DEPTH AND REGULARITY MODULO A PRINCIPAL IDEAL

arXiv:1706.09675v3 [] 29 Jan 2018

GIULIO CAVIGLIA, HUY TÀI HÀ, JÜRGEN HERZOG, MANOJ KUMMINI,
NAOKI TERAI, AND NGO VIET TRUNG
Abstract. We study the relationship between depth and regularity of a
homogeneous ideal I and those of (I, f ) and I : f , where f is a linear form or
a monomial. Our results have several interesting consequences on depth and
regularity of edge ideals of hypergraphs and of powers of ideals.

1. Introduction
Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k and let X = Proj R/I
be a projective scheme in Pn−1 , where I is a homogeneous ideal. The main goal
of our work is to understand the connection between algebraic invariants and
properties of X and its hypersurface sections. This is a classical topic that has
inspired many important results in both algebraic geometry and commutative
algebra. We shall focus on invariants that govern the computational complexity,
namely the depth and the Castelnuovo-Mumford regularity (or simply, regularity). More specifically, we shall examine the relationship between the depth and
the regularity of I and (I, f ), and their powers, when f ∈ R is a linear form or
when I is a monomial ideal and f is a monomial.
Our work is inspired by a recent paper of Dao, Huneke and Schweig [8], in
which it was proved that if I ⊆ R is a monomial ideal and x is a variable
appearing in the generators of I then
reg R/I ∈ {reg R/(I : x) + 1, reg R/(I, x)},
and if, in addition, I is a squarefree monomial ideal then
depth R/I ∈ {depth R/(I : x), depth R/(I, x)}.
These results give recursive formulas for regularity and depth of squarefree
monomial ideals and played a key role in their work on edge ideals of graphs.
Our first result extends these formulas of Dao, Huneke and Schweig to a large
class of homogeneous ideals.
Theorem 3.1. Let I = J + f H, where J and H are homogeneous ideals in R
and f is a linear form that is a non-zerodivisor of R/J. Then
2010 Mathematics Subject Classification. Primary: 13C20; Secondary: 13D45, 14B05,
05C65.
Key words and phrases. depth, regularity, monomial ideal, powers of an ideal, edge ideal,
very well-covered graph, chordal graph.
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(i) depth R/I ∈ depth R/(I : f ), depth R/(I, f ) .
(ii) reg R/I ∈ reg R/(I : f ) + 1, reg R/(I, f ) .
Note that I : f = J +H and (I, f ) = (J, f ) under the assumptions of Theorem
3.1. We also obtain similar formulas for the case where f is any form of higher
degree that is a non-zerodivisor of R/J.
Obviously, we can choose I to be any monomial ideal and f to be any variable.
This particularly shows that the depth formula of Dao, Huneke and Schweig also
holds for arbitrary monomial ideals. We can even extend their depth formula to
the case f being an arbitrary monomial as follows.
Theorem 4.3. Let I be a monomial ideal and let f be an arbitrary monomial
in R. Then
(i) depth R/I ∈ {depth R/(I : f ), depth R/(I, f )},
(ii) depth R/I = depth R/(I : f ) if depth R/(I, f ) ≥ depth R/(I : f ).
We will give examples showing that Theorem 4.3 is no longer true if I is not
a monomial ideal or if f is not a monomial, and that there is no similar formula
for reg R/I if f is not a variable.
We can also make the regularity formula of Dao, Huneke and Schweig more
precise as follows.
Theorem 4.7. Let I be a monomial ideal and x a variable of R. Then
(i) reg R/I = reg R/(I : x) + 1 if reg R/(I : x) > R/(I, x),
(ii) reg R/I ∈ {reg R/(I, x) + 1, reg R/(I, x)} if reg R/(I : x) = reg R/(I, x),
(iii) reg R/I = reg R/(I, x) if reg R/(I : x) < reg R/(I, x).
From Theorem 4.7 it follows that reg I is either max{reg(I : x), reg(I, x)} or
max{reg(I : x), reg(I, x)} + 1. In particular, if reg(I : x) ≤ r for all variables
x of R, then reg I ≤ r + 1. This implication is very useful because one can use
induction to estimate the regularity of a monomial ideal.
Our approach can be used to study the behavior of the functions depth R/I t
and reg R/I t for t ≥ 1. If I is the edge ideal of a hypergraph which contains a
good leaf and f is the monomial of the associated leaf, we show that I t+1 : f = I t
and I t+1 : f = I t for all t ≥ 1, where I t denotes the integral closure of I t . Using
these relations we derive the following result.
Theorem 5.1. Let I be the edge ideal of a hypergraph which contains a good
leaf. Then
(i) the functions depth R/I t and depth R/I t are non-increasing for all t ≥ 1,
(ii) the functions reg R/I t and reg R/I t are non-decreasing for all t ≥ 1.
Notice that the function depth R/I t of the edge ideal of a hypergraph needs
not be non-increasing (see [15, 22]), and that it is still an open question (see [1])
whether the depth function of the edge ideal of a graph is non-increasing.
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In general, the functions depth R/I t and reg R/I t need not be monotone; see
[1, 2, 13, 14]. However, if we can find a common non-zerodivisor f on I t for all
t ≥ 1, then depth R/(I, f )t and reg R/(I, f )t are monotone functions regardless
of the behavior of depth R/I t and reg R/I t . This follows from the following
result, whose proof is based on the same ideas as that of Theorem 3.1.
Theorem 5.2. Let R be a positively graded algebra over a field, and let I be a
graded ideal in R. Let f 6= 0 be a form in R which is a non-zerodivisor of R/I t
for all t ≤ s. Then
t
(i) depth R/(I, f )s = mint≤s depth R/I
 − 1, t
s
(ii) reg R/(I, f ) − s deg f = maxt≤s reg R/I − t deg f .

As a consequence, if f = x is a new variable then depth R[x]/(I, x)s is a nonincreasing function and reg R[x]/(I, x)s is a non-decreasing function, while the
functions depth R/I s and reg R/I s need not be so.
The relationship between depth and regularity of the ideals I, I : x, (I, x) is
very useful in finding combinatorial characterizations of these invariants for edge
ideals of hypergraphs. We will demonstrate this approach in two applications.
Let I(H) denote the edge ideal of a hypergraph H in a polynomial ring R. We
call H a Cohen-Macaulay hypergraph if R/I(H) is a Cohen-Macaulay ring, i.e.,
depth R/I(H) = dim R/I(H). Moreover, for simplicity, we set reg H = reg I(H)
and call it the regularity of H.
The first application is a short proof for the following characterization of
Cohen-Macaulay very well-covered graphs, which is essentially due to Crupi,
Rinaldo, and Terai [6] (see also Constantinescu and Varbaro [5] and Mahmoudi
et al [26] for other variants of this characterization). This characterization is a
generalization of a criterion for Cohen-Macaulay bipartite graphs of Herzog and
Hibi [17].
Theorem 6.3. Let H be a simple graph on 2n vertices which has a minimal
vertex cover of n elements. Then H is a Cohen-Macaulay graph if and only if
H is a twin-free very well-covered graph.
The second application is a sufficient condition for a hypergraph to have regularity ≤ 3. Since the maximal generating degree of a homogeneous ideal is
bounded above by its regularity, the edge ideal of a hypergraph of regularity 2 is
that of a graph. Fröberg [12] characterized graphs of regularity ≤ 2 to be those
whose complements are chordal.
For every vertex x of a hypergraph H, we denote by H : x the hypergraph of
all minimal sets of the form F ∩ (V \ {x}), where F is an edge of H.
Theorem 6.4. Let H be a hypergraph such that H : x is a graph whose
complement is chordal for all vertices x of H. Then reg H ≤ 3.
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A special case of Theorem 6.4 is a result of Nevo on the regularity of gap-free
and claw-free graphs [29], which was also reproved by Dao, Huneke and Schweig
in [8].
In publishing this paper, we hope that our results and methods would stimulate further investigations on the behavior of depth and regularity between the
ideals I, I : f and (I, f ).
2. Preliminaries
In this section, we recall notations and terminology used in the paper, and
a number of auxiliary results. Generally, we will follow standard texts in this
research area (cf. [?, 9, 18]).
Throughout the paper, the two important invariants that we investigate are
the depth and the regularity. Though these notions can be defined in several
ways, we prefer their definition by means of local cohomology modules.
Let R be a positively graded algebra and let m be its maximal homogeneous
ideal. Let M be a finitely generated graded R-module. Let Hmi (M), for i ≥ 0,
denote the i-th local cohomology module of M with respect to m. We define
depth M := min{i Hmi (M) 6= 0},
reg M := max{i + j

Hmi (M)j 6= 0}.

Remark 2.1. Let ai (M) := max{j| Hmi (M)j 6= 0} with the convention that
ai (M) := −∞ if Hmi (M) = 0. Then
depth M = min{i| ai (M) 6= −∞},
reg M = max{ai (M) + i i ≥ 0}.
When R is a polynomial ring and M is a finitely generated graded R-module,
the depth and the regularity of M are closely related to the minimal free resolution and graded Betti numbers of M in the following way. Suppose that M
admits the following minimal free resolution:
M
M
0→
R(−j)βp,j (M ) → · · · →
R(−j)β0,j (M ) → M → 0.
j∈Z

j∈Z

Let pd M denote the projective dimension of M. Then
pd M = max{i βi,j (M) 6= 0 for some j ∈ Z},
depth M = n − pd M,
reg M = max{j − i βi,j (M) 6= 0}.
Note that the maximal degree of the minimal generators of M is bounded by
reg M.
The definition of depth and regularity by means of the local cohomology
modules is best suited when working with short exact sequences. In fact, using
the long derived sequence of local cohomology modules we immediately obtain
the following facts, which we shall need later in the next sections.
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Lemma 2.2. Let 0 → L → M → N → 0 be a short exact sequence of finite
graded R-modules. Then
(i) depth M ≥ min{depth L, depth N},
(ii) depth M = depth N if depth L > depth M,
(iii) reg L = reg N + 1 if reg L > reg M.
From now on, unless otherwise stated, we let R = k[x1 , . . . , xn ] be a polynomial ring over a field k and let I be a nonzero homogeneous ideal in R. From the
above characterizations of depth and regularity by means of the graded Betti
numbers we immediately have the following relationships between R/I and I:
depth R/I = depth I − 1,
reg R/I = reg I − 1.
These formulas will be used without reference in this paper.
Recall that for a finitely generated graded R-module M, an element f ∈ R is
called a filter-regular element of M if f 6∈ p for all associated primes p 6= m of
M. A sequence f1 , . . . , fs ∈ R of elements is called a filter-regular sequence of
M if fi is a filter-regular element of M/(f1 , . . . , fi−1 )M for all i = 1, . . . , s. It is
easy to see that this is equivalent to the condition that
[(f1 , ..., fi−1 )M : fi ]t = [(f1 , ..., fi−1 )M]t
for all i = 1, ..., s, and t ≫ 0 (see [33, Lemma 2.1]).
Remark 2.3. If k is an infinite field, we can use prime avoidance to find a linear
form which is filter-regular with respect to M. Applying this fact successively,
we may assume that xn , ..., x1 is a filter-regular sequence of M after a linear
change of the variables.
Let in(I) denote the initial ideal of I with respect to the degree reverse lexicographic order. When the xn , ..., x1 is a filter-regular sequence of R/I, one can
easily pass the investigation from I to in(I).
Theorem 2.4. Assume that xn , ..., x1 is a filter-regular sequence of R/I. Then
(i) depth R/I = depth R/ in(I),
(ii) reg R/I = reg R/ in(I).
Proof. This follows from Remark 2.1 and a general result [34, Theorem 1.2],
which states that
max{ai (R/I) + i| i ≤ t} = max{ai (R/ in(I)) + i| i ≤ t}
for every t ≥ 0.



Our investigation on monomial ideals is based on a formula of Takayama.
Let R = k[x1 , ..., xn ] and I a monomial ideal in R. Then R/I has a natural
Nn -graded structure inherited from that of R. Therefore, the local cohomology
module Hmi (R/I) has a Zn -graded structure. Takayama’s formula [32, Theorem
1] describes the dimension of the Zn -graded component Hmi (R/I)a , for a ∈ Zn ,
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in terms of a simplicial complexes ∆a (I). We shall recall the construction of
∆a (I), as given in [28], which is simpler than the original construction of [32].
For a = (a1 , ..., an ) ∈ Zn , set xa = xa11 · · · xann and Ga := {j ∈ [1, n] aj < 0}.
For every subset F ⊆ [1, n], let RF = R[x−1
j | j ∈ F ]. Define
∆a (I) = {F \ Ga | Ga ⊆ F, xa 6∈ IRF }.
We call ∆a (I) a degree complex of I.
Takayama’s formula is stated as follows:
e i−|Ga |−1 (∆a (I), k).
dimk H i (R/I)a = dimk H
m

The original formula in [32, Theorem 1] is slightly different. It contains additional conditions on a for Hmi (R/I)a = 0. However, the proof in [32] shows that
we may drop these conditions, which is more convenient for our investigation.
From Takayama’s formula we immediately obtain the following characterizations of depth and regularity of monomial ideals in terms of the degree complexes.
Proposition 2.5. Let I ⊆ R be a monomial ideal. Then
e i−1(∆a (I), k) 6= 0},
(i) depth R/I = min{|Ga | + i a ∈ Zn , i ≥ 0, H
e i−1 (∆a (I), k) 6= 0}.
(ii) reg R/I = max{|a| + |Ga | + i a ∈ Zn , i ≥ 0, H
3. Depth and regularity modulo a linear form

The main aim of this section is to prove the following theorem, which extends
the aforementioned results of Dao, Huneke and Schweig on monomial ideals
to a large class of homogeneous ideals. Unless otherwise specified, let R =
k[x1 , . . . , xn ] be a polynomial ring over a field and let I be a homogeneous ideal
in R.
Theorem 3.1. Let I = J + f H, where J and H are homogeneous ideals in R
and f is a linear form that is a non-zerodivisor of R/J. Then

(i) depth R/I ∈ depth R/(J + H), depth R/J − 1 .
(ii) reg R/I ∈ reg R/(J + H) + 1, reg R/J .
Proof. Let S = R[x], where x is a new variable, and Q = (J, xH). Then
R/I = S/(Q, x − f ).
We shall show that x − f is a non-zerodivisor in S/Q. Indeed, it is clear
that Q = (J, x) ∩ (J, H). Observe that every associated prime of (J, x) is of
the form (p, x), where p is an associated prime of J, and f 6∈ p because f is a
non-zerodivisor in R/J. Thus, x − f does not belong to any associated prime of
(J, x). Hence, (J, x) : (x − f ) = (J, x). Since every associated prime of (J, H) is
the extension to S of an associated prime of J + H ⊆ R, x − f does not belong
to any associated prime of (J, H). Therefore, (J, H) : (x − f ) = (J, H). Hence,
we get Q : (x − f ) = Q.
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It follows from the fact that x − f is a non-zerodivisor of S/Q that
depth R/I = depth S/Q − 1 = depth Q − 2 = depth Q/xQ − 1.
It is easy to see that Q/xQ = J ⊕ ((J + H)/J)(−1). Thus,

depth Q/xQ = min depth J, depth(J + H)/J .
By Lemma 2.2(i), from the exact sequence
0 → J → (J + H) → (J + H)/J → 0,
we obtain
depth(J + H) ≥ min{depth J, depth(J + H)/J}.
Therefore,
depth R/I = depth Q/xQ − 1 ≤ depth(J + H) − 1 = depth R/(J + H).
Now, if depth R/I 6= depth R/(J + H) then it follows that
depth R/I < depth R/(J + H) = depth R/I : f.
Applying Lemma 2.2(ii) to the exact sequence
f

0 → R/I : f → R/I → R/(I, f ) → 0,
we get depth R/I = depth R/(I, f ) = depth R/(J, f ) = depth R/J − 1 as required for Theorem 3.1(i).
The proof for Theorem 3.1(ii) proceeds similarly. Firstly, we have
reg R/I = reg S/Q = reg Q − 1 = reg Q/xQ − 1.
On the other hand,
reg Q/xQ = max{reg J, reg(J + H)/J + 1}.
Therefore,


reg R/I = max reg J, reg(J + H)/J + 1 − 1

= max reg R/J, reg(J + H)/J .

If reg R/I 6= reg R/J then
reg R/I = reg(J + H)/J > reg R/J.
By Lemma 2.2(iii), from the short exact sequence
0 → (J + H)/J → R/J → R/(J + H) → 0,
we get reg(J + H)/J = reg R/(J + H) + 1. Hence, reg R/I = reg R/(J + H) + 1
as required.

It is easy to see that (I : f ) = J + H and (I, f ) = (J, f ) under the assumption
of Theorem 3.1. Hence, we can rewrite the formulae of Theorem 3.1 as

(i) depth R/I ∈ depth R/(I : f ), depth R/(I, f ) ,
(ii) reg R/I ∈ reg R/(I : f ) + 1, reg R/(I, f ) .
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Our proof of Theorem 3.1 in fact yields a slightly more general statement
where f is of higher degree.
Corollary 3.2. Let I = J + f H, where J and H are homogeneous ideals in R
and f is any form of degree d that is a non-zerodivisor of R/J. Then

(i) depth R/I ∈ depth R/(J + H), depth R/J − 1 .
(ii) reg R/I ∈ reg R/(J + H) + d, reg R/J + d − 1, . . . , reg R/J .
Proof. As in the proof of Theorem 3.1, consider S = R[x], where x is a new
variable, and Q = (J, xd H). The same argument shows that xd − f is a nonzerodivisor in S/Q. Thus, we may again reduce to the statements for Q ⊆ S.
Observe that Q = (J, x(xd−1 H)). The assertion now follows by an iterated
application of Theorem 3.1.

The assumption of Theorem 3.1 is obviously satisfied if I is any monomial
ideal and f is any variable. Therefore, we have the following consequence.
Corollary 3.3. Let I be a monomial ideal and let x be a variable in R. Then

(i) depth R/I ∈ depth R/(I : x), depth R/(I, x) ,
(ii) reg R/I ∈ reg R/(I : x) + 1, reg R/(I, x) .
Note that Corollary 3.3(ii) was already proved by Dao, Huneke and Schweig
[8, Lemma 2.10]. They also proved Corollary 3.3(i) for squarefree monomial
ideals [8, Lemma 5.1]. Their proofs are based on a result of Kummini [24] on
multigraded Betti numbers of squarefree monomial ideals.
The following examples show that Corollary 3.3 does not hold for an arbitrary
linear form f .
Example 3.4. Let I = (x1 x3 , x2 x3 , x1 x4 ) and f = x1 −x3 in R = Q[x1 , x2 , x3 , x4 ].
Then depth R/I = 2, depth R/(I : f ) = 1 and depth R/(I, f ) = 0.
Example 3.5. Let I = (x32 , x2 x3 x5 , x3 x28 x9 , x5 x7 x39 ) and f = x3 − x5 in R =
Q[x1 , . . . , x9 ]. Then reg R/I = 6, reg R/(I : f ) = 7 and reg R/(I, f ) = 7.
Under the assumption that f is general enough, Theorem 3.1 gives the following result.
Corollary 3.6. Let f ∈ R be a linear form, which is filter-regular in R/I. Then

(i) depth R/I ∈ depth R/(I : f ), depth R/(I, f ) .
(ii) reg R/I ∈ reg R/(I : f ) + 1, reg R/(I, f ) .
Proof. Without restriction we may assume that k is an infinite field. Using prime
avoidance and a linear change of variables, we may assume that f = xn and
(1) xn , . . . , x1 form a filter-regular sequence in R/I,
(2) xn , . . . , x1 form a filter-sequence in R/(I : xn ),
(3) xn−1 , . . . , x1 form a filter-sequence in R/(I, xn ).
8

Let in(I) denote the initial ideal of I with respect to the reverse lexicographic
order. By Theorem 2.4, we have
(4) depth R/I = depth R/ in(I) and reg R/I = reg R/ in(I),
(5) depth R/I : xn = depth R/ in(I : x1 ) and reg R/I : xn = reg R/ in(I : xn ),
(6) depth R/(I, xn ) = depth R/ in(I, xn ) and reg R/(I, xn ) = reg R/ in(I, xn ).
It is also known (see [9, Proposition 15.12], [18, Lemma 4.3.7]) that
(7) in(I : xn ) = in(I) : xn ,
(8) in(I, xn ) = in(I) + (xn ).
Since in(I) is a monomial ideal, there are monomial ideals J and H such that
in(I) = J + xn H and xn is a non-zerodivisor in R/J. Obviously, we have
(9) in(I) : xn = J + H,
(10) in(I) + (xn ) = (J, xn ).
Using (4)-(10), we can express the asserted formulas as

depth R/ in(I) ∈ depth R/(J + H), depth R/J − 1 ,

reg R/ in(I) ∈ reg R/(J + H) + 1, reg R/J .
The conclusion now follows from Theorem 3.1.



Corollary 3.6 is only interesting in the case depth R/I = 0. If depth R/I > 0,
then f must be a regular element; hence depth R/I = depth R/(I : f ) and
reg R/I = reg R/(I, f ).
4. Depth and regularity colon a monomial
It is natural to ask whether one can extend the results of Dao, Huneke and
Schweig to the case where x is replaced by a form f of higher degree. We shall
see that the same relationship between the depths of R/I, R/(I : f ) and R/(I, f )
as in Corollary 3.3(i) holds for any monomial ideal I and any monomial f . We
shall also give examples showing that this is no longer true if I is not a monomial
ideal or if f is not a monomial, and that there is no similar relationship between
the regularities of R/I, R/(I : f ) and R/(I, f ) as in Corollary 3.3(ii).
The investigation in this section is based on the following relationships between R/I and R/(I : f ), which were proved in [30, Corollary 1.3] for (i) and in
[31, Lemma 4.2] for (ii) by different methods.
Lemma 4.1. Let I be a monomial ideal and let f be an arbitrary monomial in
R. Then
(i) depth R/I ≤ depth R/(I : f ),
(ii) reg R/I ≥ reg R/(I : f ).
Proof. (i) By Proposition 2.5(i), there exists a ∈ Zn such that depth R/(I :
f ) = |Ga | + i and H̃i−1 (∆a (I : f ), k) 6= 0 for some i ≥ 0. Let f = xb , b ∈ Nn .
Since Ga := {j| aj < 0}, there exists c ∈ Nn such that supp(c) ⊆ Ga and
9

Ga+b−c = Ga . For every subset F ⊂ [1, n], F ⊇ Ga , the element xc is invertible
a
a+b
in RF := k[x−1
∈ IRF if
i | i ∈ F ]. Therefore, x ∈ (I : f )RF if and only if x
a+b−c
and only if x
∈ IRF . By the definition of the degree complex, this implies
that ∆a (I : f ) = ∆a+b−c (I). Hence, H̃i−1 (∆a+b−c (I), k) 6= 0. By Proposition
2.5 (i), it follows that
depth R/I ≤ |Ga+b−c| + i = |Ga | + i = depth R/(I : f ).
(ii) By Proposition 2.5(ii), there exists a ∈ Zn such that reg(I : f ) = |a| +
|Ga | + i and H̃i−1 (∆a (I : f ), k) 6= 0 for some i ≥ 0. By the maximality of
|a| + |Ga | + i we must have aj = −1 for j ∈ Ga . Let c ∈ Nn with ci = bi for
i ∈ Ga and ci = 0 else. Then supp(c) ⊆ Ga and Ga+b−c = Ga . Moreover,
|a + b − c| ≥ |a|. Similarly as above, we have ∆a (I : f ) = ∆a+b−c (I). Hence,
H̃i−1 (∆a+b−c (I), k) 6= 0. By Proposition 2.5(ii), this implies that
reg R/I ≥ |a + b − c| + |Ga | + i ≥ |a| + |Ga | + i = reg R/(I : f ).

One might expect that reg R/I ≥ reg R/(I : f ) + deg f . As shown by the
following example, this bound does not hold, even when f is a variable appearing
in the generators of I.
Example 4.2. Let R = Q[x1 , . . . , x9 ] and consider
I = (x2 x3 , x2 x4 , x3 x5 , x4 x5 , x2 x5 x7 , x4 x8 , x3 x9 , x4 x9 , x7 x9 , x8 x9 ).
Then, for f = x8 , we have reg R/I = 2 = reg R/(I : f ). Thus, reg R/I 6≥
reg R/(I : f ) + 1.
Our next result shows that the conclusion of Corollary 3.3(i) also holds for an
arbitrary monomial f .
Theorem 4.3. Let I be a monomial ideal and let f be an arbitrary monomial
in R. Then
(i) depth R/I ∈ {depth R/(I : f ), depth R/(I, f )}.
(ii) depth R/I = depth R/(I : f ) if depth R/(I, f ) ≥ depth R/(I : f ).
Proof. Assume that depth R/I 6= depth R/(I : f ). Then depth R/(I : f ) >
depth R/I by Lemma 4.1(i). Consider the short exact sequence
f

0 −→ R/(I : f ) −→ R/I −→ R/(I, f ) −→ 0.
By Lemma 2.2(ii), this implies that depth R/I = depth R/(I, f ).
If depth R/(I, f ) ≥ depth R/(I : f ), then depth R/I ≥ depth R/(I : f ) by
Lemma 2.2(i). By Lemma 4.1(i), this implies that depth R/I = depth R/(I :
f ).

Theorem 4.3(i) does not necessarily hold if I is not a monomial ideal or if f
is not a monomial.
10

Example 4.4. Let R = Q[x, y, z, u, v] and consider
I = (x2 y 3 − z 2 u3 , y 2z 3 − x2 zuv, x3 u2 − y 2 v 3 ) and f = xyzuv.
Then depth R/I = 2, depth R/(I : f ) = 1 and depth R/(I, f ) = 0.
Example 4.5. Let R = Q[x, y, z, u, v] and consider
I = (x3 y, y 2z 5 , z 2 u4 v) and f = xy − zv.
Then depth R/I = 2, depth R/(I : f ) = 1 and depth R/(I, f ) = 0.
f

From the short exact sequence 0 −→ R/(I : f ) −→ R/I −→ R/(I, f ) −→ 0,
we might expect that
reg R/I ∈ {reg R/(I : f ) + deg f, reg R/(I, f )},
which would generalize the mentioned result of Dao, Huneke and Schweig [8,
Lemma 2.10] to the case where deg f > 1. However, the following example
shows that this is not possible, even when I is a monomial ideal and f is a
monomial.
Example 4.6. Let R = Q[x, y, z, u, v] and consider
I = (xy 2 , yz 2 , zu3 , uv 2x, v 2 xz) and f = x3 y.
Then reg R/I = 6, reg R/(I, f ) = 7 and reg R/(I : f ) + deg f = 8. In particular,
reg R/I 6∈ {reg R/(I : f ) + deg f, reg R/(I, f )}.
On the other hand, the formula for reg R/I of Dao, Huneke and Schweig can
be made more precise as follows.
Theorem 4.7. Let I be a monomial ideal and x a variable of R. Then
(i) reg R/I = reg R/(I : x) + 1 if reg R/(I : x) > R/(I, x),
(ii) reg R/I ∈ {reg R/(I, x) + 1, reg R/(I, x)} if reg R/(I : x) = reg R/(I, x),
(iii) reg R/I = reg R/(I, x) if reg R/(I : x) < reg R/(I, x).
Proof. We only need to prove (i) and (iii).
If reg R/(I : x) > reg R/(I, x), then reg R/I > reg R/(I, x) by Lemma 4.1(ii).
Hence, reg R/I = reg R/(I : x) + 1.
If reg R/(I : x) < reg R/(I, x), then reg R/I > reg R/(I : x) because reg R/I ≥
reg R/(I, x) by the proof of [8, Lemma 2.10]. Therefore, reg R/I = reg R/(I, x).

From Theorem 4.7 we immediately obtain the following estimate, which shows
that reg I is either max{reg(I : x), reg(I, x)} or max{reg(I : x), reg(I, x)} + 1.
Corollary 4.8. Let I be a monomial ideal and x a variable of R. Then
max{reg(I : x), reg(I, x)} ≤ reg I ≤ max{reg(I : x) + 1, reg(I, x)}.
In particular, we obtain the following inductive estimate for the regularity of
monomial ideals.
11

Proposition 4.9. Let I be a monomial ideal. If reg(I : x) ≤ r for all variables
x of R, then reg I ≤ r + 1.
Proof. The case n = 1 is trivial. If n > 1, we have (I, x) : y = (I : y, x) for all
variables x, y of R. Therefore, reg((I, x) : y) = reg(I : y, x) ≤ reg(I : y) ≤ r
by the lower bound of Corollary 4.8. By induction, we have reg(I, x) ≤ r + 1.
Therefore, the upper bound of Corollary 4.8 implies reg I ≤ r + 1.

Using Proposition 4.9, we can easily recover the following result.
Corollary 4.10. [21, Theorem 3.1] Let I be a monomial ideal. Let lcm(I) denote
the least common multiple of the minimal monomial generators of I. Then
reg I ≤ deg lcm(I) − ht I + 1.
Proof. The case deg lcm(I) = 1 is trivial. Without restriction we may assume
that every variable x appears in the minimal monomial generators of I. Then
deg lcm(I : x) ≤ deg lcm(I) − 1 and ht(I : x) ≥ ht I. By induction, we have
reg(I : x) ≤ deg lcm(I : x) − ht(I : x) + 1 ≤ deg lcm(I) − ht I.
Therefore, the conclusion follows from Proposition 4.9.



5. Depth and regularity functions
In this section, we investigate the depth and the regularity of powers of an
ideal. Our approach allows us to study the non-increasing and non-decreasing
properties of the depth and the regularity functions for particular classes of
ideals.
Let I be a homogeneous ideal in a polynomial ring R. It is usually difficult to
compute depth R/I t and reg R/I t for all t ≥ 1. As a consequence, the behavior
of depth R/I t and reg R/I t , as functions in t, often remains mysterious. We
only know that for t sufficiently large, depth R/I t and reg R/I t are constant or
a linear function, respectively [4, 7, 23].
Random examples show that depth R/I t (respectively, reg R/I t ) tends to be
a non-increasing (respectively, non-decreasing) function, though that is not the
case in general [1, 2, 13, 14]. Bandari, Herzog and Hibi [1] asked whether
depth R/I t is a non-increasing function for squarefree monomial ideals. A counterexample was recently found by combinatorial methods in [22] (see also [15]).
However, it is still an open question whether depth R/I t is a non-increasing
function for the edge ideal of an arbitrary graph.
Recall that a hypergraph is a collection of sets, called edges, which are subsets
of a vertex set. Given a hypergraph H on the vertices x1 , ..., xn , the edge ideal
of H is defined to be
Y
I(H) :=
x| F is an edge of H .
x∈F

12

We will use Lemma 4.1(i) to present a large class of edge ideals of hypergraphs
for which depth R/I t (respectively, reg R/I t ) is a non-increasing (respectively,
non-decreasing) function.
An edge F of a hypergraph H is called a good leaf if the intersections of F
with other edges of H form a chain with respect to containment. This notion is
originally introduced for simplicial complexes [19]. For the study of edge ideals,
one may always assume that there are no containments among the edges of the
hypergraph, which can be considered as the facets of a simplicial complex. The
leaves of a graph are always good. Good leaves do exist in the hypergraph of
the facets of a simplicial forest [19, Corollary 3.4], which may be defined as a
hyperforest [19, Theorem 3.2].
Theorem 5.1. Let I be the edge ideal of a hypergraph which contains a good
leaf. Then
(i) the functions depth R/I t and depth R/I t are non-increasing for all t ≥ 1,
(ii) the functions reg R/I t and reg R/I t are non-decreasing for all t ≥ 1.
Proof. Let F be a good leaf and f the monomial associated with F . Let g be
an arbitrary monomial of I t+1 : f , t > 0. Then f g = hf1 · · · ft+1 , where h is a
monomial and f1 , ..., ft+1 are monomials associated with some edges F1 , ..., Ft+1
of the hypergraph. By the definition of a good leaf, we may assume that Fi ∩F ⊆
F1 ∩ F for all i = 1, ..., t + 1. Then the variables of F \ F1 do not appear in
f1 , ..., ft+1 . So they appear in h. From this it follows that hf1 is divisible by f .
Therefore, g is divisible by f2 · · · ft+1 . That means g ∈ I t . Since I t ⊆ I t+1 : f ,
we have shown that I t+1 : f = I t . Hence, depth R/I t ≥ depth R/I t+1 and
reg R/I t ≤ reg R/I t+1 by Lemma 4.1.
Let g ∈ I t+1 : f . Then (gf )m ∈ I m(t+1) for some m > 0. Similarly as
above, we can show that g m ∈ I mt . Hence g ∈ I t . From this it follows that
I t+1 : f = I t . Since I t is also a monomial ideal, depth R/I t ≥ depth R/I t+1 and

reg R/I t ≤ reg R/I t+1 by Lemma 4.1.
In the following we give formulas for depth R/(I, f )s and reg R/(I, f )s, when
f is a non-zerodivisor of R/I t for all t ≤ s. The proof for these formulas follows
the same line of arguments as that of Theorem 3.1.
Theorem 5.2. Let R be a positively graded algebra over a field, and I be a
graded ideal in R. Let f be a form in R which is a non-zerodivisor of R/I t for
all t ≤ s. Then
t
(i) depth R/(I, f )s = mint≤s depth R/I
− 1,

s
(ii) reg R/(I, f ) − s deg f = maxt≤s reg R/I t − t deg f .

Proof. Let S = R[x] with deg x = deg f = d. If we consider (I, x) as a graded
module over R, we have a decomposition
(I, x)s = I s ⊕ I s−1 x ⊕ · · · ⊕ Ixs−1 ⊕ Rxs ⊕ Rxs+1 · · ·
13

From this it follows that
S/(I, x)s = (R/I s ) ⊕ (R/I s−1 )(−d) ⊕ · · · ⊕ (R/I)(−(s − 1)d).
Thus, we have
depth S/(I, x)s = min depth R/I t ,
t≤s

s
reg S/(I, x) − sd = max reg R/I t − td .
t≤s

Since R/(I, f )s ≃ R[x]/((I, x)s , x − f ), it remains to show that x − f is a
non-zerodivisor of R[x]/(I, x)s . Let g ∈ (I, x)s−1 such that (x − f )g ∈ (I, x)s .
Without restriction we may assume that g = c0 + c1 x + · · · cs−1 xs−1 , ci ∈ R.
Then c0 f ∈ I s . Hence, c0 ∈ I s : f = I s . From this it follows that (c1 x +
+ · · · cs−1 xs−1 )(x − f ) ∈ (I, x)s . This implies c1 f ∈ I s−1 . Hence, c1 ∈ I s−1 : f =
I s−1 . Proceed in this fashion, we eventually get ci ∈ I s−i for i = 0, ..., s − 1.
Therefore, g ∈ (I, x)s , as required.

Theorem 5.2 does not hold if f 6= 0 is not a non-zerodivisor of R/I t for all
t ≤ s, as illustrated in the following example.
Example 5.3. Let R = k[x, y, z, u, v] and consider
I = (x5 , x4 y, xy 4, y 5 , x2 y 2 (xz 6 − yu6), x3 y 3) and f = z − u.
Then f is a non-zerodivisor of R/I, but f is a zerodivisor R/I 2 .
We now have depth R/(I, f )2 = 1, depth R/I = 2, and depth R/I 2 = 1.
Thus,
depth R/(I, f )2 6= min depth R/I t − 1.
1≤t≤2

2

We also have reg R/(I, f ) = 15, reg R/I = 10, and reg R/I 2 = 20. Therefore,
reg R/(I, f )2 6= max {reg R/I t + (s − t) deg f }.
1≤t≤2

From Theorem 5.2(i) we immediately obtain the following consequence.
Corollary 5.4. Let R be a positively graded algebra over a field and I a graded
ideal of R. Let f be a form in R which is a non-zerodivisor of R/I t for all t ≥ 0.
Then
(i) depth R/(I, f )t is a non-increasing function,
(ii) reg R/(I, f )t − t deg f is a non-decreasing function.
As an example, we may let f be a new variable x, then depth R[x]/(I, x)t is
a non-increasing function and reg R[x]/(I, x)t − t deg f is a non-decreasing function. This displays an interesting phenomenon because the functions depth R/I t
and reg R/I t needs not be so (see [1, Theorem 0.1], [13, Theorem 6.7] and [14,
Example 2.8]).
Remark 5.5. Theorem 5.2(i) and Corollary 5.4(i) also hold for ideals in a local
ring as it can be seen from the proof of Theorem 5.2(i).
14

In the following we denote by Ass I the set of associated primes of I in a
Noetherian ring R. Brodmann [4] showed that Ass I t = Ass I t+1 for t ≫ 0. In
general, we may have Ass I t 6⊆ Ass I t+1 (see e.g. [1, 13, 15]). It has been of great
interest to know when Ass I t ⊆ Ass I t+1 for all t ≥ 1 (see e.g. [11, 20, 27]).
The following consequence of Theorem 5.2(i) shows that if there is a nonzerodivisor f modulo I t for all t ≥ 1, then Ass(I, f )t ⊆ Ass(I, f )t+1 for all
t ≥ 1.
Corollary 5.6. Let R be a Noetherian ring and I an ideal of R. Let f ∈ R be
an element which is a non-zerodivisor of R/I t for all t ≤ s. Then Ass(I, f )t ⊆
Ass(I, f )t+1 for all t < s.
Proof. Let P ∈ Ass(I, f )t , t < s. Then depth(R/(I, f )t)P = 0. By the local
version of Theorem 5.2(i), we have depth(R/I j )P = 1 for some j ≤ t, and hence
depth(R/(I, f )t+1 )P = min depth(R/I j )P − 1 = 0.
j≤t+1

This implies that P ∈ Ass(I, f )t+1 , as required.



6. Depth and regularity of edge ideals
The aim of this section is to show that our approach can be used to study edge
ideals of hypergraphs. Throughout this section we assume that all hypergraphs
are simple and have no isolated vertices.
We will first investigate very well-covered graphs. Recall that a graph is
well-covered if all minimal vertex covers have the same size. A graph is very
well-covered if all minimal vertex covers contain exactly half of the vertices. In
these definitions we can replace minimal vertex covers by maximal independent
sets because they are complements of one another.
It is well known that every associated prime of the edge ideal of a graph is
generated by the elements of a minimal vertex cover of the graph, and that this
gives a correspondence between these two notions. Therefore, the edge ideal a
graph is (height) unmixed if and only if the graph is well-covered. An unmixed
edge ideal of height n in a polynomial ring of 2n variables must come from a
very well-covered graph. The interest in edge ideals of very well-covered graphs
originates from bipartite graphs because an unmixed bipartite graph must be
very well-covered.
A graph H is called Cohen-Macaulay if R/I(H) is a Cohen-Macaulay ring,
i.e., depth R/I(H) = dim R/I(H). It is well known that the edge ideal of a
Cohen-Macaulay graph is always unmixed. Herzog and Hibi [17, Theorem 3.4]
first gave a characterization of Cohen-Macaulay bipartite graphs. This result
was extended to very well-covered graphs by Crupi, Rinaldo and Terai [6] (see
also Constantinescu and Varbaro [5] and Mahmoudi et al [26]). We will give
a short proof of their results. Our proof is based on the following effective
characterizations of very-well covered graphs, due to Favaron [10].
15

Lemma 6.1. [10, Theorem 1.2] A graph H is very well-covered if and only if it
has a perfect matching M with the following two properties:
(i) no edge of M belongs to a triangle in H,
(ii) if an edge of M is the central edge of a path of length 3 in H, then the
two vertices at the ends of the path must be adjacent.
Moreover, if H is very well covered, then every perfect matching in H satisfies
these properties.
Recall that two vertices x, y of a graph are twin if they have the same (open)
neighborhood. A twin-free graph is sometimes referred to as irreducible by
authors in graph theory.
Lemma 6.2. [10, Theorem 2.5] A very well-covered graph H is twin-free if and
only if it has a perfect matching M such that there is no cycle C4 in H with two
edges of M.
Our result on the Cohen-Macaulayness of very well-covered graphs is formulated in a different way than those of Constantinescu and Varbaro [5, Theorem
2.3], Crupi, Rinaldo and Terai [6, Theorem 3.4] and Mahmoudi et al [26, Theorem 3.2].
Theorem 6.3. Let H be a simple graph on 2n vertices which has a minimal
vertex cover of n elements. Then H is a Cohen-Macaulay graph if and only if
H is a twin-free very well-covered graph H.
Proof. Without restriction, we may assume that H is a very well-covered graph.
If H is not twin-free, then H has a cycle C4 which contains two edges of a
perfect matching of H. It follows from Lemma 6.1(i) that the opposite vertices
of this C4 are not adjacent. Therefore, each maximal independent set in H
contains exactly one pair of opposite vertices of this C4 . This implies that the
independence complex of H is not strongly connected and, thus, is not CohenMacaulay; see [3, Proposition 11.7]. Hence, H is not Cohen-Macaulay.
Conversely, suppose that H is twin-free. Let M = {{x1 , y1 }, ..., {xn , yn }} be
a perfect matching in H. Using Lemmas 6.1 and 6.2, it is easy to check that
for any proper subset K ⊆ [1, n], the induced subgraph of H on the vertices
{xi , yi }i∈K is also a twin-free very well-covered graph. By induction, we may
assume that these induced subgraphs are Cohen-Macaulay.
Consider a vertex x in H and let y be the vertex matched with x in M. Let
N(x) denote the set of vertices in H adjacent to x. Observe that if there exists
z ∈ N(x) with z 6= y then by Lemma 6.1(ii) we have N(y) ⊆ N(z). Furthermore,
this inclusion is strict since y and z are not twin. Thus, by taking x such that
|N(y)| is largest possible, we may choose x so that N(x) = {y}. Without loss
of generality, suppose N(xn ) = {yn }.
Let R = k[x1 , . . . , xn , y1 , . . . , yn ] and I = I(H). Let H′ be the induced subgraph of H on {x1 , . . . , xn−1 , y1, . . . , yn−1 } and R′ = k[x1 , . . . , xn−1 , y1, . . . , yn−1 ].
16

Then (I, yn ) = (I(H′ ), yn ). Since R′ /I(H′ ) is a Cohen-Macaulay ring with
dim R′ /I(H′ ) = n − 1, we have
depth R/(I, yn ) = depth S/I(H′ ) + 1 = n.
Without loss of generality we may assume that {y1 , . . . , yn } is a maximal
independent set of H. Suppose N(yn ) = {xi1 , . . . , xis }. Let G denote the induced
subgraph of H on Z := {xi , yi | i 6= i1 , ..., is } and S = k[Z]. By Lemma 6.1(ii),
the vertices yi1 , . . . , yis are not adjacent to any vertex z ∈ Z. This implies that
(I : yn ) = I(G) + (xi1 , . . . , xis ). Since S/I(G) is a Cohen-Macaulay ring with
dim S/I(G) = n − s, we have
depth R/(I : yn ) = depth S/I(G) + s = n.
By Theorem 4.3, we have depth R/I ∈ {depth R/(I : yn ), depth R/(I, yn )}.
Therefore, depth R/I = n = dim R/I.

We shall now discuss an application of Proposition 4.9 on the regularity of edge
ideals. For a hypergraph H we set reg H = reg I(H) and call it the regularity of
H.
If reg H ≤ 2, the degree of the minimal monomial generators of I(H) is
bounded by 2. Hence, I(H) is actually the edge ideal of a graph. In particular,
H is a graph if there are no containments among the edges. If H is a graph, we
know that reg H ≤ 2 if and only if the complement of H is chordal [12]. Recall
that a graph is called chordal if it has no induced cycle Cm with m ≥ 4. In the
following we give a necessary condition for reg H ≤ 3.
For a vertex x of a hypergraph H, we denote by H : x the hypergraph of all
minimal sets of the form F ∩ (V \ {x}), where F is an edge of H.
Theorem 6.4. Let H be a hypergraph such that H : x is a graph whose complement is chordal for all vertices x of H. Then reg H ≤ 3.
Proof. Let I = I(H). Then I : x = I(H : x) for all vertices x of H. By the
above result of Fröberg, the assumption on H : x implies that reg(I : x) ≤ 2.
Therefore, reg I ≤ 3 by Proposition 4.9.

The above necessary condition for reg H ≤ 3 is not so far from being a
sufficient condition because due to Corollary 4.8, reg H ≤ 3 if and only if
reg(H − x) ≤ 3 or reg(H : x) ≤ 2 for some vertex x. Here, H − x denotes
the hypergraph of the edges of H not containing x. If reg(H : x) ≤ 2, then
H : x is a graph (there are no containments among the edges of H : x) whose
complement is chordal by [12].
As a consequence, we obtain the following result of Nevo [29, Theorem 5.1],
which was also reproved by Dao, Huneke and Schweig [8, Theorem 3.4]. Recall
that a claw of a graph is an induced complete bipartite graph K1,3 and a gap is
an induced graph whose complement is a cycle C4 .
Corollary 6.5. Let H be a claw-free and gap-free graph. Then reg H ≤ 3.
17

Proof. By Theorem 6.4, we only need to show that the complement of H : x is
a chordal graph for all x ∈ V . Assume that there is a vertex x such that the
complement of H : x is not chordal. Then the complement of H : x contains
an induced cycle Cm , m ≥ 4. Let G denote the induced subgraph of H whose
complement is this Cm . Let y be an adjacent vertex of x. Then y is not a vertex
of G. If y were not adjacent to an edge of G, the induced graph on x, y and the
vertices of this edge were a gap. Therefore, y must be adjacent to every edge
of G. Let z be a vertex of G adjacent to y. Let u, v be the two vertices of G
non-adjacent to z. Then {u, v} is an edge of G. Hence, y is adjacent to one of
them, say u. Since y is adjacent to x, z, u and since x, z, u are not adjacent to
each other, the induced subgraph on x, y, z, u is a claw, a contradiction.

Acknowledgement. Giulio Caviglia is partially supported by the Simons Foundation (Grant #209661). Huy Tai Ha is partially supported by the Simons Foundation (Grant #279786). Ngo Viet Trung is partially supported by Vietnam
National Foundation for Science and Technology Development (Grant #101.042017.19) and the project VAST.HTQT.NHAT.1/16-18. The main part of this
work was done during research stays of the authors at the American Institute of
Mathematics in the SQuaRE program “Ordinary powers and symbolic powers”
during the period 2012-2014. The authors are grateful to S. A. Seyed Fakhari
for pointing out a mistake of Theorem 5.1 in the first version of this paper and
to an anonymous referee for mentioning that Lemma 4.1(i) and (ii) were already
proved in [30] and [31], respectively.
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Soc. 129 (2001), 9–18.

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Department of Mathematics, Purdue University, 150 N. University Street,
West Lafayette, IN 47907-2067, USA
E-mail address: gcavigli@math.purdue.edu
Department of Mathematics, Tulane University, 6823 St. Charles Ave., New
Orleans, LA 70118, USA
E-mail address: tha@tulane.edu
Fachbereich 6, Mathematik, Universität Duisburg-Essen, Campus Essen, 45117
Essen, Germany
E-mail address: juergen.herzog@gmail.com
Chennai Mathematical Institute, Siruseri, Tamilnadu 603103, India
E-mail address: mkummini@cmi.ac.in
Faculty of Culture and Education, Saga University, Saga 8408502, Japan
E-mail address: terai@cc.saga-u.ac.jp
Institute of Mathematics, Vietnam Academy of Science and Technology, 18
Hoang Quoc Viet, Hanoi, Vietnam
E-mail address: nvtrung@math.ac.vn

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