ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES
OLGUR CELIKBAS, MOHAMMAD T. DIBAEI,

arXiv:1602.08625v2 [] 6 Apr 2017

MOHSEN GHEIBI, ARASH SADEGHI AND RYO TAKAHASHI
Abstract. Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local
ring R, if a Cohen-Macaulay R-module M of grade g is linked to an R-module N by a Gorenstein ideal
c, such that AssR (M ) ∩ AssR (N ) = ∅, then M ⊗R N is isomorphic to direct sum of copies of R/a, where
a is a Gorenstein ideal of R of grade g + 1. We give a criterion for the depth of a local ring (R, m, k) in
terms of the homological dimensions of the modules linked to the syzygies of the residue field k. As a
result we characterize a local ring (R, m, k) in terms of the homological dimensions of the modules linked
to the syzygies of k.

1. Introduction
Peskine and Szpiro [28], introduced the notion of linkage and geometric linkage of projective algebraic
varieties; recall that two unmixed projective varieties X and Y of codimension g are geometrically linked
if X and Y have no common components and the scheme theoretic union X ∪Y is a complete intersection.
More generally, two non-zero proper ideals I and J in a local ring R are said to be linked by an ideal
c contained in I ∩ J if I = (c :R J) and J = (c :R I) (denoted I ∼ J). If in addition, AssR (R/I) ∩
c

AssR (R/J) = ∅, then I and J are said to be geometrically linked by c. If c = 0 we just say I is linked
(or geometrically linked) to J. In their landmark paper [28], Peskine and Szpiro prove the following:
Theorem [Peskine-Szpiro]. Let R be a Gorenstein local ring and let I be a Cohen-Macaulay ideal of
grade g (i.e. R/I is Cohen-Macaulay ring). Suppose that I is geometrically linked to an ideal J by a
Gorenstein ideal c (i.e. R/c is a Gorenstein ring). Then I + J is a Gorenstein ideal of grade g + 1.
Therefore a new Gorenstein ideal can be obtained from the sum of geometrically linked ideals.
Another important result about geometrically linked ideals was proved by Ulrich in [31]. Recall that
an ideal I of a local ring R is called licci (in the linkage class of a complete intersection) if there exists a
chain of linked ideals I ∼ I1 ∼ · · · ∼ In , where c1 , · · · , cn , and In can be generated by a regular sequence.
c1

c2

cn

Ulrich showed that if R is a Gorenstein local ring, then the sum of two geometrically linked licci ideals
of R is again a licci ideal. Also Huneke [21], showed that if R is Gorenstein and I is a licci ideal then the
Koszul homologies Hi (I) of I are Cohen-Macaulay, for all i ≥ 0. So over a Gorenstein local ring, finding
Date: December 14, 2016.
2010 Mathematics Subject Classification. 13D07, 13D02, 13C40, 13D05.
Key words and phrases. Linkage of Modules, Geometrically linked ideals, Huneke-Wiegand conjecture, torsion-free
module, vanishing of Tor, homological dimension.
M. T. Dibaei was supported in part by a grant from IPM (No. 94130110). M. Gheibi was supported in part by a grant
from IPM (No. 92130025). A. Sadeghi’s research was supported by a grant from IPM. R. Takahashi was partly supported
by JSPS Grant-in-Aid for Scientific Research (C) 25400038.
1

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CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

geometrically linked ideals is useful and important for finding other nice ideals. For more results about
geometrically linked ideals, see [28], [31], [24], [20] and [29].
Linkage of ideals has been generalized to modules by Yoshino and Isogawa [32], by Marsinkovsky and
Strooker [26] and by Nagel [27], in different ways and, based on these generalizations, several works have
been done on studying classic results of linkage in the context of modules; see for example [10], [11], [12]
and [23]. In this paper, we are interested in linkage of modules in the sense of [26].
f

Let R be a commutative Noetherian local ring, M a finitely generated R-module, and F1 → F0 →
M → 0 a minimal free presentation of M . The image of f is called the first syzygy of M and it is
denoted ΩM . The transpose of M (denoted TrM ) is defined to be the cokernel of the induced map
f ∗ : HomR (F0 , R) → HomR (F1 , R). According to [26], M is horizontally linked to an R-module N if
M ∼
= ΩTrN and N ∼
= ΩTrM . The transpose and syzygy functors and their compositions already have
been studied by Auslander and Bridger; see [2]. Two R–modules M and N are said to be linked by an
ideal c ⊆ AnnR (M ) ∩ AnnR (N ) (denoted M ∼ N ), if M is horizontally linked to N over R/c. It turns
c

out that two ideals I and J of R are linked by an ideal c ⊆ I ∩ J if and only if the R-module R/I is
linked to R/J by c; see [26].
According to the notion of the geometrically linked ideals, the first purpose of this paper is to study
linked modules that have no common associated primes. In section 3, we give a generalization of the
above theorem. More precisely, we prove the following:
Theorem 1.1. Let R be a Cohen-Macaulay local ring, K a semidualizing R-module and M a GK - perfect
R-module of grade n < dim R. Assume M is linked to N by a GK - Gorenstein ideal c, and AssR (M ) ∩
AssR (N ) = ∅. Then M ⊗R N is free over S := R/(AnnR (M ) + AnnR (N )) and AnnR (M ) + AnnR (N ) is
a GK - Gorenstein ideal of grade n + 1.
For definition of semidualizing module, GK - perfect module, and GK - Gorenstein ideal, see section 2.
Along with studying the results of geometrically linked ideals in the context of modules, also we prove
the following:
Theorem 1.2. Let R be a generically Gorenstein unmixed local ring of dimension d ≥ 1, and let M be
a finitely generated R-module. Assume M is horizontally linked to λM . Then the following statements
are equivalent:
(i) AssR (M ) ∩ AssR (λM ) = ∅.
(ii) AnnR (M ) is geometrically linked to AnnR (λM ).
(iii) AnnR (M ) is linked to AnnR (λM ) and TorR
1 (M, λM ) = 0.
(iv) M is free over R/ AnnR (M ) and Ext1R (M, M ) = 0.
Our next aim in this paper is to study the properties of a local ring (R, m, k) by looking at linkage of
syzygy modules of k. Dutta [13] showed that if some syzygy module of the residue field k has a non-zero
direct summand of finite projective dimension, then R is regular. Motivated by Dutta’s Theorem, a
natural question is what properties does the ring R have if an R-module linked to a syzygy of k, has a
non-zero direct summand of finite projective dimension? We prove the following result in section 5:
Theorem 1.3. Let (R, m, k) be a local ring and let n ≥ 0 be an integer. Let X be a non-zero direct
summand of λΩn k. Then the following statements are equivalent.

ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

3

(i) pd(X) < ∞.
(ii) pd(X) = n.
(iii) depth R > n.
Also, we show that a similar assertion holds if a module horizontally linked to Ωn k has finite complete
intersection dimension or finite Gorenstein dimensions.
2. Preliminaries
In the present section we recall some preliminary results from the literature that we need for the rest
of the paper.
Throughout R is a commutative Noetherian local ring and all modules over R are assumed to be
finitely generated. For a finitely generated R–module M , let F1 → F0 → M → 0 be a minimal free
presentation of M . Then by applying (−)∗ = HomR (−, R), there exists an exact sequence
(2.1)

f∗

0 −→ M ∗ → F0∗ → F1∗ → TrM → 0.

If M is stable (i.e. has no free direct summand) then TrTrM ∼
= M , and 2.1 gives a minimal presentation
of TrM (see [4, Theorem 32.13]). As in [26], for a non-zero R-module M we will set λM to be the first
syzygy module of TrM i.e. λM = ΩTrM . Hence M is horizontally linked to λM if and only if M ∼
= λ2 M .
In this case, we briefly say M is horizontally linked. Also, if an ideal I is linked (geometrically linked) to
(0 :R I), we just say I is linked (geometrically linked) ideal of R. Note that M is a free R-module if and
only if λM = 0.
In the following, we recall some basic properties of linkage of modules that we need for the rest of
paper.
Proposition 2.1. [26, Proposition 3] Let R be a local ring and let M be a horizontally linked R-module.
Then M and λM are stable R-modules.
Let R be a ring. An R-module M is called a syzygy module if there exists an injective R-homomorphism
ι : M → F , where F is a free R-module.
Proposition 2.2. Let R be a ring and let M be an R-module. Then M is a syzygy module if and only
if Ext1R (TrM, R) = 0 (see [2, Theorem 2.17]).
Theorem 2.3. [26, Theorem 2] Let R be a local ring and let M be an R-module. Then the following
statements are equivalent.
(i) M is horizontally linked.
(ii) M is stable and Ext1R (TrM, R) = 0.
(iii) M is stable and a syzygy module.
Proposition 2.4. [14, Theorem 3.5] Let R be a generically Gorenstein local ring, i.e., Rp is Gorenstein
ring, for all p ∈ Ass R. An R-module M is a syzygy module if and only if AssR (M ) ⊆ Ass(R).
Let R be a local ring. Two R-modules M and N are called stably isomorphic (denoted M ∼
= N ) if
st
∼
there exist free R-modules F and G such that M ⊕ F = N ⊕ G. The next proposition follows from the
exact sequence (2.1) and Theorem 2.3.

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CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

Proposition 2.5. Let R be a local ring and M be an R-module. Then
(i) ΩλM ∼
= (λM )∗ ,
= M ∗ , and if M is horizontally linked, then ΩM ∼
st
(ii) (λΩn M )∗ ∼
= Ωn+1 M and λ2 Ωn M ∼
= Ωn M , for all n ≥ 1.
st

st

The following result describes an interesting relation between the set of associated primes of horizontally linked modules and associated primes of the base ring.
Proposition 2.6. [26, Proposition 6] Let R be an unmixed (i.e. Ass(R) consists of minimal prime ideals)
local ring and let M be a horizontally linked R-module. Then AssR (M ) ∪ AssR (λM ) = Ass(R).
Next, we recall the definition of Gorenstein dimension which has been introduced by Auslander and
Bridger [2].
Definition 2.7. A finitely generated R-module M is called totally reflexive or of G-dimension zero if the
natural homomorphism M → M ∗∗ is an isomorphism and ExtiR (M, R) = 0 = ExtiR (M ∗ , R) for all i > 0.
We recall that the infimum of n for which there exists an exact sequence
0 → Xn → · · · → X0 → M → 0,
such that each Xi is totally reflexive, is called the Gorenstein dimension of M . If M has Gorenstein
dimension n, we write G-dim(M ) = n. Therefore M is totally reflexive if and only if G-dim(M ) ≤ 0,
where it follows by convention that G-dim(0) = −∞.
More generally, let M and K be R–modules, and set M † = HomR (M, K). The module M is called
K-reflexive if the canonical map M → M †† is bijective. The Gorenstein dimension has been extended to
GK - dimension by Foxby in [15] and by Golod in [17].
Definition 2.8. The module M is said to have GK - dimension zero if
(i) M is K-reflexive,
(ii) ExtiR (M, K) = 0, for all i > 0,
(iii) ExtiR (M † , K) = 0, for all i > 0.
A GK - resolution of a finite R–module M is a right acyclic complex of modules of GK - dimension zero
whose 0th homology module is M . The module M is said to have finite GK - dimension, denoted by
GK -dim R (M ), if it has a GK - resolution of finite length.
Definition 2.9. An R–module K is called semidualizing (or suitable), if
(i) the homothety morphism R → HomR (K, K) is an isomorphism,
(ii) ExtiR (K, K) = 0 for all i > 0.
Semidualizing modules are studied in [15] and [17]. It is clear that R itself is a semidualizing R–module.
Also if R is Cohen-Macaulay then its canonical module (if it exists) is a semidualizing module. We recall
the following definitions from [17].
Definition 2.10. An R–module M is called GK - perfect if gradeR (M ) = GK -dim R (M ). An ideal I is
called GK - perfect if R/I is GK - perfect R–module. An R–module M is called GK - Gorenstein if M is
GK - perfect and ExtnR (M, K) is cyclic, where n = GK -dim R (M ). An ideal I is called GK - Gorenstein if
R/I is GK - Gorenstein R–module.

ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

5

Note that if K is a semidualizing R–module and I is a GK - Gorenstein ideal of GK - dimension n, then
∼ R/I(see [17, 10]).
Extn (R/I, K) =
R

Lemma 2.11. [17, Corollary] Let I be an ideal of R. Assume that K is an R–module and that n
If ExtjR (R/I, K) = 0 for all j 6= n then there is an isomorphism of functors
n+i
ExtiR/I (−, ExtnR (R/I, K)) ∼
= ExtR (−, K) on the category of R/I–modules for all i ≥ 0.
is a fixed integer.

Theorem 2.12. [17, Proposition 5]. Let I be a GK - perfect ideal, and let K be a semidualizing R–module.
grade(I)

Set C = ExtR

(R/I, K). Then the following statements hold.

(i) C is a semidualizing R/I–module.
(ii) If M is a R/I–module then GK -dim R (M ) < ∞ if and only if GC -dim R/I (M ) < ∞, and if these
dimensions are finite then GK -dim R (M ) = grade(I) + GC -dim R/I (M ).
3. Linkage of modules and geometrically linked ideals
The purpose of this section is to study the conditions under which the annihilator of a linked module is a geometrically linked ideal. We start with the following well-known proposition which gives a
characterization of geometrically linked ideals, and motivation for the results of sections 3 and 4.
Proposition 3.1. Let R be an unmixed ring, I an ideal of grade zero and linked to an ideal J. Then the
following statements are equivalent.
(i) I is geometrically linked to J.
(ii) gradeR (I + J) > 0.
(iii) I ∩ J = 0.
(iv) TorR
1 (R/I, R/J) = 0.
(v) Ext1R (R/I, R/I) = 0.
(vi) Ext1R (R/J, R/J) = 0.
Proof. The equivalence of (i), (ii), (iii) and (iv) can be easily checked; see [28] and [29, Proposition 2.3].
(v)⇐⇒(i) Consider the exact sequence 0 −→ I −→ R −→ R/I −→ 0. Applying HomR (−, R/I), we
get the exact sequence
α

0 → HomR (R/I, R/I) → HomR (R, R/I) → HomR (I, R/I) → Ext1R (R/I, R/I) → 0.
As α is an isomorphism, one has Ext1R (R/I, R/I) ∼
= HomR (I, R/I) ∼
= HomR (HomR (R/J, R), R/I). It
1
follows that ExtR (R/I, R/I) = 0 if and only if HomR (HomR (R/J, R), R/I) = 0 if and only if AssR (R/I) ∩
AssR (R/J) = ∅; see [6, Exercise 1.2.27].



We will need the following elementary lemma:
Lemma 3.2. Let R be an unmixed local ring, and let M be a horizontally linked R–module. Then
AssR (M ) = AssR (R/ AnnR (M )).
Proof. Set I = AnnR (M ). There is an inclusion R/I ֒→ HomR (M, M ) which implies that AssR (R/I) ⊆
AssR (M ). Conversely let p ∈ AssR (M ). By 2.6, AssR (M ) ⊆ Ass(R), and so p ∈ Ass(R). Hence p is
minimal over I and so p ∈ AssR (R/I).



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CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

Proposition 3.3. Let R be an unmixed local ring and let M be an R-module. Assume M is horizontally
linked to λM , and that AssR (M ) ∩ AssR (λM ) = ∅. Then M and λM are free over R/ AnnR (M ) and
R/ AnnR (λM ), respectively, and AnnR (M ) is geometrically linked to AnnR (λM ).
Proof. Let F1 → F0 → M → 0 be a minimal free presentation of M . Then we have an exact sequence
0 → M ∗ → F0∗ → λM → 0. Set I = AnnR (M ) and J = AnnR (λM ). Applying − ⊗R R/J gives the exact
sequence
f

M ∗ ⊗R R/J −→ F0∗ ⊗R R/J −→ λM ⊗R R/J −→ 0.
Since AssR (M )∩AssR (λM ) = ∅, we have AssR (M )∩AssR (R/J) = ∅ by 3.2. Hence we have Supp R (M ∗ ⊗R
R/J) ∩ AssR (R/J) = ∅. This implies that HomR (M ∗ ⊗R R/J, R/J) = 0; see [6, Exercise 1.2.27]. Hence
f = 0 and the exact sequence above implies that F ∗ ⊗R R/J ∼
= λM ⊗R R/J ∼
= λM . Symmetrically,
0

F0 ⊗R R/I ∼
= M.
For the second part, note that as M is a free R/I-module we may assume that M = R/I. Since M is
horizontally linked, I is linked to J; see [26, Lemma 3], and as AssR (R/I) ∩ AssR (R/J) = ∅, I and J are
geometrically linked.



Now we prove Theorem 1.1
Proof of Theorem 1.1. Set R := R/c. As c is a GK - Gorenstein ideal, R is a Cohen-Macaulay ring and
∼ R. Note that grade(M ) = grade(c) by [11, Lemma 5.8]. Therefore M and hence N , are
ExtnR (R, K) =
totally reflexive R-modules, by Theorem 2.12. Note that p ∈ AssR (M ) if and only if p/c ∈ AssR M . Hence
by Proposition 3.3, M and N are free over R/ AnnR (M ) and R/ AnnR (N ), respectively, and AnnR (M ) is
geometrically linked to AnnR (N ). Thus M ⊗R N is a free T -module, where T =

R
(AnnR (M)+AnnR (N )) .

But

as M is totally reflexive, it follows that R/ AnnR (M ) and R/ AnnR (N ) are totally reflexive R-modules.
Set I = AnnR (M ) and J = AnnR (N ). By 3.1 (iii), I ∩ J = 0 and hence one has gradeR (I + J) > 0, by
3.1 (ii). There is an exact sequence 0 → R → R/I ⊕ R/J → T → 0. Since R/I and R/J are totally
i
(T, R) = 0 for all i 6= 1. Also, by applying (−)∗ = HomR (−, R)
reflexive R-modules, it follows that ExtR

to the above exact sequence, we get the commutative diagram with exact rows:
1
0 −−−−→ (R/I)∗ ⊕ (R/J)∗ −−−−→ (R)∗ −−−−→ ExtR
(T, R) −−−−→ 0




∼
∼
=y
=y

0 −−−−→

It follows that

Ext1R (T, R)

by Lemma 2.11.

I +J

−−−−→

R

−−−−→

T

−−−−→ 0.

∼
= T . Therefore AnnR (M ) + AnnR (N ) is a GK - Gorenstein ideal of grade n + 1


Let R be an unmixed local ring of dimension d ≥ 1 and let M be a horizontally linked R-module. It
follows from 3.3 and 3.1 that if AssR (M ) ∩ AssR (λM ) = ∅, then TorR
1 (M, λN ) = 0. Consider the exact
sequence
∗
∗
0 −→ TorR
1 (M, λM ) −→ M ⊗R M −→ M ⊗R F0 .
∗
Therefore if TorR
1 (M, λM ) = 0 then depthR (M ⊗R M ) > 0. In particular, if R is a domain, then
∗
∗
TorR
1 (M, λM ) = 0 if and only if M ⊗R M is a torsion-free R-module if and only if λM ⊗R (λM ) is a

torsion-free R-module. Huneke and Wiegand [22] conjectured that if R is a one-dimensional Gorenstein

ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

7

domain and M is a finitely generated torsion-free R-module such that M ⊗R M ∗ also is torsion-free
then M is free. The Huneke-Wiegand conjecture holds over one-dimensional hypersurface rings and
also some classes of numerical semigroup rings; see [22], [16], and [18], and it is still open for complete
intersection rings of higher codimension. Therefore, the Huneke-Wiegand conjecture states that over a
one-dimensional Gorenstein local domain R, there exists no horizontally linked R-module M such that
TorR
1 (M, λM ) = 0.
Without assuming R is domain, we are interested in conditions on the base ring R and a horizontally linked R-module M such that vanishing of TorR
1 (M, λM ) implies that AssR (M ) ∩ AssR (λM ) = ∅,
and hence M is a free R/ AnnR (M )-module. However, in general vanishing of TorR
1 (M, λM ) does not
guarantee that AssR (M ) ∩ AssR (λM ) = ∅; see Example 4.4.
An R-module M is called generically free if Mp is a free Rp -module, for all p ∈ Ass(R).
Lemma 3.4. Let R be a generically Gorenstein unmixed local ring and let M be a horizontally linked
R–module. Then the following statements are equivalent.
(i) Supp R (TorR
1 (M, λM )) ∩ Ass(R) = ∅.
(ii) M is generically free.
Proof. (i)⇒(ii). Let p ∈ Ass(R). Since R is unmixed, p is a minimal prime ideal of R, and so Rp
R

is Artinian Gorenstein local ring. By assumption Tor1 p (Mp , (λM )p ) = 0 and since (λM )p is stably
isomorphic to λRp Mp , we get

R
Tor1 p (Mp , λRp Mp )

= 0. Hence it is enough to show that if R is an

Artinian Gorenstein local ring and M is an R-module such that TorR
1 (M, λM ) = 0 then M is free.
R
Indeed, since TorR
1 (M, λM ) = 0 so HomR (Tor1 (M, λM ), R) = 0 and as R is Artinian Gorenstein ring,
1
the isomorphism HomR (TorR (M, λM ), R) ∼
= Ext (M, (λM )∗ ); see for example [14, page 7]; implies that
R

1

Ext1R (M, (λM )∗ ) = 0. But by 2.5, (λM )∗ ∼
= ΩM . Hence Ext1R (M, ΩM ) = 0 and so M is free R-module.
st

R

Conversely, if M is generically free R–module, then Tor1 p (Mp , (λM )p ) = 0, for all p ∈ Ass R. Hence
Supp R (TorR
1 (M, λM )) ∩ AssR (R) = ∅ and we are done.



Definition 3.5. An R-module M is said to have constant rank n on a subset S of Spec R if Mp is a free
Rp -module with rank n for all p ∈ S.
Proposition 3.6. Let R be a generically Gorenstein unmixed local ring and let M be a finitely generated
R-module. Assume M is horizontally linked to λM with TorR
1 (M, λM ) = 0. If AnnR (M ) 6= 0, then
AnnR (M ) is geometrically linked to (0 :R AnnR (M )). Moreover if λM is of constant rank on AssR (λM ),
then AssR (M ) ∩ AssR (λM ) = ∅.
Proof. Set I = AnnR (M ). By 3.2 and 2.6, AssR (R/I) = AssR (M ) ⊆ Ass(R). Hence, by 2.4, R/I is
a horizontally linked R-module and so I is a linked ideal. Let p ∈ AssR (M ) = AssR (R/I), and set
J = (0 :R I). Since by Lemma 3.4, Mp is a free Rp –module, IRp = 0, and since J = (0 :R I), we have
JRp = Rp . Hence J * p and so p ∈
/ AssR (R/J). This means that I and J have no common associated
prime ideal.
Now assume that λM has constant rank on AssR (λM ). There is an exact sequence 0 → M ∗ →
F → λM → 0, where F is a free module. Choose p, q ∈ Ass(R) such that p ∈ AssR M and q ∈
Ass(R) \ AssR (M ). Note that as AnnR (M ) is geometrically linked, Lemma 3.2 implies that such q exists.
Also note that by 2.6, we have AssR (M ) ⊆ Ass(R) and AssR (λM ) ⊆ Ass(R). Hence, localizing the exact

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CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

sequence above at p and q, we get rank (λM )p 6= rank (λM )q . Since λM has constant rank on AssR (λM )
we must have (λM )p = 0. Thus AssR (M ) and AssR (λM ) have empty intersection.



The assumption AnnR (M ) 6= 0 in Proposition 3.6 is crucial. In section 4, we give an example of a
module M over a one-dimensional Gorenstein local ring R, such that M is horizontally linked to λM ,
TorR
1 (M, λM ) = 0, and λM has constant rank on AssR (λM ), but AssR (M ) ∩ AssR (λM ) 6= ∅; see 4.4.
Now we prove Theorem 1.2.
Proof of Theorem 1.2. Set I = AnnR (M ) and J = AnnR (λM ).
(i)⇒(ii) follows from Proposition 3.3.
(ii)⇒(i) and (ii)⇒(iv). We have AssR (R/I) ∩ AssR (R/J) = ∅. Hence by Lemma 3.2, AssR (M ) ∩
AssR (λM ) = ∅ and therefore, M is free R/I-module, by Proposition 3.3. Hence Ext1R (M, M ) = 0, by
Proposition 3.1.
(iv)⇒(iii) follows from Proposition 3.1.
(iii)⇒(ii) By Proposition 3.6, I is geometrically linked to (0 :R I) and as (0 :R I) = J, we are done. 
As an application of Proposition 3.3 and Proposition 3.6, we prove the following result, which is related
to Huneke-Wiegand conjecture in a sense.
Theorem 3.7. Let R be a Cohen-Macaulay local domain of dimension d ≥ 1 admitting a canonical
module ωR . Let M be a torsion-free R-module such that M ⊗R HomR (M, ωR ) also is torsion-free R∼ S/p, where S is a Gorenstein local ring and p is a geometrically linked prime ideal
module. Assume R =
of S. If AnnS (ΩS M ) 6= 0, then M is a free R-module.
Before the proof, we need the following lemma:
Lemma 3.8. Let R be a generically Gorenstein unmixed local ring of dimension d ≥ 1 and let M be a
horizontally linked R-module. Assume AnnR M is a non-zero prime ideal of R. Then TorR
1 (M, λM ) = 0
if and only if AnnR M is a geometrically linked ideal of R and AssR (M ⊗R M ∗ ) ⊆ AssR (M ).
Proof. Let F1 → F0 → M → 0 be the minimal free presentation of M and consider the exact sequence
0 → M ∗ → F0∗ → λM → 0. By applying − ⊗R M we get the exact sequence
(3.8.1)

∗
∗
0 → TorR
1 (M, λM ) → M ⊗R M → M ⊗ F0 .

Now if TorR
1 (M, λM ) = 0 then by Proposition 3.6, AnnR M is a geometrically linked ideal of R and the
exact sequence 3.8.1 implies that AssR (M ⊗R M ∗ ) ⊆ AssR (M ).
Set p = AnnR M . Since p is geometrically linked to (0 :R p), one has p ∩ (0 :R p) = 0 and therefore
(0 :R p) * p. Hence pRp = 0 and so Rp is a field. Thus Mp is a free Rp -module. Hence by 3.4
R
Supp R (TorR
1 (M, λM )) ∩ Ass(R) = ∅. By the exact sequence 3.8.1 we have AssR (Tor1 (M, λM )) ⊆

AssR (M ⊗R M ∗ ) and since AssR (M ⊗R M ∗ ) ⊆ AssR (M ) = {p} ⊂ Ass R so AssR (TorR
1 (M, λM )) = ∅.
Thus Tor1R (M, λM ) = 0.



ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

9

Proof of Theorem 3.7. Since p ∈ Ass(S), one has dim R = dim S. Therefore from the natural isomorphisms
M ⊗R HomR (M, ωR ) ∼
= M ⊗S HomR (M, ωR )
∼
= M ⊗S HomR (M, HomS (R, S))
∼
= M ⊗S HomS (M ⊗R R, S)
∼
= M ⊗S HomS (M, S),
we see that AssS (M ⊗S HomS (M, S)) = {p}. Since M is a stable S-module, it is horizontally linked as Smodule; see 2.4. Denote by λS M , the linkage of M over S. Since AssS (M ) = AssS (M ⊗S HomS (M, S)) =
{p}, we get TorS1 (M, λS M ) = 0, by 3.8.
Assume AnnS (ΩS M ) 6= 0. Since by 2.5, ΩS M ∼
= HomS (λS M, S), one has AnnS (λS M ) 6= 0 and as
AssS (M ) = {p}, M has constant rank on AssS (M ). Thus, it follows from Proposition 3.6 that AssS (M )∩
AssS (λS M ) = ∅ and so M is a free S/ AnnS (M )-module, by Proposition 3.3. But as AnnS (M ) = p, M
is a free R–module.



Corollary 3.9. Let R be a Cohen-Macaulay local domain with canonical module ωR . Then R is Gorenstein if and only if R is a homomorphic image of a reduced Gorenstein local ring S with dim S = dim R
and AnnS (ΩS ωR ) 6= 0.
Proof. If R is Gorenstein then by setting S = R we are done. Conversely, if S is domain then we must
have S ∼
= R and so there is nothing to prove. Assume that S is not domain and so there is a minimal
prime ideal p of S such that R ∼
= S/p. Hence by 2.4, p is a linked ideal of S. Note that as S is reduced,
then p is a geometrically linked ideal. Indeed, by setting I = (0 :S p) we get I ∩ p = 0 because otherwise
S has a nilpotent element which is a contradiction. Since ωR ⊗R HomR (ωR , ωR ) is torsion-free, Theorem
3.7 implies that ωR is a free R-module and so R is Gorenstein.



Proposition 3.10. Let R be a generically Gorenstein unmixed local ring, and let M be a horizontally
linked R–module. Assume AnnR (M ) 6= 0 and that TorR
1 (M, λM ) = 0. If there exists a positive odd
integer n such that Ωn M has constant rank on AssR (Ωn M ), and the sequence of the Betti numbers
β0 (M ), · · · , βn−1 (M ) is non-decreasing, then AssR (M ) ∩ AssR (λM ) = ∅.
Proof. Set βi = βi (M ). Let · · · → F1 → F0 → M → 0 be the minimal free resolution of M and consider
the exact sequence
0 → Ωn M → Fn−1 → · · · → F0 → M → 0.
Note that by 3.6, Ass(R) \ AssR (M ) 6= ∅. Choose p, q ∈ Ass(R) such that p ∈ AssR (M ) and q ∈
Ass(R) \ AssR (M ). Localizing the last exact sequence at p and q, by 3.4 we get the equations
rank Rp (Ωn M )p =

(3.10.1)

n−1
X

(−1)i βi − rank Rp Mp

i=0

and
rank Rq (Ωn M )q =

(3.10.2)

n−1
X

(−1)i βi .

i=0

n

n

Therefore, as Ω M has constant rank on AssR (Ω M ), we must have either AssR (Ωn M ) ⊆ AssR (M )
or AssR (M ) ∩ AssR (Ωn M ) = ∅. If AssR (Ωn M ) ⊆ AssR (M ) then (Ωn M )q = 0 and 3.10.2 implies that

10

CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

Pn−2
i=0

(−1)i−1 βi = βn−1 ≥ βn−2 . Canceling βn−2 and continuing in this way, we eventually obtain β0 ≤ 0

which is impossible.
Hence AssR (M ) ∩ AssR (Ωn M ) = ∅ and by localizing the exact sequence
0 → Ωn M → Fn−1 → · · · → F1 → ΩM → 0
Pn−1
at any p ∈ AssR (M ) and noting that βi is non-decreasing, we get rank Rp (ΩM )p = i=1 (−1)i−1 βi ≤ 0. It
∼ (λM )∗ , by 2.5, so AssR (M )∩AssR (λM ) = ∅. 
follows that (ΩM )p = 0, for all p ∈ AssR (M ). As ΩM =
4. One-dimensional gorenstein rings
In this section we give an example of a horizontally linked module M over a one-dimensional
Gorenstein local ring R such that TorR
1 (M, λM ) = 0, λM has constant rank on AssR (λM ), but
AssR (M ) ∩ AssR (λM ) 6= ∅. First we prove a duality between Ext∗R (M, M ) and TorR
∗ (M, λM ); see 4.2,
which has its own interest.
Lemma 4.1. Let (R, m) be a Cohen-Macaulay local ring of dimension d ≥ 1, admitting a canonical module
ωR . Let M and N be maximal Cohen-Macaulay R–modules such that TorR
i (M, N ) is of finite length, for all
∨
†
i ≥ 1. Set (−) = HomR (−, E(R/m)) and (−) = HomR (−, ωR ). Then Exti+d (M, N † ) ∼
= TorR (M, N )∨ ,
i

R

for all i ≥ 1.
Proof. There is a third quadrant spectral sequence
p+q
p
R
†
Ep,q
2 = ExtR (Torq (M, N ), ωR ) =⇒ ExtR (M, N ).
p

p
R
As TorR
q (M, N ) is of finite length, for all q ≥ 1, it follows that ExtR (Torq (M, N ), ωR ) = 0, for all p 6= d

and all q ≥ 1. Therefore
Ep,q
2

=







ExtpR (M ⊗R N, ωR )
ExtdR (TorR
q (M, N ), ωR )
0

0 ≤ p ≤ d,
q > 0,
otherwise.

Therefore, as the map dp,q
: Ep,q
→ Ep+2,q−1
is of bidegree (r, 1 − r), for all r ≥ 2, one can easily
r
r
r
p,q
verify that that d = 0, for all p, q, and r. Hence Ep,q = Ep,q , for all p and q and so Extq+d (M, N † ) ∼
=

r
∞
2
d
R
ExtR (Torq (M, N ), ωR ) for all q > 0. Now, the Local Duality Theorem
R
R
∨
∨
∼
that ExtdR (TorR
q (M, N ), ωR ) = Γm (Torq (M, N )) = Torq (M, N ) .

R

(see [6, Theorem 3.5.8]) implies


Proposition 4.2. Let (R, m) be a Gorenstein local ring of dimension one, and let M be a maximal CohenR
Macaulay R–module. Assume either TorR
i (M, λM ) is of finite length, for all i ≥ 1, or Tor1 (M, λM ) = 0.

Then for all i > 0,
ExtiR (M, M ) ∼
= HomR (TorR
i (M, λM ), E(R/m)).
Proof. Note that if TorR
1 (M, λM ) = 0 then M is locally free on the punctured spectrum of R, by 3.4, and
R
∼ (λM )∗ and
so Tori (M, λM ) is of finite length, for all i ≥ 1. As R is Gorenstein, then we have (λM )† =
(λM )∗ is stably isomorphic to ΩM . Thus we get Exti+1 (M, (λM )∗ ) ∼
= Exti (M, M ), for all i > 0. Now
R

the result follows by Lemma 4.1.

R



ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

11

Corollary 4.3. Let R be a Gorenstein local ring of dimension one and M be a horizontally linked R∼ TorR (Ωn M, λΩn M ).
module with TorR (M, λM ) = 0. Then for all n ≥ 0 and all i ≥ 1, TorR (M, λM ) =
1

i

i

R
n
n
In particular, TorR
1 (M, λM ) = 0 if and only if Tor1 (Ω M, λΩ M ) = 0, for some (and hence all) n ≥ 0.

Proof. For each n ≥ 0 and i ≥ 1, one has ExtiR (M, M ) ∼
= ExtiR (Ωn M, Ωn M ). Thus the result follows from
Proposition 4.2.



In the following example, we use Macaulay 2 for the computations.
Example 4.4. Let k be a field and R = k[[x, y, z, t]]/(xy, yz, zt, xt + yt + t2 , x2 + xz + xt), which
is a Gorenstein local ring of dimension one. Let I = (x, z) and J = (y). Then R/I is horizontally
linked to R/J. One has Ass(R) = {(z, y, x + t), (z, x, y + t), (x, z, t), (t, y, x), (t, y, x + z)}, AssR (R/I) =
{(z, x, y + t), (x, z, t)} and AssR (R/J) = {(z, y, x + t), (t, y, x), (t, y, x + z)}. Hence I is geometrically
linked to J and so TorR
1 (R/I, R/J) = 0. Set M = λΩ(R/I) = λI. Then M is horizontally linked and
∼
λM = I. Hence AnnR (λM ) = J and note that λM is not free R/J-module. By the last corollary we
have TorR
1 (M, λM ) = 0. Also note that λM has constant rank equal to one on AssR (λM ) = AssR (R/J).
Therefore we must have AnnR (M ) = 0 because otherwise λM would be a free R/J-module, by 3.6.
Therefore AssR (M ) ∩ AssR (λM ) 6= ∅.
Recall that a complete resolution of a finitely generated module M over a local ring (R, m) is a diagram
ν

π

T −→ F −→ M , where F is a free resolution of M , and T is an exact complex of finitely generated free
modules
di+1

d

i
Ti−1 −→ · · · ,
T : · · · −→ Ti+1 −→ Ti −→

such that HomR (T, R) is exact, ν is a morphism of complexes, and νi is bijection for all i ≫ 0. A
complete resolution T of M is called minimal if Im (di ) ⊆ mTi−1 , for all i ∈ Z. It is well known that
if R is Gorenstein, then every finitely generated R-module admits a minimal complete resolution. In
particular, if M is maximal Cohen-Macaulay, then M ∼
= Im (d0 ).
Theorem 4.5. Let R be a Gorenstein local domain of dimension one and let M be a torsion-free Rmodule such that M ⊗R M ∗ also is a torsion-free R-module. Assume R ∼
= S/p, where S is a Gorenstein
local ring and p is a geometrically linked prime ideal of S. Let T be a minimal complete resolution of M
over S. If AnnS (Ω−1
S M ) 6= 0, then M is a free R-module.
Proof. As we saw in the proof of 3.7, M is horizontally linked over S and one has TorS1 (M, λS M ) = 0. Set
N = Ω−1
S M . Since AnnS (N ) 6= 0, N is an stable S-module and so it is horizontally linked S-module; see
∼ ΩS N , one has TorS (N, λS N ) = 0, by 4.3. Also, note that since AssS (M ) = {p},
2.3. Note that as M =
1

M has constant rank on AssS (M ). Therefore, by 3.10 and 3.3, N is free over S/ AnnS (N ) and λS N is
free over S/ AnnS (λS N ). As M ∼
= HomS (λS N, S), it follows that AnnS (λS N ) = p, and M is isomorphic
to a direct sum of copies of HomS (S/p, S). But, since R and S are Gorenstein, HomS (S/p, S) ∼
= R. Hence
M is a free R-module.



5. Modules horizontally linked to the syzygies of the residue field
As mentioned before, a module is horizontally linked if and only if it is stable and a syzygy module.
Therefore for an R-modules M and each n ≥ 1, Ωn M is horizontally linked if and only if Ωn M is stable.

12

CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

In this section we are interested in the properties of modules that are horizontally linked to the syzygies
of the residue field. We start with the following lemma that we need for the rest of the paper.
Lemma 5.1. Let R be a local ring and let M be a non-zero finitely generated R-module.
∼ λN .
(i) If M = F ⊕ N with F a free R-module, then λM =
(ii) If M is non-free and a syzygy module, then λM is stable.
Proof. We only prove (ii). Assume M = F ⊕ N where F is a free R-module and N is a non-zero stable
R-module. By (i), λM = λN and by 2.3, N is horizontally linked. Now 2.1 implies that λM is stable. 
Lemma 5.2. Let R be a local ring and let M be a non-zero R-module. If n > max{0, depth(R) −
depth(M )}, then Ωn M is stable R-module; see for example [3, Corollary 1.2.5].
Theorem 5.3. Let R be a local ring which is not a field. Let M be a non-zero R-module and let n ≥ 0.
Assume λΩn M ∼
= X ⊕ Y for some non-zero R-module X. Then:
(i) TorR
n (M, X) 6= 0.
(ii) If pd(X) < ∞ then depth R ≥ 1 and n ≤ pd(X) < depth(R) − depth(M ).
Proof. (i) If n = 0 we have nothing to prove.
TorR
n (M, X)
(5.3.1)

Suppose n ≥ 1 and assume to the contrary that

= 0. Consider the short exact sequence
0 → λΩn M → F → TrΩn M → 0

which follows from the definition of the linkage, where F is free. By using [30, Lemma 3.1], we obtain
the following short exact sequences from (5.3.1)
(5.3.2)

0 → X → B → TrΩn M → 0

(5.3.3)

0 → F → A ⊕ B → TrΩn M → 0,

where A and B are R-modules. Since M and X are non-zero, and that n ≥ 1, Ext1R (TrΩn M, R) = 0, by
2.2. This implies that (5.3.3) splits, so A ⊕ B ∼
= F ⊕ TrΩn M . It follows from the Auslander- Bridger four
term exact sequence [2, Theorem 2.8] that Ext1R (TrΩn M, X) ֒→ TorR
n (M, X) = 0. Thus (5.3.2) splits too,
n
∼
and so B = X ⊕ TrΩ M . Hence
(5.3.4)

F ⊕ TrΩn M ∼
= A ⊕ X ⊕ TrΩn M

Now the cancelation property (see [25, Corollary 1.16]) applied to (5.3.4) shows that F ∼
= A ⊕ X, i.e., X
is free; which is a contradiction, by 5.1 (ii). This proves (i).
(ii) Assume pd(X) < ∞. If depth R = 0 then Auslander-Buchsbaum formula implies that X is free
and by the last part n = 0. By Lemma 5.1 (i), we may assume that M is stable. Let F1 → F0 → M → 0
be the minimal presentation of M . Then F0∗ → F1∗ → TrM → 0 is the minimal presentation of TrM .
But since X ⊕ Y = λM = ΩTrM and X is free, this contradicts 5.2. Thus depth R ≥ 1.
For the next part, we proceed by induction on depth M . If depthR M = 0 then we need to show
that pd(X) < depth R. But, as depth R ≥ 1 and X is a syzygy module, depthR X ≥ 1. Hence by the
Auslander-Buchsbaum formula pd(X) < depth R.
As TorR
i (R/(x), M ) =
R
n
n
n
∼
Tori (R/(x), λΩ M ) = 0 for all i > 0, it follows that λR/(x) Ω (M/xM ) = λΩ M/xλΩn M ∼
=
Let depth M ≥ 1 and choose x to be regular on R, M and λΩn M .

ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

13

X/xX ⊕ Y /xY . By induction, pdR/(x) (X/xX) < depth R/(x) − depthR/(x) M/xM = depth R − depthR M .
Since pdR/(x) (X/xX) = pd(X), we are done.



Now Theorem 1.3 follows from the next corollary.
Corollary 5.4. Let (R, m, k) be a local ring, L a finite length R-module, and n ≥ 0 an integer. Assume
that λΩn L ∼
= X ⊕ Y for some non-zero R-module X. Then the following statements are equivalent.
(i) pd(X) < ∞.
(ii) pd(X) = n.
(iii) depth R > n.
Proof. (i)=⇒(iii) Follows from 5.3 (ii).
(iii)=⇒(ii) Assume depth R > n. Note that as TorR
n (X, L) 6= 0, by 5.3 (i), therefore pd(X) ≥ n. We
show that pd(λΩn L) ≤ n and hence pd(X) ≤ n. Consider the exact sequence which is part of a minimal
free resolution of L:
G = (0 → Ωn L → Fn−1 → · · · → F1 → F0 → L → 0)
Since depth R > n and L is finite length, by dualizing G, we obtain the exact sequence:
∗
G∗ = (0 → F0∗ → F1∗ → · · · → Fn−1
→ (Ωn L)∗ → 0).

It follows that pd(Ωn L)∗ ≤ n − 1 and so pd(λΩn L) ≤ n.
(ii)=⇒(i) It is clear.



Corollary 5.5. Let R be a local ring and let M be a non-zero R-module of finite injective dimension. Let
∼ X ⊕ Y for some non-zero R-module X with G-dim(X) < ∞.
n ≥ 0 be a positive integer such that Ωn M =
Then pd(X) < ∞.
Proof. Choose i ≥ max{n, depth R−depth M }. Then Ωi M ∼
= Ωi−n X ⊕Ωi−n Y . Hence λΩi M ∼
= λΩi−n X ⊕
i−n
λΩi−n Y and by 5.3 (i), if λΩi−n X 6= 0 then TorR
X) 6= 0. But as id(M ) < ∞ and Ωi−n X is
i (M, λΩ
i−n
totally reflexive, TorR
X) = 0; see [9, Corollary 2.4.4]. Thus we must have λΩi−n X = 0 and
i (M, λΩ

therefore Ωi−n X is zero or free.



Corollary 5.6. ([19, Theorem 3.2 (i)]) Let R be a local ring and let M be a non-zero R-module. Assume
id(M ) < ∞ and G-dim(M ) < ∞. Then R is Gorenstein.
Proof. We conclude from 5.5 that pd(M ) < ∞. Now Foxby’s result [15], implies that R is Gorenstein. 
Our aim in the rest of the paper is to prove a counterpart of 5.4 in case of finite Gorenstein and
complete intersection dimension. Recall from [5] that a diagram of local ring maps R → R′ և Q is called
a quasi-deformation provided that R → R′ is flat and the kernel of the surjection R′ և Q is generated
by a Q-regular sequence. The complete intersection dimension of M is:
CI-dim(M ) = inf{pdQ (M ⊗R R′ ) − pdQ (R′ ) | R → R′ և Q is a quasi-deformation}.
Lemma 5.7. Let R be a local ring, M an R-module, and let n ≥ depth(R). Then the following conditions
are equivalent:
(i) G-dim(TrΩn M ) < ∞.

14

CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

(ii) G-dim(M ) < ∞.
(iii) G-dim(TrΩn M ) ≤ 0.
(iv) G-dim(Ωn M ) ≤ 0.
Proof. Note that (ii) ⇐⇒ (iv) and (iii) =⇒ (i) follow by definition. Moreover as the transpose of a totally
reflexive module is also totally reflexive, (iii) ⇐⇒ (iv) is clear. Hence it is enough to show that (i) =⇒
(iii).
Set t = depth R and assume G-dim(TrΩn M ) < ∞. Note that depth(Ωn M )p ≥ min{n, depth(Rp )},
for all p ∈ Supp R (Ωn M ). It follows from [7, Theorem 5.8 (1)] that ExtiR (TrΩn M, R) = 0 for all i =
1, . . . , t. Since G-dim(TrΩn M ) = sup{i ∈ Z : ExtiR (TrΩn M, R) 6= 0}; see [9, Theorem 1.2.7 (iii)], if
G-dim(TrΩn M ) > 0, then we must have G-dim(TrΩn M ) > t, which is impossible, by Auslander-Bridger
formula; see [9]. Consequently G-dim(TrΩn M ) ≤ 0 and (iii) follows.



Theorem 5.8. Let (R, m, k) be a local ring, let M be an R-module of finite length, and let n ≥ 0 be
an integer. Let H-dim stand for one of projective, complete intersection or Gorenstein dimension, and
assume H-dimR (M ) = ∞. Then the following conditions are equivalent:
(i) depth(R) ≤ n.
(ii) H-dim(λΩn M ) = ∞.
In particular depth R = inf{n| H-dim(λΩn M ) = ∞}.
Proof. (i)=⇒(ii) Let n ≥ depth R and assume to the contrary that H-dim(λΩn M ) < ∞.

If

H-dim(λΩn M ) = G-dim(λΩn M ), then it follows that G-dim(TrΩn M ) < ∞ and by Lemma 5.7,
G-dim(Ωn M ) < ∞ and hence G-dim(M ) < ∞ which is a contradiction.
If H-dim(λΩn M ) = CI-dimR (λΩn M ), then G-dim(λΩn M ) < ∞ and so G-dim(TrΩn M ) < ∞. Hence by
Lemma 5.7, G-dimR (TrΩn M ) ≤ 0 and so CI-dimR (TrΩn M ) ≤ 0. Thus CI-dimR (Ωn M ) = 0, by [8, Lemma
3.2 (i)]. Hence CI-dimR (M ) < ∞, which is a contradiction.
(ii)=⇒(i) First assume n = 0 and H-dim(λM ) = ∞. We show that depth(R) = 0. Consider the exact
sequence 0 → M ∗ → F → λM → 0, where F is a free R-module. Since H-dim(λM ) = ∞, one has
M ∗ 6= 0. Therefore, as M is a finite length R-module, one gets depth(R) = 0.
Now assume n > 0. It is enough to show that if depth(R) > n then H-dim(λΩn M ) < ∞ and this
follows by the same argument of 5.4 (iii)=⇒(ii).



Corollary 5.9. Let (R, m, k) be a local ring, n ≥ depth R be a positive integer. Then the following
statements are true.
(i) R is Gorenstein if and only if G-dimR (λΩn k) < ∞.
(ii) R is a complete intersection if and only if CI-dimR (λΩn k) < ∞.
Acknowledgements. Part of this work was completed when Celikbas and Takahashi visited Centre
de Recerca Matemàtica during the conference; Opening Perspectives in Algebra, Representations, and
Topology, in May 2015. The authors acknowledge the kind hospitality of CRM and thank the institute
for providing an excellent research environment.
Part of this work was completed when Gheibi visited Celikbas at the University of Connecticut in
October 2015. Gheibi thanks Jerzy Weyman and Roger Wiegand for supporting his visit, and is grateful
for the kind hospitality of the UConn Department of Mathematics.

ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES

15

The authors thank Roger Wiegand for his careful comments and suggestions on the manuscript, and
Thanh Vu for his help with Example 4.4.

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CELIKBAS, DIBAEI, GHEIBI, SADEGHI, TAKAHASHI

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Olgur Celikbas, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, U.S.A
E-mail address: olgur.celikbas@math.wvu.edu
Mohammad T. Dibaei, Faculty of Mathematical Sciences and Computer, Kharazmi University, and School
of Mathematics Institute for Research in Fundamental Sciences (IPM), P.O. Box19395-5746, Tehran, Iran
E-mail address: dibaeimt@ipm.ir
Mohsen Gheibi, Department of Mathematics, University of Nebraska, Lincoln, NE, 68588,USA
E-mail address: mohsen.gheibi@huskers.unl.edu
Arash Sadeghi, School of Mathematics, Institute for Research in Fundamental Sciences, (IPM), P.O. Box:
19395-5746, Tehran, Iran
E-mail address: sadeghiarash61@gmail.com
Ryo Takahashi, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku,, Nagoya 4648602, Japan
E-mail address: takahashi@math.nagoya-u.ac.jp

