DEPTH AND REGULARITY OF POWERS OF SUMS OF IDEALS

arXiv:1501.06038v2 [] 1 Jan 2016

HUY TÀI HÀ, NGO VIET TRUNG, AND TRAN NAM TRUNG
Abstract. Given arbitrary homogeneous ideals I and J in polynomial rings A and
B over a field k, we investigate the depth and the Castelnuovo-Mumford regularity of
powers of the sum I + J in A ⊗k B in terms of those of I and J. Our results can be
used to study the behavior of the depth and regularity functions of powers of an ideal.
For instance, we show that such a depth function can take as its values any infinite
non-increasing sequence of non-negative integers.

Introduction
The motivation of our work is the following general problem: given an ideal Q and a
natural number n, what can be said about the power Qn ? Asymptotically, as n gets large,
Qn often exhibits nice behaviors. However, for small or particular values of n, there is no
general approach to investigate properties of Qn . To address this problem, one usually
considers special classes of ideals that possess further structures. An instance is the
class of squarefree monomial ideals which are associated to combinatorial objects, such as
(hyper)graphs and simplicial complexes; these are unions of disconnected structures that
correspond to sums of ideals in different variables. We are naturally led to the following
question.
Question 1. Let A = k[x1 , . . . , xr ] and B = k[y1 , . . . , ys ] be polynomial rings over a field
k, and let R = k[x1 , . . . , xr , y1, . . . , ys ]. Let I ⊂ A and J ⊂ B be nonzero proper ideals.
What properties of (I + J)n ⊂ R can be described in terms of those of I and J?
Question 1 is also related to the study of singularities of the fiber product X ×k Y of
two algebraic schemes X and Y , which are encoded in properties of powers of the defining
ideal. As far as we know, this seemingly basic question has not yet been addressed for
arbitrary ideals I and J.
In this paper, we initiate our study of Question 1 by focusing on the depth and the
Castelnuovo-Mumford regularity of (I +J)n when I and J are homogeneous ideals. Depth
and Castelnuovo-Mumford regularity of powers of ideals have been the main objects of
studies of various authors in the last few decades. For instance, see [15, 16, 17, 19, 22, 24,
25, 28] for studies on depth of powers of ideals, and see [1, 3, 6, 7, 8, 12, 11, 14, 21, 23, 26]
for studies on regularity of powers of ideals. Aside from giving answers to Question 1,
2000 Mathematics Subject Classification. 13C05, 14H20.
Key words and phrases. power of ideals, sum of ideals, depth, regularity, asymptotic behavior.
We would like to thank Vietnam Institute for Advanced Study in Mathematics for the hospitality
during our visit in 2014, when we started to work on this paper. The first named author is partially
supported by the Simons Foundation (grant #279786). The second author is supported by Vietnam
National Foundation for Science and Technology Development under grant number 101.04-2014.52.
1

our results also contribute to a better understanding of the asymptotic behavior of depth
and regularity.
Our first main result gives the following bounds for the depth and the regularity of
R/(I + J)n .
Theorem 2.4. For all n ≥ 1, we have

(i) depth R (I + J)n ≥

{depth A/I n−i + depth B/J i + 1, depth A/I n−j+1 + depth B/J j },

min

i∈[1,n−1], j∈[1,n]


(ii) reg R (I + J)n ≤
max

{reg A/I n−i + reg B/J i + 1, reg A/I n−j+1 + reg B/J j }.

i∈[1,n−1], j∈[1,n]

The proof of Theorem 2.4 is based on an approximation of (I + J)n by a sequence
of subideals whose successive quotients have good representations. The depth and the
regularity of these quotients can be expressed in terms of those of I and J by a recent
work of Hoa and Tam on mixed products of ideals [20]. We also present results showing
that the minimum and maximum values of the two terms in the bounds of Theorem 2.4
are attainable (Propositions 2.6, 2.7, and 2.9). These results particularly show that there
are no general formulae for the depth and regularity of R/(I + J)n .
On the other hand, our next main result gives the following general formulae for the
depth and regularity of (I + J)n /(I + J)n+1 .
Theorem 3.3. For all n ≥ 1, we have
(i) depth(I + J)n /(I + J)n+1 = min {depth I i /I i+1 + depth J j /J j+1},
i+j=n

n

(ii) reg(I + J) /(I + J)

n+1

= max {reg I i /I i+1 + reg J j /J j+1 }.
i+j=n

Theorem 3.3 is a consequence of the decomposition of (I +J)n /(I +J)n+1 as a direct sum
of modules of the form I i /I i+1 ⊗k J j /J j+1 , whose depth and regularity can be estimated
by invoking a formula of Goto and Watanabe for the local cohomology of tensor products
over a field [13].
We can use Theorems 2.4 and 3.3 to compute depth R/(I + J)n and reg(I + J)n for n
sufficiently large. It is known that the depth (resp., the regularity) of powers of an ideal
is asymptotically a constant (resp., a linear function); see for example [4, 8, 21]. However,
the exact values of this constant and linear function remain mysterious despite much effort
from various authors [6, 12, 11, 16, 17]. A naı̈ve intuition might suggest that the constant
for depth and the linear function for regularity of (I + J)n could be obtained by “adding
up” corresponding invariants of I and J. The following formulae show otherwise, and
give an evidence as why the problem is difficult.
Theorem 4.6. lim depth R/(I + J)n =
n→∞

min lim depth A/I i + min depth B/J j , min depth A/I i + lim depth B/J j .
i→∞

j≥1

i≥1

j→∞

Theorem 5.6. Suppose that reg I n = dn + e and reg J n = cn + f for n ≫ 0, where
c ≥ d. Let lin(I) and lin(J) denote the least integer m such that reg I n = dn + e
2

and reg J n = cn + f for n ≥ m, respectively. Set e∗ = maxi≤lin(I) {reg I i − ci} and
f ∗ = maxj≤lin(J) {reg J j − dj}. For n ≫ 0 we have
(
c(n + 1) + f + e∗ − 1
if c > d,
reg(I + J)n =
∗
∗
d(n + 1) + max{f + e , e + f } − 1 if c = d.
To prove Theorems 4.6 and 5.6 we make use of the fact that for n sufficiently large,
depth R/(I +J)n = depth(I +J)n−1 /(I +J)n and reg(I +J)n = reg(I +J)n−1 /(I +J)n +1.
Here, the first equality is due to Herzog and Hibi [16]. We also give an estimate for the
least number m such that reg(I + J)n becomes a linear function for n ≥ m. However, we
are unable to do the same for the index of stability of the function depth R/(I + J)n .
Our results have interesting applications on the depth function of powers of an ideal.
For example, we show that R/(I + J)i is Cohen-Macaulay for all i ≤ n if and only if A/I i
and B/J i are Cohen-Macaulay for all i ≤ n (Proposition 3.7). We also show that when
I, J are squarefree monomial ideals, depth R/(I + J)n is a constant function if and only
if so are depth A/I n and depth B/I n (Proposition 4.7). This property improves a recent
result of Herzog and Vladiou in [19] substantially. Furthermore, in [16], Herzog and Hibi
conjectured that the function depth R/Qn of a monomial ideal Q in a polynomial ring R
over k can be any convergent non-negative integer valued function, and gave the proof for
all non-decreasing functions and a large class of non-increasing functions. We complete
the statement for all non-increasing functions (Theorem 4.9).
We should point out that even though stated for homogeneous ideals, our results on
the depth in Theorem 2.4, Theorem 3.3, and Theorem 4.6 aso hold for non-homogeneous
ideals I, J if A, B, and R are replaced by their localizations at maximal homogeneous
ideals.
The paper is outlined as follows. In Section 1, we collect notations, terminology and
basic results on depth and regularity. In Section 2, we give bounds for depth R/(I + J)n
and reg R/(I + J)n . The formulae for depth(I + J)n /(I + J)n+1 and reg(I + J)n /(I + J)n+1
are proved in Section 3. Section 4 and Section 5 are devoted to the asymptotic behavior
of depth R/(I + J)n and reg(I + J)n , respectively.
1. Preliminaries
In this section, we collect notations, terminology and basic results used in the paper.
We follow standard texts [5, 9].
Let (A, m) be a local ring and let M be a finitely generated A-module. An M-sequence is
a sequence z1 , . . . , zr ∈ m such that (x1 , . . . , xi−1 )M : xi = (x1 , . . . , xi−1 )M for i = 1, . . . , r.
The depth of M, denoted by depth M, is the length of a (or any) maximal M-sequence.
We always have depth M ≤ dim M, and if depth M = dim M then M is called a CohenMacaulay module.
Let Hmi (M) denote the ith local cohomology module of M with support m, for i ≥ 0.
By Grothendieck’s theorem [5, Theorem 3.5.7], we have
depth M = min{i | Hmi (M) 6= 0}.
In this paper we shall use this characterization as the definition of depth M.
3

Now, let A be a standard graded algebra and let m be the maximal homogeneous ideal
of A. It is well-known that the notion of depth can also be defined similarly for finitely
generated graded A-modules, and that these notions of depth over local and graded rings
share the same properties and characterizations [5, Remark 3.6.18]. This allows us to refer
to depth of a module without specifying whether the underlying ring is local or graded.
In the graded setting, local cohomology modules Hmi (M) are also graded A-modules,
and we can introduce the invariants ai (M) = max{t [Hmi (M)]t 6= 0}, where by convention
ai (M) = −∞ if Hmi (M) = 0. The Castelnuovo-Mumford regularity (or simply regularity)
of M is defined by
reg M := max{ai (M) + i}.
i≥0

If A is a polynomial ring over a field then reg M controls the complexity of the graded
structure of M in the following sense. Let
M
M
0→
A(−j)βp,j (M ) → · · · →
A(−j)β0,j (M ) → M → 0
j∈Z

j∈Z

be a minimal free resolution of M. Then

reg(M) := max{j − i | βi,j (M) 6= 0}

by a result of Goto and Eisenbud [10].
Given a short exact sequence of finitely generated graded A-modules
0 → M → N → P → 0,

by taking the derived long exact sequence of local cohomology modules, we can easily
verify the following lemmas.
Lemma 1.1.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)

depth N ≥ min{depth M, depth P },
depth M ≥ min{depth N, depth P + 1},
depth P ≥ min{depth M − 1, depth N},
depth M = depth P + 1 if depth N > depth P ,
depth P = depth M − 1 if depth N > depth M,
depth P = depth N if depth N < depth M,
depth N = depth M if depth P ≥ depth M.

Lemma 1.2.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)

reg N ≤ max{reg M, reg P },
reg M ≤ max{reg N, reg P + 1},
reg P ≤ max{reg M − 1, reg N},
reg M = reg P + 1 if reg N < reg P ,
reg P = reg M − 1 if reg N < reg M,
reg P = reg N if reg N > reg M,
reg N = reg M if reg P + 1 < reg M.

These basic lemmas will be used in this paper without references. Using the fact that
a polynomial ring over a field is Cohen-Macaulay, and the characterization of regularity
in terms of the minimal free resolution, one can deduce the following well-known lemma,
which will also be used in this paper without references.
4

Lemma 1.3. Let A be a polynomial ring over a field and let I be a non-zero proper
homogeneous ideal in A. Then
(i) depth I = depth A/I + 1,
(ii) reg I = reg A/I + 1.
Let us conclude this section by fixing some notations for the remaining of the paper.
Let k be a field, and let A = k[x1 , . . . , xr ] and B = k[y1 , . . . , ys ] be polynomial rings
over k. Let I ⊂ A and J ⊂ B be arbitrary nonzero proper homogeneous ideals. For
convenience, we also use I and J to denote the extensions of I and J in the polynomial
ring R = k[x1 , . . . , xr , y1, . . . , ys ]. The main objectives of our investigation are the depth
and the regularity of powers of the sum I + J in R.
2. Depth and regularity of R/(I + J)n
The aim of this section is to give bounds for the depth and the regularity of (I + J)n
in terms of those of powers of I and J. We shall start with the following observation.
Lemma 2.1. [20, Lemmas 1.1] IJ = I ∩ J.
Lemma 2.1 allows us to show that (I + J)n possesses an ascending sequence of subideals
whose quotients of successive elements have a nice representation.
P
Lemma 2.2. Let Qi = ij=0 I n−j J j , i = 0, . . . , n. For i ≥ 1 we have
Qi /Qi−1 ∼
= I n−i J i /I n−i+1 J i .
Proof. It can be seen that
Qi /Qi−1 = I n−i J i + Qi−1 /Qi−1 ∼
= I n−i J i /Qi−1 ∩ I n−i J i .

Thus, it suffices to show that Qi−1 ∩ I n−i J i = I n−i+1 J i .
Note that Qi−1 = I n + I n−1 J + · · · + I n−i+1J i−1 . Then Qi−1 ∩ I n−i J i ⊇ I n−i+1 J i . On
the other hand, since Qi−1 ⊆ I n−i+1 , we have
Qi−1 ∩ I n−i J i ⊆ I n−i+1 ∩ I n−i ∩ J i = I n−i+1 ∩ J i = I n−i+1 J i ,

where the last equality follows from Lemma 2.1. Therefore, Qi−1 ∩ I n−i J i = I n−i+1 J i and
we are done.

Thanks to Lemma 2.2, the depth and regularity of Qi /Qi−1 can be estimated by invoking
the following results of Hoa and Tam [20].
Lemma 2.3. [20, Lemma 2.2 and Lemma 3.2]
(i) depth R/IJ = depth A/I + depth B/J + 1.
(ii) reg R/IJ = reg A/I + reg B/J + 1.
We are now ready to give bounds for the depth and the regularity of R/(I + J)n in
terms of those of powers of I and J in A and B, respectively.
Theorem 2.4. For all n ≥ 1, we have

(i) depth R (I + J)n ≥
min
{depth A/I n−i + depth B/J i + 1, depth A/I n−j+1 + depth B/J j },
i∈[1,n−1], j∈[1,n]

5


(ii) reg R (I + J)n ≤
max
{reg A/I n−i + reg B/J i + 1, reg A/I n−j+1 + reg B/J j }.
i∈[1,n−1], j∈[1,n]

Proof. Let Qi =
exact sequence

Pi

j=0 I

n−j

0 → Qi /Qi−1 → R/Qi−1 → R/Qi → 0

we can deduce that
(2.1)

J j , for i = 0, . . . , n. Note that Qn = (I + J)n . Using the short

depth R/Qn ≥ min{depth R/Q0 , depth Qi /Qi−1 − 1| i = 1, . . . , n},

(2.2)

reg R/Qn ≤ max{reg R/Q0 , reg Qi /Qi−1 − 1| i = 1, . . . , n}.

Since Q0 = R/I n , we have
depth R/Q0 = depth A/I n + s ≥ depth A/I n + depth B/J,
reg R/Q0 = reg A/I n ≤ reg A/I n + reg B/J.

By Lemma 2.2, Qi /Qi−1 = I n−i J i /I n−i+1 J i . Hence, we have the short exact sequence
It then follows that

0 → Qi /Qi−1 → R/I n−i+1 J i → R/I n−i J i → 0.

depth Qi /Qi−1 ≥ min{depth R/I n−i J i + 1, depth R/I n−i+1J i },

(2.3)

reg Qi /Qi−1 ≤ max{reg R/I n−i J i + 1, reg R/I n−i+1J i }.

(2.4)

For i = 1, . . . , n − 1, by applying Lemma 2.3 to compute the depth and regularity in
the left hand side of (2.3) and (2.4), we get
depth Qi /Qi−1 −1 ≥ min{depth A/I n−i + depth B/J i + 1, depth A/I n−i+1 + depth B/J i },
reg Qi /Qi−1 −1 ≤ max{reg A/I n−i + reg B/J i + 1, reg A/I n−i+1 + reg B/J i }.

For i = n, notice that
depth R/J n = r + depth B/J n ≥ depth A/I + depth B/J n + 1 = depth R/IJ n ,
Thus,

reg R/J n = reg B/J n ≤ reg A/I + reg B/J n + 1 = reg R/IJ n .

depth Qn /Qn−1 −1 ≥ depth A/I + depth B/J n + 1,
reg Qn /Qn−1 −1 ≤ reg A/I + reg B/J n + 1.

The conclusion now follows by combining the above estimates for depth R/Q0 , reg R/Q0 ,
depth Qi /Qi−1 −1, and reg Qi /Qi−1 −1, for i = 1, . . . , n, together with (2.1) and (2.2). 
The bounds in Theorem 2.4 are given by the minimum and maximum values of two
terms. The following propositions show that the values of both terms are attainable and,
thus, both terms are essential in the statement of Theorem 2.4. For that we shall need
the following result on tensor products of modules over a field.
Lemma 2.5. Let M and N be graded module over A and B, respectively. Then
(i) depth M ⊗k N = depth M + depth N,
(ii) reg M ⊗k N = reg M + reg N.
6

Proof. Let M denote the maximal ideal of R. Note that
k
depth M ⊗k N = min{k| HM
(M ⊗k N) 6= 0},

k
reg M ⊗k N = max{n + k| HM
(M ⊗k N)n 6= 0, k ≥ 0}.

The assertions follow from the following formula for the local cohomology modules of
tensor products of Goto-Watanabe [13, Theorem 2.2.5]:
M
k
Hmi (M) ⊗k Hnj (N),
HM
(M ⊗k N) =
i+j=k

where m and n denote the maximal ideals of A and B.



Proposition 2.6. Assume that depth A/I 2 ≥ depth A/I + 1.
(i) If depth B/J 2 ≥ depth B/J + 1 then depth R/(I + J)2 = depth A/I + depth B/J + 1.
(ii) If depth B/J 2 < depth B/J then depth R/(I + J)2 = depth A/I + depth B/J 2 .
Proof. From the proof of Theorem 2.4, we have the following short exact sequences
(2.5)
(2.6)

0 → IJ/I 2 J → R/I 2 → R/(I 2 + IJ) → 0,

0 → J 2 /IJ 2 → R/(I 2 + IJ) → R/(I + J)2 → 0.

Note that IJ = I ⊗k J for arbitrary ideals I ⊆ A and J ⊆ B. Thus, IJ/I 2J ∼
= (I/I 2 ) ⊗k J
2
2
2 ∼
and J /IJ = (A/I) ⊗k J . Therefore, it follows from Lemma 2.5 that
depth IJ/I 2J = depth I/I 2 + depth J = depth I/I 2 + depth B/J + 1,

depth J 2 /IJ 2 = depth A/I + depth J 2 = depth A/I + depth B/J 2 + 1.
By considering the short exact sequence 0 → I/I 2 → A/I 2 → A/I → 0, it follows from
our hypotheses that depth I/I 2 = depth A/I + 1. Since depth B/J + 1 ≤ s, we get
depth R/I 2 = depth A/I 2 + s ≥ depth I/I 2 + depth B/J + 1 = depth IJ/I 2 J.

By (2.5), this implies that

depth R/(I 2 + IJ) ≥ depth IJ/I 2 J − 1 = depth A/I + depth B/J + 1.
(i) If depth B/J 2 ≥ depth B/J + 1 then depth B/J + 1 ≤ dim B/J < s. Hence,
depth R/I 2 > depth IJ/I 2 J. We thus have
depth R/(I 2 + IJ) = depth IJ/I 2J − 1 = depth A/I + depth B/J + 1
By (2.6), this implies that

≤ depth A/I + depth B/J 2 = depth J 2 /IJ 2 − 1.

depth R/(I + J)2 = depth R/(I 2 + IJ) = depth A/I + depth B/J + 1.
(ii) If depth B/J 2 < depth B/J then
depth R/(I 2 + IJ) ≥ depth A/I + depth B/J + 1

> depth A/I + depth B/J 2 + 1 = depth J 2 /IJ 2 .

By (2.6), this implies that
depth R/(I + J)2 = depth J 2 /IJ 2 − 1 = depth A/I + depth B/J 2 .

7



To find ideals I which satisfy the conditions of Proposition 2.6 we refer the reader to [16,
Theorem 4.1], where for any bounded non-decreasing function f : N → N, one constructs
a monomial ideal I such that depth A/I n = f (n) for all n ≥ 1.

Proposition 2.7. Assume that reg A/I 2 ≤ reg A/I + 1.
(i) If reg B/J 2 ≤ reg B/I + 1 then reg R/(I + J)2 = reg A/I + reg B/J + 1.
(ii) If reg B/J 2 > reg B/I then reg R/(I + J)2 = reg A/I + reg B/J 2 .
Proof. The proof is similar to that of Proposition 2.6.



It is easy to find ideals I with reg A/I 2 > reg A/I. However, to find ideals I with
reg A/I 2 ≤ reg A/I + 1 is hard. We are thankful to A. Conca, who communicated to us
the following example.
Example 2.8. Let A = k[x1 , x2 , x3 ] and I = (x41 , x31 x2 , x1 x32 , x42 , x21 x22 x53 ). Using Macaulay2,
we get reg A/I = 8 and reg A/I 2 = 7.
If J is generated by linear forms, we have the following formulae. Note that in this
case, depth B/J n = n − 1 for n ≥ 1.
Proposition 2.9. Assume that J is generated by linear forms. Then
(i) depth R/(I + J)n = mini≤n depth A/I i + dim B/J,
(ii) reg R/(I + J)n = maxi≤n {reg A/I i − i} + n.
Proof. Without restriction we may assume that J = (y1 , ..., yt), t ≤ s. Then dim B/J =
s − t. Set B ′ = k[y1 , ..., yt ], J ′ = (y1 , ..., yt )B ′ , and R′ = k[x1 , ..., xr , y1, ..., yt ]. Then
depth R/(I + J)n = depth R′ /(I + J ′ )n + s − t,
reg R/(I + J)n = reg R′ /(I + J ′ )n .

Therefore, we only need to prove the case t = s.
If t = s = 1, we set y = y1 . Then B = k[y] and J = (y). Write R = ⊕i≥0 Ay i and
(I, y)n = I n ⊕ I n−1y ⊕ · · · ⊕ Ay n ⊕ Ay n+1 ⊕ · · · .

Then R/(I, y)n = ⊕i≤n (A/I i )y n−i . From this it follows that
depth R/(I, y)n = min depth A/I i ,
i≤n

n

reg R/(I, y) = max{reg A/I i + n − i}.
i≤n

If t = s > 1, we set A′ = k[x1 , ..., xr , y1, ..., ys−1 ] and I ′ = (I, y1, ..., ys−1 )A′ . Using
induction we may assume that
depth A′ /(I ′ )n = min depth A/I i ,
i≤n

′

′ n

reg A /(I ) = max{reg A/I i − i} + n.
i≤n

′

Note that I + J = (I , ys ). Then we have
depth R/(I + J)n = min depth A′ /(I ′ )i = min depth A/I i ,
i≤n

i≤n

′

n

′ i

reg R/(I + J) = max{reg A /(I ) − i} + n = max{reg A/I i − i} + n.
i≤n

i≤n

8


Propositions 2.6, 2.7 and 2.9 show that there is no general formulae for depth R/(I +J)n
and reg R/(I + J)n , and that both terms in the statement of Theorem 2.4 are essential.
3. Depth and regularity of (I + J)n /(I + J)n+1
In this section, we shall see that the depth and regularity of (I + J)n /(I + J)n+1 can
be nicely related to those associated to I and J. In particular, we shall provide bounds
for the depth and regularity of R/(I + J)n in terms of those of I i /I i+1 and J j /J j+1 .
We start by making the following observation, which is essential for our results in this
section.
Lemma 3.1. I i /I i+1 ⊗k J j /J j+1 ∼
= I i J j /(I i+1 J j + I i+1 J j ).
Proof. It is clear that
I i /I i+1 ⊗k J j /J j+1 ∼
= (I i ⊗k J j /I i ⊗k J j+1 )/(I i+1 ⊗k J j /I i+1 ⊗k J j+1 ).
∼
= (I i J j /I i J j+1 )/(I i+1 J j /I i+1J j+1 ).
Using Lemma 2.1, we have
I i+1 J j /I i+1 J j+1 = I i+1 ∩ J j /I i+1 ∩ J j+1 = I i+1 ∩ J j /(I i ∩ J j+1 ) ∩ (I i+1 ∩ J i )
∼
= I i+1 ∩ J j + I i ∩ J j+1 /I i ∩ J j+1 = I i+1 J j + I i J j+1/I i J j+1 .

Therefore,

I i /I i+1 ⊗k J j /J j+1 ∼
= (I i J j /I i J j+1 )/(I i+1 J j + I i J j+1 /I i J j+1 )
∼
= I i J j /(I i+1 J j + I i+1 J j ).

Proposition 3.2. (I + J)n /(I + J)n+1 =

M

i+j=n

I i /I i+1 ⊗k J j /J


j+1

.

P
Proof. Note that (I + J)n = i+j=n I i J j . Let (i, j) and (h, t) be two different pairs of
non-negative integers such that i + j = h + t = n. By Lemma 2.1, we have
I i J j ∩ I h J t = (I i ∩ J j ) ∩ (I h ∩ J t ) = I max{i,h} ∩ J max{j,t}

It then follows that

= J max{i,h} J max{j,t} ⊆ (I + J)n+1 .

(I + J)n /(I + J)n+1 =

M

I i J j + (I + J)n+1 /(I + J)n+1

i+j=n

(3.1)

∼
=

M

i+j=n


I i J j /(I + J)n+1 ∩ I i J j .



It is easy to verify that (I + J)n+1 ∩ I i J j ⊇ I i+1 J j + I i+1 J j . On the other hand, it can
be seen that (I + J)n+1 ⊆ I i+1 + J j+1 . Hence,

(I + J)n+1 ∩ I i J j ⊆ (I i+1 + J j+1 ) ∩ I i ∩ J j ⊆ I i+1 + I i ∩ J j+1 ∩ J j
= I i+1 ∩ J j + I i+1 ∩ J j = I i+1 J j + I i+1 J j ,
9

where the last equality follows by applying Lemma 2.1. Therefore,
(I + J)n+1 ∩ I i J j = I i+1 J j + I i+1 J j .

By Lemma 3.1, I i J j / I i+1 J j + I i+1 J j ∼
= I i /I i+1 ⊗k J j /J j+1 . Thus,
I i J j /(I + J)n+1 ∩ I i J j ∼
= I i /I i+1 ⊗k J j /J j+1.

The conclusion now follows from (3.1).



The decomposition of (I +J)n /(I +J)n+1 in Proposition 3.2 yields the following formulae
for its depth and the regularity.
Theorem 3.3. For all n ≥ 1, we have

(i) depth(I + J)n /(I + J)n+1 = min {depth I i /I i+1 + depth J j /J j+1 },
i+j=n

n

(ii) reg(I + J) /(I + J)

n+1

= max {reg I i /I i+1 + reg J j /J j+1}.
i+j=n

Proof. By Proposition 3.2, we have
depth(I + J)n /(I + J)n+1 = min depth(I i /I i+1 ⊗k J j /J j+1 ),
i+j=n

n

reg(I + J) /(I + J)

n+1

= max reg(I i /I i+1 ⊗k J j /J j+1).
i+j=n

Hence, the assertions follow by applying Lemma 2.5.



Now, we can relate the depth and regularity of R/(I + J)n to those of successive
quotients associated to I and J.
Corollary 3.4. For all n ≥ 1, we have

(i) depth R/(I + J)n ≥ min {depth I i /I i+1 + depth J j /J j+1 },
i+j≤n−1

n

(ii) reg R/(I + J) ≤ max {reg I i /I i+1 + reg J j /J j+1 }.
i+j≤n−1

Proof. Using the short exact sequences
0 → (I + J)t /(I + J)t+1 → R/(I + J)t+1 → R/(I + J)t → 0

for t < n, we deduce that

depth R/(I + J)n ≥ min depth(I + J)t /(I + J)t+1 ,
t≤n−1

n

reg R/(I + J) ≤ max reg(I + J)t /(I + J)t+1 .
t≤n−1

Hence, the assertions follow from Theorem 3.3.
Remark 3.5. For an arbitrary ideal I ⊆ A, by considering the short exact sequence
it can be seen that

0 −→ I i /I i+1 −→ A/I i+1 −→ A/I i −→ 0,

depth I i /I i+1 ≥ min{depth A/I i + 1, depth A/I i+1 },
reg I i /I i+1 ≤ max{reg A/I i + 1, reg A/I i+1 }.
10



Thus, from Corollary 3.4, one can derive bounds for depth R/(I + J)n and reg R/(I + J)n
in terms of depth A/I i , depth B/J j , reg A/I i , and reg B/J j . However, these bounds are
worse than what was given in Theorem 2.4, since they involve all indices i, j with i+j ≤ n.
It often happens that depth I i−1 /I i ≥ depth I i /I i+1 and, if I is generated by forms
of degree ≥ 2, reg I i−1 /I i + 2 ≤ reg I i /I i+1 . In these situations, we have the following
formulae for depth R/(I + J)n and reg R/(I + J)n .
Corollary 3.6.
(i) If depth I i−1 /I i ≥ depth I i /I i+1 for i ≤ n − 1, then
depth R/(I + J)n = min {depth I i /I i+1 + depth J j /J i+1 }.
i+j=n−1

(ii) If reg I i−1 /I i + 2 ≤ reg I i /I i+1 for i ≤ n − 1, then

reg R/(I + J)n = max {reg I i /I i+1 + reg J j /J i+1 }.
i+j=n−1

Proof. (i) By Theorem 3.3.(i) we only need to prove that
depth R/(I + J)n = depth(I + J)n−1 /(I + J)n .
For n = 1, this is trivial. For n > 1, we may assume that depth R/(I + J)n−1 = depth(I +
J)n−2 /(I + J)n−1 . By Theorem 3.3.(i) and the assumption depth I i−1 /I i ≥ depth I i /I i+1 ,
i ≤ n − 1, we can see that depth(I + J)n−2 /(I + J)n−1 ≥ depth(I + J)n−1 /(I + J)n . Thus,
depth R/(I + J)n−1 ≥ depth(I + J)n−1 /(I + J)n . Now, from the exact sequence
0 → (I + J)n−1 /(I + J)n → R/(I + J)n → R/(I + J)n−1 → 0

we get depth R/(I + J)n = depth(I + J)n−1 /(I + J)n .
(ii) As in the proof of (i) we only need to prove that

reg R/(I + J)n = reg(I + J)n−1 /(I + J)n
for n > 1. By induction we may assume that reg R/(I + J)n−1 = reg(I + J)n−2/(I + J)n−1.
From Theorem 3.3.(ii) and the assumption reg I i−1 /I i + 2 ≤ reg I i /I i+1 , i ≤ n − 1,
we can deduce that reg(I + J)n−2 /(I + J)n−1 + 2 ≤ reg(I + J)n−1 /(I + J)n . Thus,
reg R/(I + J)n−1 + 2 ≤ reg(I + J)n−1 /(I + J)n . Hence, from the above exact sequence we
get reg R/(I + J)n = reg(I + J)n−1 /(I + J)n .

Theorem 3.3.(i) has a nice application on the Cohen-Macaulayness of R/(I + J)n .
Proposition 3.7. R/(I + J)i is Cohen-Macaulay for all i ≤ n if and only if A/I i and
B/J i are Cohen-Macaulay for all i ≤ n.
Proof. It is obvious that
dim(I +J)n−1 /(I +J)n = dim R/(I +J) = dim A/I +dim B/J = dim I i /I i+1 +dim J j /J j+1.
By Corollary 3.4.(i), it can be seen that depth(I + J)n−1/(I + J)n = dim R/(I + J) if and
only if depth I i−1 /I i = dim A/I and depth J i−1 /J i = dim B/J for all i ≤ n. It follows
that (I + J)n−1 /(I + J)n is Cohen-Macaulay if and only if I i−1 /I i and J i−1 /J i are CohenMacaulay for all i ≤ n. In particular, these conditions imply that (I + J)i−1 /(I + J)i are
Cohen-Macaulay for i ≤ n.
11

On the other hand, using induction and the exact sequence
0 → (I + J)i−1 /(I + J)i → R/(I + J)i → R/(I + J)i−1 → 0
we can prove that (I +J)i−1 /(I +J)i is Cohen-Macaulay for i ≤ n if and only if R/(I +J)i
is Cohen-Macaulay for i ≤ n. Similarly, I i−1 /I i and J i−1 /J i are Cohen-Macaulay for i ≤ n
if and only if A/I i and B/J i are Cohen-Macaulay for i ≤ n. The result is proved.

The proof of Theorem 3.7 exhibits an interesting phenomenon that if (I +J)n−1 /(I +J)n
is Cohen-Macaulay, then so is R/(I + J)i for i ≤ n. This implication does not hold in
general for an arbitrary ideal.
Example 3.8. Let I = A = K[x, y, z] and I = (x4 , x3 y, xy 3, y 4, x2 y 2z). Then dim A = 1,
depth A/I = 0, and depth A/I 2 = 1 (see [16, Theorem 4.1]). From the exact sequence
0 → I/I 2 → A/I 2 → A/I → 0 one can deduce that depth I/I 2 = 1. Hence, I/I 2 is
Cohen-Macaulay but A/I is not.
4. Asymptotic behavior of depth
For an arbitrary ideal I, it is a well celebrated result of Brodman [4] that depth(A/I n ) is
a constant function for n ≫ 0. There has been much interest in estimating the stable value
limn→∞ depth A/I n and the stability index of the function depth A/I n [16, 19, 22, 28].
In this section, we describe the stable value of depth R/(I + J)n in terms of those of
depth A/I n and depth B/J n .
We first recall the following result of Herzog and Hibi.
Lemma 4.1. [16, Theorem 1.2] For an arbitrary homogeneous ideal I, depth I i−1 /I i is a
constant for i ≫ 1 and
lim depth A/I i = lim depth I i−1 /I i .

i→∞

i→∞

Let us denote by s(I) the stability index of the function depth I i−1 /I i , that is, s(I) is
the least integer m such that depth I i−1 /I i = depth I i /I i+1 for i ≥ m. Note that
min depth I i−1 /I i = min depth I i−1 /I i .
i≥1

i≤s(I)

For large enough n, we obtain the following formula for depth(I + J)n−1 /(I + J)n .
Proposition 4.2. For n ≥ s(I) + s(J) − 1, we have

depth(I + J)n−1 /(I + J)n =

min lim depth I i−1 /I i + min depth J j−1 /J j , min depth I i−1 /I i + lim depth J j−1 /J j .
i→∞

j≤s(J)

i≤s(I)

j→∞

Proof. It follows from Theorem 3.3.(i) that

depth(I + J)n−1 /(I + J)n = min {depth I i−1 /I i + depth J j−1/J j }.
i+j=n+1

By the definition of s(I) and s(J), we have that depth I i−1 /I i = limi→∞ depth I i−1 /I i for
i ≥ s(I) and depth J i−1 /J j = limj→∞ depth J j−1 /J j for j ≥ s(J).
12

Consider n ≥ s(I) + s(J) − 1 and i + j = n + 1. Then, i ≥ s(I) if j ≤ s(J). Therefore,

 lim depth I i−1 /I i + depth J i−1 /J j if j ≤ s(J),
i−1
i
i−1
j
depth I /I + depth J /J = i→∞ i−1 i
depth I /I + lim depth J j−1 /J j if j ≥ s(J).
j→∞

Note that i ≤ n − s(J) + 1 if and only if j ≥ s(J), and that
min

i≤n−s(J)+1

depth I i−1 /I i = min depth I i−1 /I i
i≤s(I)

since n − s(J) + 1 ≥ s(I). Then we have

min {depth I i−1 /I i + depth J j−1/J j } =

min lim depth I i−1 /I i + min depth J j−1/J j , min depth I i−1 /I i + lim depth J j−1 /J j
i+j=n+1

i→∞

j≤s(J)

j→∞

i≤s(I

for n ≥ s(I) + s(J) − 1, which yields the assertion.



Proposition 4.2 immediately
gives the following bound for the index of stability of the

function depth(I + J)n−1 (I + J)n .
Corollary 4.3. s(I + J) ≤ s(I) + s(J) − 1.

To derive a formula for limn→∞ depth R/(I + J)n in terms of those of A/I i and B/J j ,
we shall need the following observation.
Lemma 4.4. min depth A/I i = min depth I i−1 /I i .
i≥1

i≥1

Proof. Let m be the least integer such that
depth I m−1 /I m = min depth I i−1 /I i .
i≥1

Using the short exact sequences 0 → I

j−1

j

/I → A/I j → A/I j−1 → 0 for j ≤ i, we get

depth A/I i ≥ min depth I j−1 /I j ≥ depth I m−1 /I m
j≤i

for all i ≥ 1. In particular, depth A/I m−1 ≥ depth I m−1 /I m . Hence, from the short exact
sequence 0 → I m−1 /I m → A/I m → A/I m−1 → 0, we get depth A/I m = depth I m−1 /I m .
Thus, mini≥1 depth A/I i = depth I m−1 /I m .

Remark 4.5. In general, we do not have mini≥1 depth A/I i = limi→∞ depth A/I i (see,
for example, [2, Theorem 0.1]).
Theorem 4.6. lim depth R/(I + J)n =
n→∞

min lim depth A/I i + min depth B/J j , min depth A/I i + lim depth B/J j .
i→∞

j≥1

i≥1

j→∞

Proof. It follows from Lemma 4.1 and Proposition 4.2 that

lim depth R/(I + J)n = lim depth(I + J)n−1 /(I + J)n =

n→∞

min



lim depth I

i→∞

i−1

n→∞

i

/I + min depth J
j≥1

j−1

/J j , min depth I i−1 /I i + lim depth J j−1 /J j .
j→∞

i≥1

By Lemma 4.1 and Lemma 4.4, we can replace I
obtain the assertion.
13

i−1

i

/I , J

j−1

/J j by A/I i , B/J j to


We are unable to deduce from Corollary 4.3 a bound for the stability index of the
function depth R/(I + J)n . In general, there seems to be no relationships between the
stability indices of the functions depth A/I n and depth I n−1 /I n .
We shall now give two interesting applications on the depth functions. Recall that
an ideal is said to have a constant depth function if the depth of all its powers are the
same. This notion was introduced by Herzog and Vladiou in [19]. The following result
was proved in [19] under the additional assumption that the Rees rings of I and J are
Cohen-Macaulay.
Proposition 4.7. (cf. [19, Theorem 1.1]) Let I and J be squarefree monomial ideals.
Then I + J has a constant depth function if and only if so do I and J.
Proof. Assume that I + J has a constant depth function. Let d = limn→∞ depth A/I n
and e = limn→∞ depth B/J n . By Theorem 4.6, we have depth R/(I + J) ≤ d + e. By the
proof of [18, Theorem 2.6], depth A/I n ≤ depth A/I and depth B/J n ≤ depth B/J for all
n ≥ 1. As a consequence,
d + e ≤ depth A/I + depth B/J = depth R/(I + J).

Therefore, depth R/(I + J) = d + e, depth A/I = d and depth B/J = e. By Theorem 4.6,
this implies that depth A/I = minn≥1 depth A/I n and depth B/J = minn≥1 depth B/J n .
Thus, we must have depth A/I n = depth A/I and depth B/J n = depth B/J for all n ≥ 1.
Conversely, assume that I and J have constant depth functions. By Theorem 2.4, we
obtain
depth R/(I + J)n ≥ depth A/I + depth B/J = depth R/(I + J)
for all n ≥ 1. Following the proof of [18, Theorem 2.6], we also get that depth R/(I +J)n ≤
depth R/(I + J). Therefore, depth R/(I + J)n = depth R/(I + J) for all n ≥ 1.

It is easy to construct examples showing that Theorem 4.7 does not hold for nonsquarefree monomial ideals.
Example 4.8. Take I to be a monomial ideal with depth I n /I n+1 = 0 for all n ≥ 1 (e.g.
dim A/I = 0) and J a monomial ideal with depth B/J = 0 such that J does not have a
constant depth function (see e.g. [16]). Then depth(I + J)n−1 /(I + J)n = 0 for all n ≥ 1
by Theorem 3.3. This implies that depth R/(I + J)n = 0 for all n ≥ 1.
Given a convergent non-negative integer valued function f , Herzog and Hibi [16] conjectured that there exists a monomial ideal Q in a polynomial ring R such that depth R/Qn =
f (n) for all n ≥ 1. They proved the conjecture for all bounded non-decreasing functions
and a special class of non-increasing functions. We complete the statement for all nonincreasing functions in the following theorem.
Theorem 4.9. For any non-increasing function f : N → N there exist a monomial ideal
Q in a polynomial ring R such that depth R/Qn = f (n) for all n ≥ 1.
Proof. We shall first show that there exists a monomial ideal Q in a polynomial ring R
over k such that depth Qn−1 /Qn = f (n) for all n ≥ 1. Let t be the least integer such that
f (n) = f (n + 1) for all n ≥ t.
Assume that f (t) = 0. Then f (n) = 0 for all n ≥ t. If t = 0 then choose Q to be any
monomial ideal with dim R/Q = 0. If t > 0 then consider the function g(n) = f (n) − 1
14

for n < t, g(n) = 0 for n ≥P
t. Let v be the least integer such that g(n) = g(n + 1) for all
n ≥ v. Using induction on tn=1 f (n), we may assume that there exists a monomial ideal
I in a polynomial ring A such that depth I n−1 /I n = g(n) for all n ≥ 1. Choose J to be
a monomial ideal in a polynomial ring B such that depth J n−1 /J n = 1 for n < t − v and
depth J n−1 /J n = 0 for n > t − v (see Example 4.10). Set Q = I + J. By Theorem 3.3.(i),
it can be easily verified that depth Qn−1 /Qn = f (n).
If f (t) > 0 then then f (n) > 0 for all n ≥ 1. Consider g(n) = f (n) − 1 for all
n ≥ 1. Using induction on f (t), we may assume that there exists a monomial ideal
I in a polynomial ring A such that depth I n−1 /I n = g(n) for all n ≥ 1. Choose J
to be a monomial ideal in a polynomial ring B such that depth J n−1 /J n = 1 for all
n ≥ 1 (e.g. J = (y1 ) ⊂ k[y1 , y2] = B). Put Q = I + J. Then Theorem 3.3(i) yields
depth Qn−1 /Qn = f (n) for all n ≥ 1.
Once there exists a homogeneous ideal Q such that depth Qn−1 /Qn = f (n) for all n ≥ 1,
we can proceed to show that depth R/Qn = depth Qn−1 /Qn as follows. The assertion is
trivial for n = 1. For n ≥ 2, we may assume that
depth R/Qn−1 = depth Qn−2 /Qn−1 < depth Qn−1 /Qn .

Then, using the short exact sequence 0 → Qn−1 /Qn → R/Qn → R/Qn−1 → 0, we deduce
that depth R/Qn = depth Qn−1 /Qn .

3t−2
x3 )3 , for
x3 , x3t−1
Example 4.10. Let I be the integral closure of the ideal (x3t
2
1 , x1 x2
n
some t ≥ 1, in the ring A = k[x1 , x2 , x3 ]. Then, dim A/I = 1 and I is integrally closed
for n ≥ 1.
By [27, Proposition 4], Ass I n−1 /I n = Ass A/I n , where Ass denotes the set of the
associated primes. By [27, Example, p. 54], (x1 , x2 , x3 ) ∈ Ass A/I n if and only if n ≥ r.
Thus, depth I n−1 /I n = 1 for n < t and depth I n−1 /I n = 0 for n ≥ t.
Note that examples with this property do not belong to the class of ideals with nonincreasing depth functions constructed by Herzog and Hibi in [16].

5. Asymptotic behavior of regularity
For an arbitrary homogeneous ideal I in A, it is well known that reg I n = dn + e for
n ≫ 0, where d is the minimum of the maximal generating degree of a homogeneous
reduction of I and e is a non-negative integer [8, 21]. Recall that a homogeneous ideal
Q ⊆ I is called a reduction of I if I n+1 = QI n for some n ≥ 1. We denote by lin(I) the
least natural number m such that reg I n = dn + e for n ≥ m. In general, it is hard to
estimate e and lin(I); see, for example, [3, 6, 12, 11]. In this section, we investigate these
invariants of (I + J) in terms of those of I and J.
We shall need the following lemmas on the relation between reg I n and reg I n−1 /I n .
Lemma 5.1. reg I n−1 /I n = reg I n − 1 for n ≥ lin(I) + 1.
Proof. We have reg I n = dn+e > d(n−1)+e = reg I n−1 for n ≥ lin(I)+1. Therefore, using
the short exact sequence 0 → I n → I n−1 → I n−1 /I n → 0, we deduce that reg I n−1 /I n =
reg I n − 1 for n ≥ lin(I) + 1.

In general, lin(I) is not the least integer m such that reg I n−1 /I n becomes a linear
function for n ≥ m + 1.
15

Example 5.2. Let A = k[x1 , x2 ] and I = (x31 , x21 x2 , x1 x22 , x32 ). Then I n = (x1 , x2 )n for
n ≥ 2. Therefore, it is easy to check that reg I = 4 and reg I n = 3n for n ≥ 2, while
reg A/I = 3 and reg I n−1 /I n = 3n − 1 for n ≥ 2.
Lemma 5.3. For any positive integers c and t we have
max{reg I n−1 /I n − cn} = max{reg I n − cn} − 1.
n≤t

n≤t

Proof. Let m ≤ t be the least positive integer such that

reg I m−1 /I m − cm = max{reg I n−1 /I n − cn}.
n≤t

Using the short exact sequences 0 → I i−1 /I i → A/I i → A/I i−1 → 0 for i ≤ n, we deduce
that reg A/I n ≤ maxi≤n reg I i−1 /I i . Thus,
reg A/I n − cn ≤ max{reg I i−1 /I i − ci} ≤ reg I m−1 /I m − cm
i≤n

for n ≤ t. In particular,

reg A/I m−1 − c(m − 1) ≤ max {reg I i−1 /I i − ci} < reg I m−1 /I m − cm.
i≤m−1

Therefore, reg A/I m−1 +1 < reg I m−1 /I m . Using the above short exact sequence for i = m,
it follows that reg A/I m = reg I m−1 /I m . Hence, reg A/I m − cm = reg I m−1 /I m − cm. This
implies that
max{reg A/I n − cn} = reg I m−1 /I m − cm.
n≤t

Note that reg I n = reg A/I n + 1. Then maxn≤t {reg I n − cn} = reg I m−1 /I m − cm + 1 and
the conclusion follows.

Remark 5.4. If c = d and t = lin(I) we have
max {reg I n − dn} = max{reg I n − cn}.
n≥0

n≤lin(I)

If dim A/I = 0 and I is generated in a single degree, Eisenbud and Harris [12, Proposition
1.1] proved that the function reg I n − dn is non-decreasing (see also [3]). In this case, if
reg I n −dn is not a constant function, reg I n −dn has its maximum value at n = 1 < lin(I).
This shows that maxn≤lin(I) {reg I n − cn} needs not be attained at n = lin(I).
Using Theorem 3.3 we find the following asymptotic formula for reg(I + J)n−1/(I + J)n .
Proposition 5.5. Assume that reg I n = dn + e and reg J n = cn + f for n ≫ 0, c ≥ d.
Set e∗ = maxi≤lin(I) {reg I i − ci} and f ∗ = maxj≤lin(J) {reg J j − dj}. For n ≥ lin(I) + lin(J)
we have
reg(I + J)n−1 /(I + J)n = max{d(n + 1) + e + f ∗ , c(n + 1) + f + e∗ } − 2.
Proof. By Theorem 3.3.(ii), we have
reg(I + J)n−1 /(I + J)n = max {reg I i−1 /I i + reg J j−1 /J j }.
i+j=n+1

Consider n ≥ lin(I)+lin(J) and i+j = n+1. It can be seen that i ≥ lin(I)+1 if j ≤ lin(J).
By Lemma 5.1, reg I i−1 /I i = di + e − 1 if i ≥ lin(I) + 1 and reg J j−1 /J j = cj + f − 1 if
j ≥ lin(J) + 1. Therefore,
16

(
di + e − 1 + reg J j−1 /J j if
reg I i−1 /I i + reg J j−1 /J j =
reg I i−1 /I i + cj + f − 1 if
(
d(n + 1) + e − 1 + reg J j−1 /J j − dj if j
=
c(n + 1) + f − 1 + reg I i−1 /I i − ci if j

j ≤ lin(J),
j ≥ lin(J) + 1.

≤ lin(J),
≥ lin(J) + 1.

Note that j ≥ lin(J) + 1 if and only if i ≤ n − lin(J). By Lemma 5.3,

max {reg J j−1 /J j − dj} = max {reg J j − dj} − 1 = f ∗ − 1,

j≤lin(J)

j≤lin(J)

max {reg I i−1 /I i − ci} =

i≤n−lin(J)

max {reg I i − ci} − 1.

i≤n−lin(J)

For i ≥ lin(I), we have reg I i −ci = (d−c)i+e, which is a non-increasing function because
c ≥ d. Since n − lin(J) ≥ lin(I), this property implies
max {reg I i − ci} = max {reg I i − ci} = e∗ .

i≤n−lin(J)

i≤lin(I)

Therefore,
max {reg I i−1 /I i − ci} = e∗ − 1.

i≤n−lin(J)

Taking into account the maximum in the two cases j ≤ lin(I) and j ≥ lin(J) + 1, we
get

max {reg I i−1 /I i + reg J j−1 /J j } = max d(n + 1) + e + f ∗ − 2, c(n + 1) + f + e∗ − 2
i+j=n+1

= max d(n + 1) + e + f ∗ , c(n + 1) + f + e∗ − 2
for n ≥ lin(I) + lin(J).



We are now ready to give the asymptotic linear function of reg(I + J)n .
Theorem 5.6. Assume that reg I n = dn + e and reg J n = cn + f for n ≫ 0, c ≥ d. Set
e∗ = maxi≤lin(I) {reg I i − ci} and f ∗ = maxj≤lin(J) {reg J j − dj}. Then, for n ≫ 0, we have
(
c(n + 1) + f + e∗ − 1
if c > d,
reg(I + J)n =
∗
∗
d(n + 1) + max{f + e , e + f } − 1 if c = d.
Proof. By Lemma 5.1, we have reg(I + J)n = reg(I + J)n−1 /(I + J)n + 1 for n ≫ 0. Hence,
the conclusion follows by applying Proposition 5.5 for n ≫ 0.

We can also give an upper bound for lin(I + J) in terms of related invariants of I and
J. This follows from the following formula for reg(I + J)n .
Proposition 5.7. Assume that reg I n = dn+ e and reg J n = cn+ f for n ≫ 0, c ≥ d. Set
e∗ = maxi≤lin(I) {reg I i − ci} and f ∗ = maxj≤lin(J) {reg J j − dj}. For n ≥ lin(I) + lin(J) + 1,
we have
reg(I + J)n = max{d(n + 1) + e + f ∗ , c(n + 1) + f + e∗ } − 1.
Proof.√If d = 1, then I has a reduction Q generated by linear forms. Since Q is a prime
ideal, I = Q ⊆ I. This implies I = Q. Hence, reg I n = n for all n ≥ 1. Thus, lin(I) = 1,
e = 0 and e∗ = 1 − c. By Proposition 2.9,
reg(I + J)n = max{reg J i − i} + n
i≤n

17

for n ≥ 1. For lin(J) ≤ i ≤ n, reg J i − i + n = ci + f − i + n = cn + f + (1 − c)(n − i).
Hence maxlin(J)≤i≤n {reg J i − i} + n = cn + f. From this it follows that

max{reg J i − i} + n = max{f ∗ + n, cn + f } = max{n + 1 + e + f ∗, c(n + 1) + f + e∗ } − 1.
i≤n

for n ≥ lin(J).
If d ≥ 2, we will show below that reg R/(I + J)n = reg(I + J)n−1 /(I + J)n for n ≥
lin(I) + lin(J) + 1. By Proposition 5.5, this will imply the assertion.
By Theorem 2.4, we have
reg R/(I + J)n−1 ≤
max
{reg A/I n−i−1 + reg B/J i + 1, reg A/I n−j + reg B/J j } =
i∈[1,n−2], j∈[1,n−1]

max

{reg I n−i−1 + reg J i − 1, reg I n−j + reg J j − 2}.

i∈[1,n−2], j∈[1,n−1]

Similar to the proof of Proposition 5.5, we can show that
max {reg I n−j + reg J j } ≤ max{nd + e + f ∗ , nc + f + e∗ }

j∈[1,n−1]

for n ≥ lin(I) + lin(I). Therefore,

max {reg I n−i−1 + reg J i } ≤ max{(n − 1)d + e + f ∗ , (n − 1)c + f + e∗ }

i∈[1,n−2]

for n ≥ lin(I) + lin(I) + 1. These bounds imply that

reg R/(I + J)n−1 ≤ max{nd + e + f ∗ , nc + f + e∗ } − 2.

On the other hand, by Proposition 5.5 we have
reg(I + J)n−1 /(I + J)n = max{(n + 1)d + e + f ∗ , (n + 1)c + f + e∗ } − 2.

for n ≥ lin(I) + lin(I). Since c ≥ d ≥ 2, it follows that

reg R/(I + J)n−1 + 1 < reg(I + J)n−1 /(I + J)n

for n ≥ lin(I) + lin(I) + 1. Therefore, from the short exact sequence

0 → (I + J)n−1 /(I + J)n → R/(I + J)n → R/(I + J)n−1 → 0

we get reg R/(I + J)n = reg(I + J)n−1 /(I + J)n for n ≥ lin(I) + lin(I) + 1.



Corollary 5.8. If c = d then lin(I + J) ≤ lin(I) + lin(J) + 1.
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Tulane University, Department of Mathematics, 6823 St. Charles Ave., New Orleans,
LA 70118, USA
E-mail address: tha@tulane.edu
URL: http://www.math.tulane.edu/∼tai/
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang
Quoc Viet, Hanoi, Vietnam
E-mail address: nvtrung@math.ac.vn
URL: http://math.ac.vn/en/component/staff/?task=getProfile&staffID=64
Institute of Mathematics, Vietnam Academy of Science and Technology,, 18 Hoang
Quoc Viet, Hanoi, Vietnam
E-mail address: tntrung@math.ac.vn
URL: http://math.ac.vn/en/component/staff/?task=getProfile&staffID=65
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