\leq r$. In particular, if $I$ has a linear
resolution, then $I$ is also componentwise linear.

###### Proof.

The if-and-only if statement follows directly from Theorem 2.3 and the
definition. If $I$ has a linear resolution, then all the homogeneous
generators of $I$ have the same degree, which coincides with its regularity
$r$. Thus $\langle I_{d}\rangle=\langle 0\rangle$ for all $0\leq d<r$, and
$\langle I_{r}\rangle=I$. The conclusion now follows from the first part of
the statement. ∎

######  Example 2.5.

Let $J=\langle x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle$ in
$R=k[x_{1},\ldots,x_{5}]$ be the ideal of Example 2.1. From the minimal graded
free resolution of $J$ given in Example 2.1, we have

${\rm reg}(J)=\max\\{3-1,5-1,2-0,4-0\\}=4.$

So, by Corollary 2.4, we need to compute the minimal graded free resolutions
of $\langle J_{d}\rangle$ for $d=0,\ldots,4$. Since $\langle
J_{d}\rangle=\langle 0\rangle$ for $d=0,1$, we focus on the remaining cases.
For the ideal $\langle J_{2}\rangle=\langle x_{1}x_{3},x_{2}x_{3}\rangle$ we
have

$0\rightarrow R(-3)\rightarrow R(-2)^{2}\rightarrow\langle
J_{2}\rangle\rightarrow 0;$

for the ideal

$\langle J_{3}\rangle=\langle
x_{1}^{2}x_{3},x_{1}x_{2}x_{3},x_{1}x_{3}x_{4},x_{1}x_{3}x_{5},x_{2}^{2}x_{3},x_{2}x_{3}^{2},x_{2}x_{3}x_{4},x_{2}x_{3}x_{5}\rangle$

the resolution is

$0\rightarrow R(-7)^{2}\rightarrow R(-6)^{10}\rightarrow R(-5)^{20}\rightarrow
R(-4)^{20}\rightarrow R(-3)^{9}\rightarrow\langle J_{3}\rangle\rightarrow 0;$

and for the ideal $\langle J_{4}\rangle$ (we suppress the 26 minimal
generators) the resolution has the form

$0\rightarrow R(-8)^{9}\rightarrow R(-7)^{43}\rightarrow R(-6)^{80}\rightarrow
R(-5)^{71}\rightarrow R(-4)^{26}\rightarrow\langle J_{4}\rangle\rightarrow 0.$

Thus, the ideal $J$ is componentwise linear.

We give some alternative ways to characterize when an ideal is componentwise
linear. One such characterization is in terms of the generic initial ideal. We
review the relevant terminology; for more on generic initial ideals see [29].

Let $GL_{n}(k)$ denote the general linear group of order $n$ over $k$, i.e.,
all the $n\times n$ invertible matrices with entries in $k$. Any matrix $g\in
GL_{n}(k)$ acts on the variables $x=\\{x_{1},\ldots,x_{n}\\}$ by a linear
change of variables, i.e., $x_{i}$ is sent to
$g_{i,1}x_{1}+g_{i,2}x_{2}+\cdots+g_{i,n}x_{n}$ for $i=1,\ldots,n$, where
$(g_{i,1},\ldots,g_{i,n})$ is the $i$-th row of $g$. Given $g\in GL_{n}(k)$
and a polynomial $f=f(x_{1},\ldots,x_{n})\in R$, then $g$ acts on $f$ by
$g\cdot f:=f(g\cdot x).$ Now fix an ideal $I$ of $R$ and monomial order $>$.
Every matrix $g\in GL_{n}(k)$ results in an initial ideal
$\operatorname{in}_{>}(g\cdot I)$ where $g\cdot I=\langle g\cdot f~{}|~{}f\in
I\rangle.$ We say that two matrices $g$ and $g^{\prime}$ are _equivalent_ if

$\operatorname{in}_{>}(g\cdot I)=\operatorname{in}_{>}(g^{\prime}\cdot I).$

This definition induces an equivalence relation on $GL_{n}(k)$, and thus, the
equivalence classes partition the group $GL_{n}(k)$.

One of these partitions is quite “large” in the following sense:

######  Lemma 2.6 ([34, Theorem 4.1.2]).

For a fixed $I$ and term order $>$, one of the equivalence classes is a
nonempty Zariski open subset $U$ inside $GL_{n}(k)$.

Although we do not go into the details here, the Zariski open set inside of
$GL_{n}(k)$ refers to the zero set of some ideal in $k[y_{i,j}~{}|~{}1\leq
i,j\leq n]$. Note that for all $g\in U$, the Zariski open subset in Lemma 2.6,
the initial ideal $\operatorname{in}_{>}(g\cdot I)$ is the same ideal.

######  Definition 2.7.

Fix a term order $>$ on $R$, let $I$ be an ideal of $R$, and let $g$ be any
element of the open Zariski subset of Lemma 2.6. The initial ideal
$\operatorname{in}_{>}(g\cdot I)$ is called the _generic initial ideal_ of $I$
for the term order $>$. It is denoted
$\operatorname{gin}_{>}(I)=\operatorname{in}_{>}(g\cdot I)$.

Roughly speaking, the generic initial ideal is the ideal we should expect if
we pick a “random” matrix $g\in GL_{n}(k)$ and form
$\operatorname{in}_{>}(g\cdot I).$

With this terminology, we then have the following equivalent statements. In
particular, an ideal $I$ is componentwise linear if $I$ and the generic
initial ideal $I$ with respect to the reverse lexicographical order have the
same number of generators in each degree. We want to highlight that this
statement also requires the hypothesis that the field $k$ has characteristic
zero.

######  Theorem 2.8.

Let $I$ be a homogeneous ideal of $R$. Assume ${\rm char}(k)=0$. Then the
following are equivalent:

  1. (1)

$I$ is componentwise linear

  2. (2)

$\beta_{i,j}(I)=\beta_{i,j}({\rm gin}_{>}(I))$ for all $i,j\geq 0$ where $>$
is the reverse lexicographical order.

  3. (3)

$\beta_{0,j}(I)=\beta_{0,j}({\rm gin}_{>}(I))$ for all $j\geq 0$ where $>$ is
the reverse lexicographical order.

###### Proof.

The equivalence of (1) and (2) is [2, Theorem 1.1]. The equivalence of (1) and
(3) first appears in the paper of Conca [11, Theorem 1.2], although Conca
points out that this equivalence is implicit in the proof of [2, Theorem 1.1].
∎

######  Example 2.9.

Theorem 2.8 gives an alternative way to determine if an ideal is componentwise
linear. The computer algebra package Macaulay2 [28] is able to compute the
generic initial ideal (although the algorithm is only probabilistic in the
sense that it computes the initial ideal $g\cdot I$ for a random $g\in
GL_{n}(k)$).

Here is a sample session applied to the componentwise linear ideal $J$ of
Example 2.1.

    i1 : R = QQ[x_1..x_5]

    i2 : j = monomialIdeal(x_1*x_3,x_2*x_3,x_1*x_2*x_4*x_5)

    i3 : loadPackage "GenericInitialIdeal"

    i4 : gin j

    o4 = ideal (x_1^2, x_1*x_2, x_2^4)

Since the ideal and its generic initial ideal have the same number of
generators in each degree, the ideal is componentwise linear.

As mentioned in the introduction, componentwise linear ideals are related to
the notion of Koszulness. We record this equivalence.

######  Theorem 2.10.

Let $I$ be a homogeneous ideal of $R$. Assume ${\rm char}(k)=0$. Then $I$ is
componentwise linear if and only if the $R$-module $I$ is a Koszul module.

###### Proof.

See Römer [49, Theorem 3.2.8] or Yanagawa [62, Proposition 4.9]. ∎

######  Remark 2.11.

Because of the equivalence of Theorem 2.10, componentwise linear ideals are
sometimes called Koszul ideals; for example, see [16].

We end this section by introducing linear quotients, an extremely useful
technique to show that an ideal is componentwise linear. Ideals with linear
quotients were first defined by Herzog and Takayama [39] for monomial ideals;
the more general definition is given below.

######  Definition 2.12.

A homogeneous ideal $I$ has linear quotients if the minimal generators of $I$
can be ordered as $f_{1},\ldots,f_{s}$ such that for each $i=2,\ldots,s$, the
ideal $\langle f_{1},\ldots,f_{i-1}\rangle:\langle f_{i}\rangle$ is generated
by linear forms.

Linear quotients can then be used to verify that an ideal is componentwise
linear:

######  Theorem 2.13 ([34, Theorem 8.2.15]).

Let $I$ be a homogeneous ideal, and suppose that $I$ has linear quotients with
respect to a minimal set of generators of $I$. Then $I$ is componentwise
linear.

######  Example 2.14.

If we consider the ideal $J=\langle
x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle$ of Example 2.1, we have

$\langle x_{1}x_{3}\rangle:\langle x_{2}x_{3}\rangle=\langle
x_{1}\rangle~{}~{}~{}\mbox{and}~{}~{}~{}\langle
x_{1}x_{3},x_{2}x_{3}\rangle:\langle x_{1}x_{2}x_{4}x_{5}\rangle=\langle
x_{1},x_{2}\rangle.$

The ideal $J$ has linear quotients, thus giving another way of seeing that $J$
is componentwise linear.

######  Remark 2.15.

We want to stress that the property of being componentwise linear (or having a
linear resolution) depends upon the characteristic of the field $k$. A well
known example of this phenomenon, due to Reisner [48], is the square-free
monomial ideal

$I=\langle
x_{1}x_{2}x_{3},x_{1}x_{2}x_{6},x_{1}x_{3}x_{5},x_{1}x_{4}x_{5},x_{1}x_{4}x_{6},x_{2}x_{3}x_{4},x_{2}x_{4}x_{5},x_{2}x_{5}x_{6},x_{3}x_{4}x_{6},x_{3}x_{5}x_{6}\rangle$

in $R=k[x_{1},\ldots,x_{6}]$. The ideal $I$ has a linear resolution if and
only if ${\rm char}(k)\neq 2$. So $I$ is componentwise linear if and only if
${\rm char}(k)\neq 2$. This example also shows that the converse of Theorem
2.13 cannot hold, since linear quotients is a property that does not “see” the
characteristic of the field.

##  3\. Componentwise linearity of (square-free) monomial ideals

In this section we recall properties of (square-free) monomial ideals that are
componentwise linear. A highlight of this section is Herzog and Hibi’s
classification of square-free monomial ideals that are componentwise linear,
which was one of main results in the paper that introduced componentwise
linearity [32].

Let $V=\\{x_{1},\ldots,x_{n}\\}$ be a collection of vertices. A simplicial
complex $\Delta$ on $V$ is a subset of the power set of $V$ that satisfies the
following two properties: (1) if $F\in\Delta$ and if $G\subseteq F$, then
$G\in\Delta$; and (2) $\\{x_{i}\\}\in\Delta$ for all $i=1,\ldots,n$. The
maximal elements of $\Delta$ ordered with respect to inclusion are called the
facets of $\Delta$. If $F_{1},\ldots,F_{s}$ is a complete list of the facets
of $\Delta$, then we usually write $\Delta=\langle F_{1},\ldots,F_{s}\rangle$.
In this case, we say $\Delta$ is generated by $F_{1},\ldots,F_{s}$. An element
$F\in\Delta$ is called a face of $\Delta$. The dimension of $F$ is $\dim
F=|F|-1$ (we use the convention that $\dim\emptyset=-1$). The dimension of a
simplicial complex is $\dim\Delta=\max\\{\dim F~{}|~{}F\in\Delta\\}$. A
simplicial complex is pure if all of its facets have the same dimension.

We associate with $\Delta$ a square-free monomial ideal $I_{\Delta}$ in the
polynomial ring $R=k[x_{1},\ldots,x_{n}]$ as follows:

$I_{\Delta}=\langle x_{i_{1}}\cdots
x_{i_{s}}~{}|~{}\\{x_{i_{1}},\ldots,x_{i_{s}}\\}\not\in\Delta\rangle.$

The ideal $I_{\Delta}$ is the Stanley-Reisner ideal of $\Delta$; it captures
many of the combinatorial properties of $\Delta$. This construction can be
reversed, that is, given any square-free monomial ideal $I$, we can construct
its _Stanley-Reisner simplicial complex_

$\Delta_{I}=\\{\\{x_{i_{1}},\ldots,x_{i_{s}}\\}~{}|~{}\mbox{$x_{i_{1}}\cdots
x_{i_{s}}$ is a square-free monomial not in $I$}\\}.$

######  Example 3.1.

Consider our running example (see Example 2.1)

$J=\langle x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle\subseteq
k[x_{1},\ldots,x_{5}].$

On the vertex set $V=\\{x_{1},\ldots,x_{5}\\}$, we then have

$\Delta_{J}=\langle\\{x_{3},x_{4},x_{5}\\},\\{x_{2},x_{4},x_{5}\\},\\{x_{1},x_{4},x_{5}\\},\\{x_{1},x_{2},x_{5}\\},\\{x_{1},x_{2},x_{4}\\}\rangle.$

We say that $\Delta$ is a (sequentially) Cohen-Macaulay simplicial complex if
the quotient ring $R/I_{\Delta}$ is (sequentially) Cohen-Macaulay.
Particularly, sequentially Cohen-Macaulay simplicial complexes are described
as follows:

######  Definition 3.2.

Let $\Delta$ be a simplicial complex with $\dim\Delta=d$. For each
$i=-1,\ldots,d$, let $\Delta(i)=\langle F\in\Delta~{}|~{}\dim F=i\rangle,$
i.e., the simplicial complex generated by all the faces of dimension $i$ in
$\Delta$. Then $\Delta$ is sequentially Cohen-Macaulay if $\Delta(i)$ is
Cohen-Macaulay for all $i=-1,\ldots,d$.

Given a simplicial complex $\Delta$, the Alexander dual of $\Delta$ is the
simplicial complex

$\Delta^{\vee}=\langle V\setminus F~{}|~{}F\not\in\Delta\rangle.$

Using the above terminology, square-free monomial ideals that are
componentwise linear can be classified. This result, which is due to Herzog
and Hibi, is one of the first major results about componentwise linear ideals.

######  Theorem 3.3 ([32, Theorem 2.1]).

Let $I$ be a square-free monomial ideal and let $\Delta=\Delta_{I}$ be its
Stanley-Reisner simplicial complex. Then $I$ is componentwise linear if and
only if $\Delta^{\vee}$ is sequentially Cohen-Macaulay.

######  Example 3.4.

As noted multiple times, the ideal $J=\langle
x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle$ is componentwise linear. So
the simplicial complex

$\Delta^{\vee}_{J}=\langle\\{x_{3}\\},\\{x_{2},x_{4},x_{5}\\},\\{x_{1},x_{4},x_{5}\\}\rangle$

is sequentially Cohen-Macaulay.

The above theorem generalizes an important result of Eagon and Reiner. We also
record this result.

######  Theorem 3.5 ([17, Theorem 3]).

Let $I$ be a square-free monomial ideal and let $\Delta=\Delta_{I}$ be its
Stanley-Reisner simplicial complex. Then $I$ has a linear resolution if and
only if $\Delta^{\vee}$ is Cohen-Macaulay.

######  Example 3.6.

If $I$ is the square-free monomial ideal of Remark 2.15, then the ideal $I$
has a linear resolution if and only if ${\rm char}(k)\neq 2$. So
$\Delta^{\vee}_{I}$ is Cohen-Macaulay if and only if ${\rm char}(k)\neq 2$.

Theorem 3.3 provides a new strategy to prove that an ideal is componentwise
linear. In particular, instead of showing that $I$ is componentwise linear, it
is enough to show that $\Delta^{\vee}_{I}$ is sequentially Cohen-Macaulay.
There are two combinatorial ways to determine if a simplicial complex is
sequentially Cohen-Macaulay; we first give some relevant definitions.

For a vertex $x$ in $\Delta$, the _deletion_ of $x$ in $\Delta$, denoted by
$\operatorname{del}_{\Delta}(x)$, is the simplicial complex obtained by
removing $x$ and all faces containing $x$ from $\Delta$. Also, the _link_ of
$x$ in $\Delta$, denoted by $\operatorname{link}_{\Delta}(x)$, is the
simplicial complex whose faces are

$\\{F\in\Delta~{}\big{|}~{}x\not\in F\text{ and }F\cup\\{x\\}\in\Delta\\}.$

######  Definition 3.7.

Let $\Delta=\langle F_{1},\ldots,F_{s}\rangle$ be a simplicial complex.

  1. (1)

The complex $\Delta$ is shellable if there exists a linear order of its facets
$F_{1},\dots,F_{s}$ such that for all $i=2,\dots,s$, the subcomplex $\langle
F_{1},\dots,F_{i-1}\rangle\cap\langle F_{i}\rangle$ is pure and of dimension
$(\dim F_{i}-1)$.

  2. (2)

The complex $\Delta$ is vertex decomposable if either:

    1. (a)

$\Delta$ is a simplex (i.e., it has a unique facet); or the empty complex; or

    2. (b)

there exists a vertex $x$ in $\Delta$ such that all facets of
$\operatorname{del}_{\Delta}(x)$ are facets of $\Delta$ (i.e.,$x$ is a
shedding vertex), and both $\operatorname{link}_{\Delta}(x)$ and
$\operatorname{del}_{\Delta}(x)$ are vertex decomposable.

######  Remark 3.8.

The notions of shellability and vertex decomposability were first given for
pure simplicial complexes, that is, when all the facets have the same
dimension. These notions were generalized by Björner and Wachs [5, 6] to
include nonpure simplicial complexes.

One approach in the literature (which we will return to later in the paper) to
show a square-free monomial ideal is componentwise linear is to show that the
associated simplicial complex has one of the above properties.

######  Theorem 3.9.

Let $I$ be a square-free monomial ideal with associated simplicial complex
$\Delta=\Delta_{I}$. If $\Delta^{\vee}$, the Alexander dual of $\Delta$, is
either vertex decomposable or shellable, then $I$ is componentwise linear.

###### Proof.

By [6, Theorem 11.3], a vertex decomposable simplicial complex $\Gamma$ is
also shellable. Stanley (see page 87 of [56]) first observed that if $\Gamma$
is a shellable simplicial complex, then $\Gamma$ is sequentially Cohen-
Macaulay. The conclusion then follows from Theorem 3.3 since both hypotheses
imply that $\Delta^{\vee}$ is sequentially Cohen-Macaulay. ∎

For the case of square-free monomial ideals, there is an alternative way to
verify that the ideal is componentwise linear. Given a square-free monomial
ideal $I$, let $I_{[d]}$ be the ideal generated by all the square-free
monomials of degree $d$ in $I$. We then say $I$ is square-free componentwise
linear if $I_{[d]}$ has a linear resolution for all $d$. Checking whether a
square-free monomial ideal is componentwise linear then reduces to checking
whether or not it is square-free componentwise linear.

######  Theorem 3.10 ([32, Proposition 1.5]).

Let $I$ be a square-free monomial ideal. Then $I$ is componentwise linear if
and only if $I$ is square-free componentwise linear.

######  Remark 3.11.

If $I$ is a square-free monomial ideal in $R=k[x_{1},\ldots,x_{n}]$, then $I$
cannot have any square-free monomials of degree $>n$. So, to check if $I$ is
square-free componentwise linear, we only need to check that $I_{[d]}$ has a
linear resolution for $0\leq d\leq n$. In fact, since there is only one
square-free monomial of degree $n$, namely $x_{1}\cdots x_{n}$, either
$I_{[n]}=\langle 0\rangle$ or $I_{[n]}=\langle x_{1}\cdots x_{n}\rangle$, and
so $I_{[n]}$ always has a linear resolution. So we only need to check for
$0\leq d<n$.

We expand our scope to now say a few words about monomial ideals more
generally, and not just the square-free case. One common approach to studying
monomial ideals is to use the process of polarization to turn the monomial
ideal into a square-free monomial ideal in a much larger polynomial ring. In
many instances, properties of the original ideal are preserved in the new
ideal, and vice versa. It turns out that the linear quotient property is
preserved under this operation.

We formally define the polarization procedure. Let $I=\langle
x_{1}^{a_{1,1}}\cdots x_{n}^{a_{1,n}},\ldots,x_{1}^{a_{t,1}}\cdots
x_{n}^{a_{t,n}}\rangle$ be a monomial ideal in $R=k[x_{1},\ldots,x_{n}]$. For
$j=1,\ldots,n$, set $b_{j}=\max\\{a_{i,j}~{}|~{}1\leq i\leq t\\}$, that is,
$b_{j}$ is the highest power of $x_{j}$ that appears among the generators of
$I$. In a polynomial ring

$S=k[x_{1,1},\ldots,x_{1,b_{1}},x_{2,1},\ldots,x_{2,b_{2}},\ldots,x_{n,1},\ldots,x_{n,b_{n}}]$

we define the polarization of $I$ to be the ideal

$I^{{\rm pol}}=\langle x_{1,1}\cdots x_{1,a_{1,1}}x_{2,1}\cdots
x_{2,a_{1,2}}\cdots x_{n,1}\cdots x_{n,a_{1,n}},\ldots,x_{1,1}\cdots
x_{1,a_{t,1}}x_{2,1}\cdots x_{2,a_{t,2}}\cdots x_{n,1}\cdots
x_{n,a_{t,n}}\rangle.$

That is, we replace $x_{j}^{a_{k,j}}$ with $x_{j,1}\cdots x_{j,a_{k,j}}$ in
each generator of $I$.

As we saw earlier, linear quotients is a technique that can be used to check
if an ideal is componentwise linear. The following result of Seyed Fakhari
shows that for an arbitrary monomial ideal, we can use the polarization of the
ideal to check if the original ideal has linear quotients.

######  Theorem 3.12 ([53, Lemma 3.5]).

Let $I$ be a monomial ideal. Then $I$ has linear quotients if and only $I^{\rm
pol}$, the polarization of $I$, has linear quotients.

######  Example 3.13.

Let $I=\langle y_{1}y_{3},y_{2}y_{3},y_{1}y_{2}y_{4}^{2}\rangle$ in
$R=k[y_{1},y_{2},y_{3},y_{4}]$. We then have $b_{1}=1,~{}b_{2}=1,~{}b_{3}=1$,
and $b_{4}=2$. In the polynomial ring
$S=k[y_{1,1},y_{2,1},y_{3,1},y_{4,1},y_{4,2}]$, the polarization of $I$ is

$I^{\rm pol}=\langle
y_{1,1}y_{3,1},y_{2,1}y_{3,1},y_{1,1}y_{2,1}y_{4,1}y_{4,2}\rangle.$

If we relabel the variables so that
$x_{1}=y_{1,1},x_{2}=y_{2,1},x_{3}=y_{3,1},x_{4}=y_{4,1}$ and $x_{5}=y_{4,2}$,
then $I^{\rm pol}$ is the ideal $J$ of our running example. So $I$ has linear
quotients (which can also be checked directly).

There are many classes of monomial ideals which have been identified as
componentwise linear. While we do not survey all of this literature (since we
wish to focus on powers of ideals), we highlight some families that are
relevant for our future discussions.

A monomial ideal $I$ is weakly polymatroidal if for every pair of minimal
generators $m_{1}=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ and
$m_{2}=x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}$ with $m_{1}>m_{2}$ with respect to
the the lexicographical ordering, and if $a_{1}=b_{1},\ldots,a_{t-1}=b_{t-1}$
but $a_{t}>b_{t}$, then there exists a $j>t$ such that $x_{t}(m_{2}/x_{j})\in
I$. The notion of weakly polymatroidal generalizes the notation of a stable
ideal. An ideal $I$ is stable if for any monomial
$m=x_{i_{1}}^{a_{i_{1}}}\cdots x_{i_{s}}^{a_{i_{s}}}$ in $I$, if $j<{i_{s}}$,
then $x_{j}(m/x_{i_{s}})\in I$. As shown by Mohammadi and Moradi, these ideals
all have linear quotients, thus providing us with a large class of
componentwise linear monomial ideals.

######  Theorem 3.14 ([44, Theorem 1.3]).

If $I$ is a weakly polymatroidal ideal, then $I$ has linear quotients, and
consequently, $I$ is componentwise linear.

We now introduce another class of ideals. For any
$J=\\{j_{1},\ldots,j_{s}\\}\subseteq\\{1,\ldots,n\\}$, we define
$\mathfrak{m}_{J}=\langle x_{j_{1}},\ldots,x_{j_{s}}\rangle$. For any integer
$a\geq 1$, we call $\mathfrak{m}_{J}^{a}$ a Veronese ideal (see [33]). Given
subsets $J_{1},\ldots,J_{s}$ of $\\{1,\ldots,n\\}$ and positive integers
$a_{1},\ldots,a_{s}$ we call

$I=\mathfrak{m}_{J_{1}}^{a_{1}}\cap\cdots\cap\mathfrak{m}_{J_{s}}^{a_{s}}$

an intersection of Veronese ideals. Ideals of this type appear throughout the
literature (for example, the primary decomposition of a square-free monomial
ideal can viewed as the intersection of Veronese ideals). In some cases, we
can determine if $I$ is componentwise linear simply from the subsets
$J_{1},\ldots,J_{s}$. One such example is the following result of Francisco
and Van Tuyl [26, Theorem 3.1], and generalized by Mohammadi and Moradi (whose
result is presented below).

######  Theorem 3.15 ([44, Theorem 2.5]).

Let $J_{1},\ldots,J_{s},K$ be subsets of $[n]=\\{1,\ldots,n\\}$. Suppose that
$J_{i}\cup J_{j}=[n]$ for all $i\neq j$ and $K\subseteq[n]$. Then

$I=\mathfrak{m}_{J_{1}}^{a_{1}}\cap\cdots\cap\mathfrak{m}_{J_{s}}^{a_{s}}\cap\mathfrak{m}_{K}^{b}$

is componentwise linear for any positive integers $a_{1},\ldots,a_{s},b$.

We end this section with a recent result of Dung, Hien, Nguyen, and Trung that
allows one to build new componentwise linear ideals from old ones. We have
only presented the monomial ideal version of this result, although the work of
[16] is more general since the focus of their work is the linear defect of an
ideal.

######  Theorem 3.16 ([16, Corollary 5.6]).

Let $R=k[x_{1},\ldots,x_{n}]$. Let $I^{\prime}$ and $T$ be non-trivial
monomial ideals and $x$ a variable such that

  1. (1)

$I^{\prime}$ is componentwise linear,

  2. (2)

$T\subseteq\langle x_{1},\ldots,x_{n}\rangle I^{\prime}$, and

  3. (3)

no generator of $T$ is divisible by $x$.

If $I=xI^{\prime}+T$, then $I$ is componentwise linear if and only if $T$ is
componentwise linear.

##  4\. Componentwise linearity of regular powers

In this section we survey the problem of determining when the regular powers
of an ideal is componentwise linear. In particular, we will focus on the
Herzog-Hibi-Ohsugi Conjecture on the behavior of cover ideals of chordal
graphs.

One theme in commutative algebra is to understand how properties of an ideal
are preserved when one takes powers of these ideals. This theme is
encapsulated into the following broad question:

######  Question 4.1.

Let $I$ be an ideal of a ring $T$. Suppose that the ideal $I$ has some
property $\mathcal{P}$. Does $I^{s}$ also have property $\mathcal{P}$ for all
integers $s\geq 1$?

The monograph [9] looks at this question for a number of ideals that arise in
either combinatorics or geometry. Given this theme, it is natural to ask if
the property of being componentwise linear is preserved by taking powers. The
answer turns out to be no in general as shown in the follow examples.

######  Example 4.2.

Let

$I=\langle
x_{1}x_{2}x_{3},x_{1}x_{2}x_{6},x_{1}x_{3}x_{5},x_{1}x_{4}x_{5},x_{1}x_{4}x_{6},x_{2}x_{3}x_{4},x_{2}x_{4}x_{5},x_{2}x_{5}x_{6},x_{3}x_{4}x_{6},x_{3}x_{5}x_{6}\rangle$

in $R=k[x_{1},\ldots,x_{6}]$ be Reisner’s example as given in Example 2.15. As
already noted, this ideal has a linear resolution if ${\rm char}(k)\neq 2$. It
was shown by Conca in [10, Remark 3] that when ${\rm char}(k)\neq 2$, the
ideal $I^{2}$ does not have a linear resolution. Since
$I^{2}=\langle(I^{2})_{6}\rangle$, the square of $I$ is not componentwise
linear. Note that in [10], this example was attributed to Terai.

Sturmfels [57] gave another example, which does not depend upon the
characteristic of the ground field. In particular, the ideal

$I=\langle
x_{1}x_{3}x_{6},x_{1}x_{4}x_{5},x_{2}x_{3}x_{4},x_{2}x_{5}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6},x_{3}x_{5}x_{6},x_{4}x_{5}x_{6}\rangle$

has the property that $I$ has a linear resolution in all characteristics, but
$I^{2}$ does not have a linear resolution.

Conca’s paper [12] also contains many other examples of (monomial) ideals with
this behavior.

The previous example shows that we will require some extra hypotheses on $I$
in order to guarantee that $I^{s}$ is componentwise linear. Restricting to the
case that $I$ has a linear resolution is a natural starting point, and it
turns out that we can say more in this case.

We say that a homogeneous ideal $I$ has linear powers (following Bruns, Conca,
and Varbaro [7]) if $I$ has a linear resolution and $I^{s}$ has a linear
resolution for all $s\geq 2$. Ideals with linear powers can be classified in
terms of their Rees algebras. If $I=\langle f_{1},\ldots,f_{s}\rangle$ is
generated by homogeneous elements of degree $d$, the Rees algebra is

${\rm Rees}(I)=\bigoplus_{s\in\mathbb{N}}I^{s}.$

This ring has a bigraded structure given by ${\rm Rees}(I)_{a,b}=(I^{b})_{a}$
with $(a,b)\in\mathbb{N}^{2}$, that is, all the elements of degree $a$ in the
$b$-th power of $I$. We can give ${\rm Rees}(I)$ a graded structure by setting
the degree $a$ part of ${\rm Rees}(I)$ to be

${\rm Rees}(I)_{(a,*)}=\bigoplus_{b\in\mathbb{N}}(I^{b})_{a}.$

With this grading ${\rm Rees}(I)$ is a graded $R$-module. We denote its
Castelnuovo-Mumford regularity with respect to this grading by ${\rm
reg}_{(1,0)}{\rm Rees}(I)$. (There are other possible $\mathbb{N}$-gradings
one can put on ${\rm Rees}(I)$, so one wants to distinguish the grading used
when taking the regularity.) We then have:

######  Theorem 4.3 ([7, Theorem 2.5]).

Let $I=\langle f_{1},\ldots,f_{s}\rangle$ be a homogeneous ideal generated by
forms of the same degree. Then $I$ has linear powers if and only if ${\rm
reg}_{(1,0)}{\rm Rees}(I)=0.$

This result is generalized to the case of $I^{s}M$ for a module $M$ in [8].
The “if” direction was first proved by Römer [49, Corollary 5.5].

While Example 4.2 shows that we should not expect arbitrary products of
componentwise linear ideals to be componentwise linear, Conca, De Negri, and
Rossi [13] gave a sufficient condition for this property.

######  Theorem 4.4 ([13, Theorem 2.20]).

Suppose that $I$ and $J$ are componentwise linear, and suppose $d$ is the
smallest degree of a generator of $I$. If $\dim R/\langle I_{d}\rangle\leq 1$,
then $IJ$ is componentwise linear. In particular, if $\dim R/\langle
I_{d}\rangle\leq 1$, then $I^{s}$ is componentwise linear for all $s\geq 1$.

Another way to approach Question 4.1 is to restrict to families of ideals that
are known to be componentwise linear, and check if the powers of ideals within
this family continue to be componentwise linear. One such family of ideals is
the cover ideals of graphs. We recall the relevant definitions and notation.

Let $G=(V,E)$ be a finite simple graph on the vertex set
$V=\\{x_{1},\ldots,x_{n}\\}$ and edge set $E$, which consists of unordered
pairs of distinct elements of $V$. By identifying the vertex $x_{i}\in V$ with
the variable in $x_{i}$ in $R=k[x_{1},\ldots,x_{n}]$, we can associate to $G$
two square-free monomials ideals, the edge ideal

$I(G)=\langle x_{i}x_{j}~{}|~{}\\{x_{i},x_{j}\\}\in E\rangle,$

and the cover ideal

$J(G)=\bigcap_{\\{x_{i},x_{j}\\}\in E}\langle x_{i},x_{j}\rangle.$

The terminology of edge ideal is used to highlight the fact that the minimal
generators of $I(G)$ correspond to the edges of the graph. For the cover
ideal, the minimal generators of $J(G)$ correspond to the minimal vertex
covers of $G$. A subset $W\subseteq V$ is a vertex cover of $G$ if $e\cap
W\neq\emptyset$ for all $e\in E$. Edge and cover ideals give us an algebraic
way to study graphs; for more on these ideals and their properties, see [31,
45, 54, 58]. By restricting to cover and edge ideals, one can exploit the
extra combinatorial information.

######  Example 4.5.

Consider the graph $G$ on five vertices as given in Figure 1. For this graph,
the edge ideal is

$I(G)=\langle x_{1}x_{2},x_{1}x_{3},x_{2}x_{3},x_{3}x_{4},x_{3}x_{5}\rangle,$

and the cover ideal is

$\displaystyle J(G)$ $\displaystyle=$ $\displaystyle\langle
x_{1},x_{2}\rangle\cap\langle x_{1},x_{3}\rangle\cap\langle
x_{2},x_{3}\rangle\cap\langle x_{3},x_{4}\rangle\cap\langle
x_{3},x_{5}\rangle=\langle x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle.$

Both ideals belong to the ring $R=k[x_{1},\ldots,x_{5}]$. Observe that $J(G)$
is our running example. Also, each generator of $J(G)$ corresponds to a vertex
cover of $G$. For example, if we look at the generator $x_{1}x_{3}$, then
every edge of $G$ has either $x_{1}$ or $x_{3}$ as an endpoint.

$x_{1}$$x_{3}$$x_{4}$$x_{5}$$x_{2}$ Figure 1. The graph $G$

It turns out that the cover ideals of chordal graphs provide a large family of
componentwise ideals. Given a graph $G=(V,E)$, the induced subgraph on the set
$W\subseteq V$ is the graph $G_{W}=(W,E_{W})$ where $E_{W}=\\{e\in
E~{}|~{}e\subseteq W\\}$, that is, an edge $e$ of $G$ also belongs to $G_{W}$
if and only if both endpoints of $e$ belong to $W$. A cycle of length $n$ is
the graph

$C_{n}=(\\{x_{1},\ldots,x_{n}\\},\\{\\{x_{1},x_{2}\\},\\{x_{2},x_{3}\\},\ldots,\\{x_{n-1},x_{n}\\},\\{x_{n},x_{1}\\}\\}).$

A graph $G$ is a chordal graph if $G$ has no induced subgraphs isomorphic to a
$C_{n}$ with $n\geq 4$. The graph in Example 4.5 is an example of chordal
graph.

The following result is due to Francisco and Van Tuyl:

######  Theorem 4.6 ([25, Theorem 3.2]).

If $G$ is a chordal graph, then the cover ideal $J(G)$ is componentwise
linear.

Note that the statement in [25] uses the equivalence of Theorem 3.3, that is,
the conclusion is that $\Delta^{\vee}_{J(G)}$ is a sequentially Cohen-Macaulay
simplicial complex. To prove Theorem 4.6, Francisco and Van Tuyl use Theorem
3.10 to show that $J(G)$ is square-free componentwise linear. Other proofs of
Theorem 4.6 exist making use of Theorem 3.9: (1) Van Tuyl and Villarreal [59]
showed that $\Delta_{J(G)}^{\vee}$ is shellable, from which we can deduce that
$J(G)$ is componentwise linear; (2) Erey [18] gave an alternative ordering of
the generators of $J(G)$ to prove that $J(G)$ has the linear quotients; and
(3) Dochtermann and Engström [15], and independently Woodroofe [61], proved
that $\Delta_{J(G)}^{\vee}$ was vertex decomposable when $G$ is chordal, which
implies $\Delta_{J(G)}^{\vee}$ is shellable.

Herzog, Hibi, and Ohsugi [36] were the first to consider the problem of when
powers of componentwise linear ideals are also componentwise linear. Using
Theorem 4.6 as their starting point, they proposed the following conjecture
[36, Conjecture 2.5].

######  Conjecture 4.7 (The Herzog-Hibi-Ohsugi Conjecture).

If $G$ is a chordal graph with cover ideal $J(G)$, then $J(G)^{s}$ is
componentwise linear for all $s\geq 1$.

Part of the difficulty of this conjecture lies in the fact that $J(G)^{s}$ is
no longer a square-free monomial ideal if $s\geq 2$. The majority of the
proofs for Theorem 4.6 as described above rely heavily on the fact that $J(G)$
is a square-free monomial ideal. Answering the Herzog-Hibi-Ohsugi Conjecture
will rely on new techniques for proving a monomial ideal is componentwise
linear.

There is a growing body of families of chordal graphs that satisfy Conjecture
4.7, thus pointing towards the validity of the conjecture. The proof
strategies broadly fall into two categories. One strategy is to find a
description or ordering of the generators of $J(G)^{s}$, and then show that
the ideal has linear quotients to apply Theorem 2.13, or $J(G)^{s}$ satisfies
a property like being weakly polymatroidal and apply a result like Theorem
3.14. The second strategy is to use properties of the Rees algebra, as in
Theorem 4.3. Current attacks on the conjecture have also exploited the
structure of chordal graphs. For example, some approaches exploit chordal
graphs with “lots” of edges, e.g., a graph with a large complete graph
(defined below) as a subgraph. At the other extreme, the conjecture has been
investigated for chordal graphs with “few” edges, e.g., trees or graphs with
very rigid structure.

Herzog, Hibi, and Ohsugi provided the initial evidence for the validity of
Conjecture 4.7. Under the extra assumption that $J(G)$ has a linear resolution
(which is equivalent to the fact that $\Delta_{J(G)}^{\vee}$ is a Cohen-
Macaulay simplicial complex by Theorem 3.5), they show that ${\rm
reg}_{(1,0)}R(J(G))=0$, and use the approach of Theorem 4.3 to verify
Conjecture 4.7 for all cover ideals of chordal graphs with a linear
resolution.

######  Theorem 4.8 ([36, Theorem 2.7]).

If $G$ is a chordal graph such that cover ideal $J(G)$ has a linear
resolution, then $J(G)^{s}$ is componentwise linear for all $s\geq 1$.

We need to introduce some special classes of chordal graphs. The complete
graph $K_{n}$ on $n$ vertices is the graph with vertex set
$\\{x_{1},\ldots,x_{n}\\}$ and edge set $\\{\\{x_{i},x_{j}\\}~{}|~{}1\leq
i<j\leq n\\}$. We call a graph $G$ on $n+m$ vertices a star graph based on
$K_{n}$ if the vertices of $G$ can be relabeled so that the induced graph on
$\\{x_{1},\ldots,x_{n}\\}$ is the complete graph $K_{n}$, and for any $n\leq
i<j\leq n+m$, the edge $\\{x_{i},x_{j}\\}\not\in E$. The graph in Figure 4.5
is an example of a star graph based on $K_{3}$ since the induced graph on
$\\{x_{1},x_{2},x_{3}\\}$ is a $K_{3}$.

Mohammadi [42] introduced a wider class of ideals that generalized this
construction that were called generalized star graphs. While we will not
recall this construction, the idea is similar in that one glues together a
collection of complete graphs in a prescribed way to form a “core”, and then
one is allowed to attach some extra edges. We then have the following result.

######  Theorem 4.9.

Conjecture 4.7 is true for the cover ideals of the following chordal graphs:

  1. (1)

[36, Theorem 2.3] Star graphs based on $K_{n}$.

  2. (2)

[42, Theorem 1.5] Generalized star graphs.

######  Example 4.10.

For any complete graph $K_{n}$, the cover ideal $J(K_{n})$ has a linear
resolution. This can be checked directly from the fact that $J(K_{n})=\langle
x_{1}\cdots\widehat{x_{i}}\cdots x_{n}~{}|~{}1\leq i\leq n\rangle$, and that
this ideal has linear quotients. So all powers of $J(K_{n})$ are componentwise
linear.

As a second example, the graph in Figure 4.5 is a star graph based on $K_{3}$.
Consequently, the ideal of our running example, that is, $J(G)=\langle
x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle$ has the property that all
of its powers has the componentwise linear property.

Herzog, Hibi, Ohsugi’s proof of Theorem 4.9 uses properties of the Rees
algebra $R(J(G))$. Mohammadi gives a different proof for star graphs based on
$K_{n}$ that shows for each graph $G$ in this family, the ideal $J(G)^{s}$ is
weakly polymatroidal. In fact, the following theorem gives a combinatorial way
to check if all powers of $J(G)$ are componentwise linear. Given a graph
$G=(V,E)$, the clique complex of $G$ is the simplicial complex

${\rm Cliq}(G)=\langle W\subseteq V~{}|~{}\mbox{$G_{W}=K_{|W|}$}\rangle,$

in other words, the clique complex consists of all the subsets of $V$ such
that induced graph on that subset is a complete graph. For any simplicial
complex $\Delta$, a facet $F\in\Delta$ has a free vertex if there is some
vertex $x_{i}\in F$ that only appears in $F$, but no other facet. We denote
the set of all facets of $\Delta$ with a free vertex by
$\mathcal{FV}(\Delta)$. We then have the following tool:

######  Theorem 4.11 ([43, Corollary 2.4]).

Let $G=(V,E)$ be a chordal graph with clique complex ${\rm Cliq}(G)$. If
$|V|-1\leq|\bigcup_{F\in\mathcal{FV}({\rm Cliq}(G))}F|$, then the ideal
$J(G)^{s}$ is componentwise linear for all $s\geq 1$.

######  Example 4.12.

The graph $G$ in Figure 1 has clique complex

${\rm
Cliq}(G)=\langle\\{x_{1},x_{2},x_{3}\\},\\{x_{3},x_{4}\\},\\{x_{3},x_{5}\\}\rangle.$

Note that each facet has a free vertex: $x_{1},x_{2}$ only appear in the first
facet, while $x_{4}$ only appears in the second, and $x_{5}$ only in the
third. Since

$|V|-1=4\leq|\\{x_{1},x_{2},x_{3}\\}\cup\\{x_{3},x_{4}\\}\cup\\{x_{3},x_{5}\\}|=5.$

all powers of the cover ideal $J(G)$ are componentwise linear by Theorem 4.11.

Erey (see [19, 20]) approached Conjecture 4.7 by finding an order of the
minimal generators of $J(G)^{s}$ that gives linear quotients. In the statement
below, the path graph $P_{n}$ is the graph with vertex set
$V=\\{x_{1},\ldots,x_{n}\\}$ and edge set
$E=\\{\\{x_{1},x_{2}\\},\\{x_{2},x_{3}\\},\ldots,\\{x_{n-1},x_{n}\\}\\}$. A
graph is $(C_{4},2K_{2})$-free if it has no induced graph isomorphic to
$C_{4}$ or two copies of $K_{2}$.

######  Theorem 4.13.

Conjecture 4.7 is true for the cover ideals of the following chordal graphs:

  1. (1)

[19, Theorem 3.7] Chordal graphs that are also $(C_{4},2K_{2})$-free graphs.

  2. (2)

[20, Theorem 4.3] The path graphs $P_{n}$.

######  Remark 4.14.

Erey showed a stronger result in [19, Theorem 3.7], namely, the cover ideal
$J(G)$ of any $(C_{4},2K_{2})$-free graph $G$ satisfies the property that
$J(G)^{s}$ is componentwise linear for all $s\geq 1$. Note that the cycle
$C_{5}$ is a $(C_{4},2K_{2})$-free graph that is not chordal.

######  Remark 4.15.

As an intermediate step, Erey and Qureshi first proved that $J(P_{n})^{2}$ was
componentwise linear in [21, Theorem 5.1]. Erey was later able to extend this
result to all powers, as noted above.

Herzog, Hibi, and Moradi were also able to prove that the same result for
$P_{n}$ as a consequence of a more general result, which again uses the Rees
algebra. We recall how one can construct the Rees algebra of an ideal
$I=\langle f_{1},\ldots,f_{s}\rangle\subseteq R=k[x_{1},\ldots,x_{n}]$ when
$I$ is not necessarily generated by terms of the same degree. In the ring
$R[t]$, consider the subring

$R[It]=k[f_{1}t,f_{2}t,\ldots,f_{s}t]\subseteq R[t].$

Let $S=k[y_{1},\ldots,y_{s},x_{1},\ldots,x_{n}]$. We define a $k$-algebra
homomorphism $\varphi:S\rightarrow R$ by

$y_{i}\mapsto f_{i}t~{}~{}\mbox{and}~{}~{}x_{j}\mapsto x_{j}$

for $i=1,\ldots,s$ and $j=1,\ldots,n$. Let $J={\rm ker}{\varphi}$. The ideal
$J$ is called the defining ideal of the Rees ring ${\rm Rees}(I)$. A new
criteria for when an ideal is componentwise linear is then given in terms of
$J$.

######  Theorem 4.16 ([35, Theorem 2.6]).

Let $I$ be a monomial ideal and let $J$ be the defining ideal of the Rees ring
${\rm Rees}(I)$. There exists a monomial order $>$ such that if the initial
ideal ${\rm in}_{>}(J)$ is generated by quadratic monomials with respect to
this order, then $I^{s}$ is componentwise linear for all $s\geq 1$.

The definition of the required monomial order can be found in [35]. Using
Theorem 4.16, Herzog, Hibi, and Moradi were able to verify Conjecture 4.7 for
more families of chordal families. A biclique graphs is a graph on the vertex
set $\\{x_{1},\ldots,x_{p},y_{1},\ldots,y_{q},z_{1},\ldots,z_{r}\\}$ such that
the induced graphs on $\\{x_{1},\ldots,x_{p},y_{1},\ldots,y_{q}\\}$ and on
$\\{y_{1},\ldots,y_{q},z_{1},\ldots,z_{r}\\}$ are complete graphs. Cameron-
Walker graphs are graphs whose induced matching number equals its matching
number; we do not formally define this family here, but point the reader to
[40].

######  Theorem 4.17 ([35, Corollary 4.7]).

Conjecture 4.7 is true for the cover ideals of the following chordal graphs:

  1. (1)

Biclique graphs.

  2. (2)

The path graphs $P_{n}$.

  3. (3)

Cameron-Walker graphs whose bipartite graph is a complete bipartite graph.

Kumar and Kumar have recently shown that Conjecture 4.7 holds for all trees.
Trees are graphs which have no induced cycles, and thus, they are examples of
chordal graphs. Kumar and Kumar’s proof uses a different strategy then the
above results. In the case that $G$ is a tree, $J(G)^{s}$ equals its
$s$-symbolic power (to be defined in the next section). It can the be shown
that $(J(G)^{s})^{\rm pol}$, the polarization of $J(G)^{s}$, is the cover
ideal of another graph. The proof for trees then shows that the cover ideal of
this new graph is also componentwise linear. This strategy will be expanded
upon in more detail when we look at symbolic powers in the next section.

######  Theorem 4.18 ([41, Corollary 3.5]).

Conjecture 4.7 is true for the cover ideals of all trees.

######  Remark 4.19.

The above result is slightly stronger since it is shown that $J(G)^{s}$ has
linear quotients for all $s\geq 0$ when $G$ is a tree.

We round out this section by describing three results not directly related to
Conjecture 4.7, but related to the more general theme of Question 4.1. The
first result concerns the cover ideals of graphs that may not be chordal. The
second result concerns edge ideals, not cover ideals, and the third is for
quadratic monomial ideals (which includes all edge ideals).

A graph $G$ is bipartite if the vertex set of $G$ can partitioned as
$V=V_{1}\cup V_{2}$ so that for every edge $e\in E$, one has $e\cap
V_{1}\neq\emptyset$ and $e\cap V_{2}\neq\emptyset$. That is, every edge has
one endpoint in $V_{1}$ and the other in $V_{2}$. We then have the following
result for the cover ideals of bipartite graphs; an earlier version of this
result appeared in [44, Theorem 2.2].

######  Theorem 4.20 ([53, Corollary 3.7]).

Let $G$ be a bipartite graph. Then $J(G)$ has linear powers if and only if
$J(G)$ has a linear resolution.

As is evident from this section, the majority of work on powers of
componentwise linear ideals has focused on cover ideals. Of course, similar
questions could be asked about edge ideals. To-date, the primary focus has
been on the linear property, as demonstrated in the next two results.

######  Theorem 4.21 ([19, Theorem 2.12]).

Let $G$ be a $(C_{4},2K_{2})$-free graph with edge ideal $I(G)$. Then
$I(G)^{s}$ is componentwise linear for all $s\geq 2$; in fact, $I(G)^{s}$ has
a linear resolution for all $s\geq 2$.

Note that in the above result the ideal $I(G)$ may or may not be componentwise
linear, but its powers are. Our final result looks at quadratic monomial
ideals that need not be square-free.

######  Theorem 4.22 ([37, Theorem 3.2]).

Let $I$ be a quadratic monomial ideal. Then $I$ has linear powers if and only
if $I$ has a linear resolution.

##  5\. Componentwise linearity of symbolic powers

In this section, we move beyond the Herzog-Hibi-Ohsugi conjecture to address
the question of when _symbolic_ powers of an ideal are componentwise linear.
For any arbitrary ideal $I\subseteq R$, the $s$-th symbolic power of $I$ is
the ideal

$I^{(s)}=\bigcap_{P\in{\rm Ass}(I)}(I^{s}R_{P}\cap R)$

where ${\rm Ass}(I)$ is the set of associated primes of $I$ and $R_{P}$ is the
ring $R$ localized at the prime ideal $P$. In the case that $I$ is a square-
free monomial ideal with primary decomposition $I=P_{1}\cap\cdots\cap P_{r}$,
its $s$-th symbolic power is given by

$I^{(s)}=P_{1}^{s}\cap\cdots\cap P_{r}^{s}.$

In particular, the $s$-th symbolic power of the cover ideal of a graph $G$
satisfies

$J(G)^{(s)}=\bigcap_{\\{x_{i},x_{j}\\}\in E}\langle x_{i},x_{j}\rangle^{s}.$

We survey a number of recent articles focusing on the class of cover ideals of
graphs that have addressed the following umbrella question:

######  Question 5.1.

For which graphs $G$ is $J(G)^{(s)}$ componentwise linear for all $s\geq 1$?

Going forward, we will employ the following terminology. For a graph $G$, its
_independent complex_ , denoted by $\text{Ind}(G)$, is the simplicial complex
whose faces are independent sets in $G$. A set $A\subseteq V$ is an
independent set if for all $e\in E$, $e\not\subseteq A$. Equivalently,
$V\setminus A$ is a vertex cover. It is not hard to show that
$\text{Ind}(G)=\Delta_{J(G)}^{\vee}$.

######  Definition 5.2.

A graph $G$ is vertex decomposable, respectively shellable, if its independent
complex ${\rm Ind}(G)$ is vertex decomposable, respectively shellable.

Note that by Theorem 3.9, if a graph $G$ is vertex decomposable or shellable,
then $J(G)$ is componentwise linear.

When $G$ is a bipartite graph, it is known (cf. [27]) that
$J(G)^{s}=J(G)^{(s)}$ for all $s\geq 1$, and so Question 5.1 reduces to the
question of when regular powers of the cover ideal of a graph are
componentwise linear — this question has been discussed in the previous
section and is closely related to the Herzog-Hibi-Ohsugi conjecture. In this
case, by combining previous work of Seyed Fakhari [53] and of Selvaraja and
Skelton [52], one obtains the following result.

######  Theorem 5.3 ([53, Theorem 3.6 and Corollary 3.7] and [52, Theorem
5.3]).

Let $G$ be a bipartite graph, and thus $J(G)^{s}=J(G)^{(s)}$.

  1. (1)

The following are equivalent:

    1. (a)

$J(G)^{s}$ is componentwise linear for all $s\geq 1$,

    2. (b)

$J(G)^{s}$ is componentwise linear for some $s>1$,

    3. (c)

$J(G)^{s}$ has linear quotients for all $s\geq 1$,

    4. (d)

$G$ is a vertex decomposable graph.

  2. (2)

The following are equivalent:

    1. (a)

$J(G)^{s}$ has a linear resolution for all $s\geq 1$,

    2. (b)

$J(G)$ has a linear resolution,

    3. (c)

$G$ is a pure vertex decomposable graph (i.e., ${\rm
Ind}(\Delta)=\Delta^{\vee}_{J(G)}$ is also a pure simplicial complex)

For an arbitrary graph $G$, the general approach to investigate symbolic
powers of the cover ideal $J(G)$ is to view the polarization of these symbolic
powers as the cover ideals of other graphs constructed from $G$. Particularly,
the following constructions, due to Seyed Fakhari [53] and Kumar and Kumar
[41], have proved to be essential in this line of work.

######  Construction 5.4 (Duplicating vertices).

Let $G$ be a graph over the vertex set $V_{G}=\\{x_{1},\dots,x_{n}\\}$ and let
$s\geq 1$ be an integer. We construct a new graph, denoted by $G_{s}$, as
follows:

$\displaystyle V_{G_{s}}$ $\displaystyle=\\{x_{i,p}~{}\big{|}~{}1\leq i\leq
n,\ 1\leq p\leq s\\},\text{ and }$ $\displaystyle E_{G_{s}}$
$\displaystyle=\left\\{\\{x_{i,p},x_{j,q}\\}~{}\big{|}~{}\\{x_{i},x_{j}\\}\in
E_{G}\text{ and }p+q\leq s+1\right\\}.$

######  Construction 5.5 (Duplicating edges).

Let $G$ be a graph with vertex set $V_{G}=\\{x_{1},\dots,x_{n}\\}$ and edge
set $E_{G}=\\{e_{1},\dots,e_{m}\\}$.

  1. (1)

Let $r\in{\mathbb{Z}}_{\geq 0}$ and $e=\\{x_{i},x_{j}\\}\in E_{G}$. Set

$\displaystyle V(e(r))$
$\displaystyle=\\{x_{l,p}~{}\big{|}~{}l\in\\{i,j\\}\text{ and }1\leq p\leq
r\\},\text{ and }$ $\displaystyle E(e(r))$
$\displaystyle=\left\\{\\{x_{i,p},x_{j,q}\\}~{}\big{|}~{}p+q\leq
r+1\right\\}.$

  2. (2)

For an ordered tuple $(s_{1},\dots,s_{m})\in{\mathbb{Z}}^{m}_{\geq 0}$, we
construct a new graph, denoted by $G(s_{1},\dots,s_{m})$, as follows:

$\displaystyle V_{G(s_{1},\dots,s_{m})}$
$\displaystyle=\bigcup_{i=1}^{m}V(e_{i}(s_{i})),\text{ and }$ $\displaystyle
E_{G(s_{1},\dots,s_{m})}$ $\displaystyle=\bigcup_{i=1}^{m}E(e_{i}(s_{i})).$

Obviously, for $s_{1}=\dots=s_{m}=s$, we have $G(s_{1},\dots,s_{m})=G_{s}$.
The use of Constructions 5.4 and 5.5 is reflected in the following lemma.

######  Lemma 5.6 ([53, Lemma 3.4]).

Let $G$ be a graph and let $J(G)$ be its cover ideal. For any integer $s\geq
1$, the polarization $\left(J(G)^{(s)}\right)^{\operatorname{pol}}$ coincides
with the cover ideal of $G_{s}$.

######  Example 5.7.

We illustrate the above ideas by using the graph of Example 4.5 for $s=2$. The
graph $G_{2}$ is then given in Figure 2.

$x_{1,1}$$x_{3,1}$$x_{4,1}$$x_{5,1}$$x_{2,1}$$x_{1,2}$$x_{3,2}$$x_{4,2}$$x_{5,2}$$x_{2,2}$
Figure 2. The graph $G_{2}$ constructed from the graph $G$ in Figure 1

Note that $J(G)=\langle x_{1}x_{3},x_{2}x_{3},x_{1}x_{2}x_{4}x_{5}\rangle$, so
the ideal $J(G)^{(2)}$ is given by

$J(G)^{(2)}=\langle
x_{1}^{2}x_{3}^{2},x_{1}x_{2}x_{3}^{2},x_{2}^{2}x_{3}^{2},x_{1}x_{2}x_{3}x_{4}x_{5},x_{1}^{2}x_{2}^{2}x_{4}^{2}x_{5}^{2}\rangle.$

The polarization of $J(G)^{(2)}$ is then the ideal

$\displaystyle(J(G)^{(2)})^{\rm pol}$ $\displaystyle=$ $\displaystyle\langle
x_{1,1}x_{1,2}x_{3,1}x_{3,2},x_{1,1}x_{2,1}x_{3,1}x_{3,2},x_{2,1}x_{2,2}x_{3,1}x_{3,2},$
$\displaystyle
x_{1,1}x_{2,1}x_{3,1}x_{4,1}x_{5,1},x_{1,1}x_{1,2}x_{2,1}x_{2,2}x_{4,1}x_{4,2}x_{5,1}x_{5,2}\rangle.$

This ideal then satisfies $(J(G)^{(2)})^{\rm pol}=J(G_{2})$.

Lemma 5.6 fits into the context of studying the componentwise linearity of
symbolic powers of the cover ideal $J(G)$ via the following result. It allows
us to, instead of looking at the componentwise linearity of $J(G)^{(s)}$,
consider when $G_{s}$ is vertex decomposable, which is a combinatorial
property and could be more natural to examine.

######  Lemma 5.8.

Let $G$ be a graph and let $s\in{\mathbb{N}}$. If $G_{s}$ is vertex
decomposable, then $J(G_{s})$ has linear quotients. Particularly, if $G_{s}$
is vertex decomposable, then $J(G)^{(s)}$ has linear quotients and is
componentwise linear.

###### Proof.

By Theorem 3.9, we know that if $G_{s}$ is vertex decomposable, then
$J(G_{s})$ has linear quotients. Thus, Lemma 5.6 implies $(J(G^{(s)})^{\rm
pol}$ has linear quotients. This, together with Theorem 3.12 implies that
$J(G)^{(s)}$ has linear quotients. The last statement follows from Theorem
2.13. ∎

By applying Lemma 5.8, Seyed Fakhari [53], Selvaraja [57], and Kumar and Kumar
[41] showed that the following special classes of graphs $G$ have the property
that $J(G)^{(s)}$ is componentwise linear for any $s\geq 1$.

######  Theorem 5.9.

Let $G$ be a graph.

  1. (1)

[53, Theorem 3.6] If $G$ is very well-covered and $J(G)$ has a linear
resolution, then $J(G)^{(s)}$ has linear quotients for all $s\geq 1$.

  2. (2)

[51, Corollary 4.7] If $G$ is a Cameron-Walker graph, then $J(G)^{(s)}$ has
linear quotients for all $s\geq 1$.

  3. (3)

[41, Theorem 3.4 and Corollary 3.5] If $G$ is a tree on $n$ vertices, then for
any tuple $(s_{1},\dots,s_{n-1})\in{\mathbb{Z}}^{n-1}_{\geq 0}$,
$G(s_{1},\dots,s_{n-1})$ is a vertex decomposable graph. Particularly,
$J(G)^{s}=J(G)^{(s)}$ has linear quotients for all $s\geq 1$.

  4. (4)

[41, Corollary 4.5] If $G$ is a uni-cyclic vertex decomposable graph, then
$J(G)^{(s)}$ is componentwise linear for all $s\geq 1$.

In the above statement, a graph is uni-cyclic if the graph has only one
induced cycle, and a graph $G$ is very well-covered if all of its maximal
vertex covers have cardinality $\frac{1}{2}|V|$.

Seyed Fakhari [55] improved his previous result [53, Theorem 3.6] (see Theorem
5.9 (1)). In particular, he classified all the graphs whose symbolic powers
have a linear resolution.

######  Theorem 5.10 ([55, Theorem 3.4]).

Let $G$ be a graph with no isolated vertices. Then the following are
equivalent:

  1. (1)

$J(G)^{(s)}$ has a linear resolution for all $s\geq 1$,

  2. (2)

$J(G)^{(s)}$ has a linear resolution for some $s\geq 1$, and

  3. (3)

$G$ is very well-covered and ${\rm Ind}(G)$ is Cohen-Macaulay.

In a different approach, Selvaraja and Skelton [52] gave the following
sufficient condition for $J(G)^{(s)}$ to fail componentwise linearity for all
$s\geq 1$.

######  Theorem 5.11 ([52, Theorem 3.1]).

Let $G$ be graph, and suppose that $J(G)^{(s)}$ is not componentwise linear
for either $s=1$ and $2$, or $s=2$ and $3$. Then $J(G)^{(s)}$ is not
componentwise linear for all $s\geq 2$.

The strategy to prove Theorem 5.11 is to use Seyed Fakhari’s construction of
$G_{s}$, and then show that in the inductive hypothesis of $G_{s}$ being
vertex decomposable, a subgraph obtained from $G_{s}$ by removing the
neighbours of a shedding vertex is isomorphic to $G_{s-2}$.

In the same spirit, finding conditions so that $J(G)^{(s)}$ fails to be
componentwise linear for all $s\geq 1$, Selvaraja and Skelton [52] gave the
following result. Note that for a vertex decomposable graph $G$ with a
shedding sequence $x_{\alpha(1)},\dots,x_{\alpha(l)}$, let
$\\{x_{\gamma(1)},\dots,x_{\gamma(r)}\\}$ be the collection of isolated
vertices remaining in $G\setminus\\{x_{\alpha(1)},\dots,x_{\alpha(l)}\\}$. The
_spanning bipartite graph_ ${\mathcal{B}}_{G}$ is defined to be the bipartite
graph with the bipartition of the vertices
$\\{x_{\alpha(1)},\dots,x_{\alpha(l)}\\}\cup\\{x_{\gamma(1)},\dots,x_{\gamma(r)}\\}$
and edges

$\\{\\{x_{\alpha(i)},x_{\gamma(j)}\\}~{}\big{|}~{}\\{x_{\alpha(i)},x_{\gamma(j)}\\}\in
E_{G},1\leq i\leq l\text{ and }1\leq j\leq r\\}.$

######  Theorem 5.12 ([52, Theorem 3.6]).

Let $G$ be a vertex decomposable graph. If there exists an independent set $A$
such that $\mathcal{B}_{G\setminus N[A]}$ is not vertex decomposable, then
$J(G)^{(s)}$ is not componentwise linear for all $s\geq 2$.

The necessary condition in Theorem 5.12 is also sufficient to achieve the
componentwise linearity of $J(G)^{(s)}$, for all $s\geq 1$, for a special
class of vertex decomposable graphs, namely, the class of $W$-graphs.
Selvaraja and Skelton [52] defined a _$W$ -graph_ $G$ to be graph such that
$G\setminus N[A]$ has a simplicial vertex for any independent set $A$. A
vertex $x$ is a simplicial vertex if the induced graph on $x$ and all of its
neighbors is a complete graph.

######  Theorem 5.13 ([52, Theorem 4.2]).

Let $G$ be a $W$-graph. Then the following are equivalent:

  1. (1)

$\mathcal{B}_{G\setminus N[A]}$ is vertex decomposable for any independent set
$A$ in $G$.

  2. (2)

$J(G)^{(s)}$ is componentwise linear for all $s\geq 1$.

  3. (3)

$J(G)^{(s)}$ is componentwise linear for some $s\geq 2$.

In addressing Question 5.1 and identifying new classes of graphs for which all
symbolic powers of the cover ideal are componentwise linear, the following
approach has been investigated: combinatorially modify a given graph $G$ to
obtain a new graph $G^{\prime}$ with the required property that
$J(G^{\prime})^{(s)}$ is componentwise linear for any $s\geq 1$. Specifically,
originating from Villarreal’s work [60], the process of adding _whiskers_ (or
_whiskering_) to the vertices of a graph has been studied and developed by
many authors and from various directions (cf. [3, 4, 14, 16, 24, 30, 41, 51,
52]).

######  Definition 5.14.

By adding a _whisker_ to a vertex $x$ of a graph $G$, one adds a new vertex
$y$ and the edge $\\{x,y\\}$ to $G$. Let $S\subseteq V_{G}$ be a subset of the
vertices in $G$. Then we denote by $G\cup W(S)$ the graph obtained by adding a
whisker to $G$ at each vertex in $S$.

The first result in this approach to Question 5.1 is due to Dung, Hien, Nguyen
and Trung [16], which shows that by adding a whisker to every vertex of any
given graph one obtains a new graph with the desired property. The case $s=1$
of the following theorem in fact implies Villarreal’s result in [60].

######  Theorem 5.15 ([16, Corollary 5.9]).

Let $G$ be a graph and let $H=G\cup W(V_{G})$ be the graph obtained by adding
a whisker at every vertex in $G$. Then $J(H)^{(s)}$ is componentwise linear
for all $s\geq 1$.

Dung, Hien, Nguyen, and Trung, in fact, proved a stronger statement than
Theorem 5.15 in [16, Theorem 5.7], where they showed that the same conclusion
holds if at least one whisker is added to every vertex of $G$.

Cook and Nagel [14] generalized the process of whiskering to that of _clique-
whiskering_ to extend Villarreal’s previous work [60].

######  Definition 5.16.

Let $G$ be a graph. A _clique partition_ $\pi$ of $G$ is a partition of the
vertices in $G$ into disjoint (possibly empty) subsets $W_{1},\dots,W_{t}$
such that the induced graphs $G_{W_{i}}$ is a complete (or empty) graph in $G$
for all $i=1,\dots,t$. A _clique-whiskering_ of $G$ associated to a clique-
partition $\pi$, denoted by $G^{\pi}$, is a graph over the vertices
$V_{G^{\pi}}=V_{G}\cup\\{y_{1},\dots,y_{t}\\}$ and has edges

$E_{G^{\pi}}=E_{G}\cup\left(\bigcup_{i=1}^{t}\left\\{\\{v,y_{i}\\}~{}\big{|}~{}v\in
W_{i}\right\\}\right).$

Selvaraja [51] proved the following theorem, of which the case where $s=1$ was
known in the previous work of Cook and Nagel [14].

######  Theorem 5.17 ([51, Theorem 4.9]).

Let $G$ be a graph and let $\pi$ be a clique vertex partition of $G$. Then
$J(G^{\pi})^{(s)}$ has linear quotients for any $s\geq 1$.

Inspired by Theorem 5.15, the following question arises naturally: for which
subset $S\subseteq V_{G}$ of the vertices in a graph $G$ do we have that
$J(G\cup W(S))^{(s)}$ is componentwise linear for all $s\in{\mathbb{N}}$? A
number of special configurations of such subsets $S$ have been identified. The
case $s=1$ in the following result of Selvaraja [51] was already known by
Francisco and Hà in [24].

######  Theorem 5.18 ([51, Corollary 4.5]).

Let $G$ be graph and let $S\subset V_{G}$ be a vertex cover of $G$. Then,
$J(G\cup W(S))^{(s)}$ has linear quotients for any $s\geq 1$.

As a consequence to Theorem 5.18, Selvaraja and Skelton [52] obtained the
following corollary; the case $s=1$ was again known in [24].

######  Corollary 5.19 ([52, Corollary 4.6]).

Let $G$ be a graph and $S\subseteq G$. If $|S|\geq|V(G)|-3$, then $J(G\cup
W(S))^{(s)}$ is componentwise linear for all $s\geq 1$.

The condition that $S$ is a vertex cover in Theorem 5.18 is improved by Gu,
Hà, and Skelton [30]. We call a subset $S$ of the vertices in $G$ a _cycle
cover_ if every cycle in $G$ contains at least a vertex in $S$. A vertex cover
is necessarily a cycle cover, but contains a lot more vertices in general.

######  Theorem 5.20 ([30, Theorem 3.10]).

Let $G$ be a graph and let $S$ be a cycle cover of $G$. Let $H$ be the graph
obtained by adding at least one whisker to each vertex in $S$. Then
$J(H)^{(s)}$ is componentwise linear for all $s\geq 1$.

Theorem 5.20 is slightly generalized further in [30, Theorem 4.6], where it is
shown that instead of adding just a whisker at each vertex in $S$, one can add
a non-pure star complete graph, a graph constructed by adjoining complete
graphs of different sizes at a single vertex, which has at least one whisker
(see [30] for more details).

We round out this section by pointing out that there has been little work into
the case of symbolic powers of edge ideals. We complete this paper with one
result in this direction that makes use of Theorem 3.15.

######  Theorem 5.21.

Let $G$ be a graph and suppose that $W_{1},\ldots,W_{t}$ is a complete list of
minimal vertex covers. Suppose that $W_{i}\cup W_{j}=V$ for all $i\neq j$.
Then $I(G)^{(s)}$ is componentwise linear for all $s\geq 1$.

###### Proof.

Suppose that $W_{1},\ldots,W_{t}$ is a complete list of minimal vertex covers
of $G$. The edge ideal $I(G)$ then has the following primary decomposition

$I(G)=\langle x~{}|~{}x\in W_{1}\rangle\cap\cdots\cap\langle x~{}|~{}x\in
W_{t}\rangle.$

For a proof, see [58, Corollary 1.35]. For a variable $x_{i}$, let ${\rm
supp}(x_{i})=\\{i\\}$. Note we can rewrite $\langle x~{}|~{}x\in
W_{i}\rangle=\mathfrak{m}_{J_{i}}$ as a Veronese ideal with $J_{i}=\\{{\rm
supp}(x)~{}|~{}x\in W_{i}\\}\subseteq[n]$. Thus, since $I(G)$ is a square-free
monomial ideal, we can write the $s$-th symbolic powers of $I(G)$ as

$I(G)^{(s)}=\mathfrak{m}_{J_{1}}^{s}\cap\cdots\cap\mathfrak{m}_{J_{t}}^{s}.$

The conclusion now follows by Theorem 3.15. ∎

A graph $G$ is a complete $m$-partite graph if the vertices $V$ can be
partitioned as $V=V_{1}\cup V_{2}\cup\cdots\cup V_{m}$ such that for all
$i\neq j$, if $x\in V_{i}$ and $y\in V_{j}$, then $\\{x,y\\}\in E$. Note that
the complete graph $K_{n}$ is the complete $n$-partite graph with
$V_{i}=\\{x_{i}\\}$ for all $i$. We then have the following result.

######  Corollary 5.22.

Let $G$ be a complete $m$-partite graph (for any $m\geq 2$). Then $I(G)^{(s)}$
is componentwise linear for any $s\geq 1$.

###### Proof.

If $V=V_{1}\cup V_{2}\cup\cdots\cup V_{m}$ is the partition of $V$, then the
minimal vertex covers of $G$ have the form
$W_{i}=V_{1}\cup\cdots\cup\widehat{V}_{i}\cup\cdots\cup V_{m}$ for
$i=1,\ldots,m$, where we mean $V_{i}$ is omitted. Now apply Theorem 5.21. ∎

##  6\. Future research directions

We finish this paper with some problems which we hope will generate future
work.

A natural way to generalize Herzog, Hibi, and Ohsugi’s conjecture is consider
objects more general than chordal graphs. Over the last decade, there has been
interest in generalizing the property of chordality of graphs to simplicial
complexes (see, for example, [1] and references therein). The notion of a
cover ideal can be generalized to simplicial complexes as follows: given a
simplicial complex $\Delta$, the cover ideal of $\Delta$ is

$J(\Delta)=\langle x_{i_{1}}\cdots
x_{i_{s}}~{}|~{}\\{x_{i_{1}},\ldots,x_{i_{s}}\\}\cap
F\neq\emptyset~{}~{}\mbox{for all facets $F\in\Delta$}\rangle.$

The following question then generalizes the Herzog-Hibi-Ohsugi Conjecture:

######  Question 6.1.

Let $\Delta$ be a “chordal” simplicial complex (using an appropriate
definition of chordal) with cover ideal $J(\Delta)$. Is $J(\Delta)$
componentwise linear? Is $J(\Delta)^{s}$ componentwise linear for all $s\geq
1?$ We can also ask similar questions for $J(\Delta)^{(s)}$.

We have put chordal in quotes since it is not clear which generalization of
chordality one will want to use. A starting point to attack this question
would be the case of simplicial trees, as defined by Faridi [23]. In fact,
when $\Delta$ is a simplicial tree, Faridi has already shown that $J(\Delta)$
is componentwise linear (see [23, Corollary 5.5]).

Theorems 4.8 and 4.20 show that if $G$ is a chordal or bipartite graph such
that $J(G)$ has a linear resolution, then all powers of $J(G)$ have a linear
resolution. This leads to the following question, which was first posed by
Mohammadi [43, Question 4.1].

######  Question 6.2.

Let $G$ be a graph such that the cover ideal $J(G)$ has a linear resolution.
Does $J(G)^{s}$ have a linear resolution for all $s\geq 1$? If not, what
hypotheses are needed on $G$ to give this conclusion?

Mohammadi has shown that the previous question is true for cactus graphs (see
[43, Theorem 4.3]). A cactus graph is one where each edge belongs to at most
one induced cycle in the graph. We are not aware of any other families of
graphs for which there is a positive (or negative!) answer to Question 6.2.

As noted in Section 4, we do not know of many examples for edge ideals whose
powers are componentwise linear. We formalize this as a question.

######  Question 6.3.

Let $G$ be a graph with edge ideal $I(G)$. What properties on $G$ imply that
$I(G)^{s}$ is componentwise linear? Similarly, what conditions imply
$I(G)^{(s)}$ is componentwise linear?

Observe that $I(G)^{s}$ is generated in a single degree. Thus, for $I(G)^{s}$
to be componentwise linear, it needs to have a linear resolution. Hence,
Question 6.3 for $I(G)^{s}$ reduces to asking when a power of an edge ideal
has a linear resolution. See Peeva and Nevo [46] for some work in this
direction.

We also add a question from Selvaraja and Skelton’s work (see [52, Question
5.8]).

######  Question 6.4.

Let $G$ be a vertex decomposable graph.

  1. (1)

If $J(G)^{(2)}$ is not componentwise linear, is it true that $J(G)^{(s)}$ is
not componentwise linear for all $s\geq 3$?

  2. (2)

If $\mathcal{B}_{G\setminus N[A]}$ is vertex decomposable for any independent
set $A$, is $J(G)^{(s)}$ a componentwise linear ideal for all $s\geq 2$?

We consider another question inspired by Selvaraja and Skelton’s work, namely,
Theorem 5.11. As was shown in this theorem, one can determine if $J(G)^{(s)}$
is not componentwise linear by checking for small values of $s$. This leads to
a much more general question:

######  Question 6.5.

Suppose that $I$ is an ideal that is componentwise linear. Does there exists
an integer $t\geq 1$ such that if $I^{(i)}$ is componentwise linear for all
$1\leq i\leq t$, then $I^{(s)}$ is componentwise linear for all $s\geq t$?

An answer to the above question even in the case that $I=I(G)$ or $J(G)$ would
be of great interest, especially if the value of $t$ is related to a graph
invariant. Note that the analogous question for regular powers has a negative
answer. In particular, Conca [12, Theorem 3.1] showed that for any integer
$d>1$, there exists an ideal $I(d)$ such that $I(d)^{k}$ has a linear
resolution (and hence, is componentwise linear) for all $1\leq k<d$, but
$I(d)^{d}$ does not have a linear resolution (and hence, is not componentwise
linear).

As seen in Theorems 5.15, 5.17, and 5.20, the operation of whiskering can turn
a graph $G$ into a new graph $H$ such that $J(H)^{(s)}$ is componentwise
linear for all $s\geq 1$. We know of no similar results for the regular
powers. We thus pose the following question:

######  Question 6.6.

Given a graph $G$, can we attach whiskers to $G$ so that the resulting graph
$H$ has the property that $J(H)^{s}$ is componentwise linear for all $s\geq
1$. Can we classify all the ways to add whiskers to $G$ to make $H$ so that
$J(H)^{s}$, respectively $J(H)^{(s)}$, is componentwise linear.

Moving beyond cover ideals, in [3, 4], the authors together with Biermann and
Francisco gave a generalization for the whiskering process that works also for
simplicial complexes. We would like to understand if this process would
produce more general monomial ideals with the property that all the symbolic
powers of the Alexander dual of its Stanley-Reisner ideal are componentwise
linear.

For a subset $W$ of the vertices of a simplicial complex $\Delta$, the
_restriction_ of $\Delta$ on $W$, denoted by $\Delta|_{W}$, is the simplicial
complex whose faces are $\\{F\in\Delta~{}\big{|}~{}F\subseteq W\\}$.

######  Definition 6.7.

Let $\Delta$ be a simplicial complex on the vertices $V$, let $W\subseteq V$,
and let $t\in{\mathbb{N}}$.

  1. (1)

A _partial $t$-coloring_ of $\Delta|_{W}$ is given by a partition
$W=W_{1}\cup\dots\cup W_{t}$ (where the set $W_{i}$ can be empty) such that no
facet of $\Delta|_{W}$ contains more than one vertex in each $W_{i}$.

  2. (2)

Let $\chi$ be a partial $t$-coloring of $\Delta|_{W}$ given by
$W=W_{1}\cup\dots\cup W_{t}$. We define a new simplicial complex
$\Delta_{\chi}$ on the vertex set $V\cup\\{y_{1},\dots,y_{t}\\},$ with faces

$\\{\sigma\cup\tau~{}\big{|}~{}\sigma\in\Delta,\tau\subseteq\\{y_{1},\dots,y_{t}\\}~{}\mbox{and}~{}\sigma\cap
W_{j}=\emptyset\text{ if }y_{j}\in\tau\\}.$

######  Question 6.8.

Let $\Delta$ be a simplicial complex and let $W$ be a subset of the vertices
in $\Delta$. For a partial $t$-coloring $\chi$ of $\Delta|_{W}$, let
$I_{\Delta_{\chi}}$ be the Stanley-Reisner ideal of $\Delta_{\chi}$ and let
$J_{\Delta_{\chi}}$ be its Alexander dual. What conditions on $W$ and $\chi$
would guarantee that $J_{\Delta_{\chi}}^{(s)}$ is componentwise linear for all
$s\geq 1$?

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