arXiv:1611.05790v1 [] 17 Nov 2016

CHAINS OF SEMIDUALIZING MODULES
ENSIYEH AMANZADEH
Abstract. Let (R, m, k) be a commutative Noetherian local ring. We study the suitable
chains of semidualizing R-modules. We prove that when R is Artinian, the existence of
a suitable chain of semidualizing modules of length n = max { i > 0 | mi 6= 0 } implies
that the the Poincaré series of k and the Bass series of R have very specific forms. Also,
in this case we show that the Bass numbers of R are strictly increasing. This gives an
insight into the question of Huneke about the Bass numbers of R.

1. Introduction
In this paper R is a commutative Noetherian local ring with maximal ideal m and
residue field k. Foxby [6], Vasconcelos [19] and Golod [9] independently initiated the
study of semidualizing modules. A finite (i.e. finitely generated) R-module C is called
semidualizing if the natural homothety map χR
C : R −→ HomR (C, C) is an isomorphism
and Ext>1
R (C, C) = 0 (see [11, Definition 1.1]). Trivial examples of semidualizing Rmodules are R itself and a dualizing R-module when one exists. The set of all isomorphism
classes of semidualizing R-modules is denoted by G0 (R), and the isomorphism class of a
semidualizing R-module C is denoted [C]. The set G0 (R) has a rich structure, for instance,
it comes equipped with an ordering based on the notion total reflexivity. For semidualizing
R-modules B and C, we write [C] E [B] whenever B is totally C-reflexive. In [8], Gerko
defines chains in G0 (R). A chain in G0 (R) is a sequence [Cn ] E · · · E [C1 ] E [C0 ], and
such a chain has length n if [Ci ] 6= [Cj ] whenever i 6= j. In [16], Sather-Wagstaff uses the
length of chains in G0 (R) to provide a lower bound for the cardinality of G0 (R).
The well-known result of Foxby [6], Reiten [14] and Sharp [18] is that R admits a
dualizing module if and only if R is Cohen-Macaulay and a homomorphic image of a
local Gorenstein ring. Then Jorgensen et. al. [13], characterize the Cohen-Macaulay local
rings which admit dualizing modules and non-trivial semidualizing modules. Recently,
Amanzadeh and Dibaei [1], characterize the Cohen-Macaulay local rings which admit
dualizing modules and suitable chains of semidualizing modules.
In [8], Gerko has shown that, when R is Artinian, the existence of collections of strongly
Tor-independent semidualizing R-modules (this condition conjecturally equivalent to the
existence of a corresponding long chain of semidualizing modules) implies that the Poincaré
2010 Mathematics Subject Classification. 13D05, 13D07.
Key words and phrases. Bass number, semidualizing, suitable chain, Tor-independent.
1

2

E. AMANZADEH

series of k has a very specific form. We show that the [8, Conjecture 4.6] holds true for
suitable chains in G0 (R) (see Theorem 3.5). This result implies that the set G0 (R) can
not contain suitable chains of arbitrary length. Indeed, the length of a suitable chain in
G0 (R) is less than the number l = min { i > 0 | mi = 0 }.
There are several works which were motivated by the following questions of Huneke
about the Bass numbers µiR (R) = rankk (ExtiR (k, R)). For instance, see [3], [5], [12] and
[15]. However, each of the following questions is still open in general.
Question. Let R be a Cohen-Macaulay ring.
(a) If the sequence {µiR (R)} is bounded above by a polynomial in i, must R be Gorenstein?
(b) If R is not Gorenstein, must the sequence {µiR (R)} grow exponentially?
In [15], Sather-Wagstaff has proved that if there exists a chain of semidualizing Rmodules of length n, then the Bass series of R is a product of n power series with positive
integer coefficients, and the Bass numbers of R are bounded below by a polynomial of
degree n − 1. In Theorem 3.8, we prove that an Artinian ring R with mn+1 = 0 is SD(n)full if and only if there is a suitable chain in G0 (R) of length n. Then by this result and
some results of [8], we show that the existence of a suitable chain in G0 (R) implies that
the Bass series of R has very specific form and the sequence {µiR (R)} is strictly increasing
(see Proposition 3.11 and Remark 3.12).
2. Preliminaries
This section contains definitions and background material.
Definition 2.1. ([11, Definition 2.7] and [17, Theorem 5.2.3 and Definition 6.1.2]) Let C
be a semidualizing R-module. A finite R-module M is totally C-reflexive when it satisfies
the following conditions :
C : M −→ Hom (Hom (M, C), C) is an isomor(i) the natural homomorphism δM
R
R

phism, and
>1
(ii) Ext>1
R (M, C) = 0 = ExtR (HomR (M, C), C).

Fact 2.2. Define the order E on G0 (R). For [B], [C] ∈ G0 (R), write [C] E [B] when B is
totally C-reflexive (see, e.g., [16]). This relation is reflexive and antisymmetric [7, Lemma
3.2], but it is not known whether it is transitive in general. Also, write [C] ⊳ [B] when
[C] E [B] and [C] 6= [B].
In the case D is a dualizing R-module, one has [D] E [B] for any semidualizing R-module
B, by [10, (V.2.1)].
If [C] E [B] then HomR (B, C) is a semidualizing and [C] E [HomR (B, C)] ([4, Theorem
2.11]).

CHAINS OF SEMIDUALIZING MODULES

3

The following notations are taken from [16].
Notation 2.3. Assume that [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a chain in G0 (R). For each i ∈
[n] = {1, · · · , n} set Bi = HomR (Ci−1 , Ci ). For each sequence of integers i = {i1 , · · · , ij }
with j > 1 and 1 6 i1 < · · · < ij 6 n, set Bi = Bi1 ⊗R · · · ⊗R Bij . (B{i1 } = Bi1 and set
B∅ = C0 ).
Fact 2.4. If [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a chain in G0 (R) such that C0 ∼
= R, then by [8,
Corollary 3.3], for each i ∈ [n] we have the following isomorphisms
Ci ∼
= C0 ⊗R HomR (C0 , C1 ) ⊗R · · · ⊗R HomR (Ci−1 , Ci ) ∼
= B1 ⊗ R · · · ⊗ R Bi .
For a semidualizing R-module C, set (−)†C = HomR (−, C).
Definition 2.5. ([16, Remark 3.4] and [1, Definition 3.1]) Let [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] be
a chain in G0 (R) of length n. For each sequence of integers i = {i1 , · · · , ij } such that j > 0
†Ci †Ci ···†Ci

and 1 6 i1 < · · · < ij 6 n, set Ci = C0

1

2

j

. (When j = 0, set Ci = C∅ = C0 ).

We say that the above chain is suitable if C0 = R and Ci is totally Ct -reflexive, for all
i and t with ij 6 t 6 n.
Note that if [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [R] is a suitable chain, then Ci is a semidualizing
R-module for each i ⊆ [n].
Proposition 2.6. Assume that [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a suitable chain in G0 (R).
(1) ([16, Lemma 4.5] and [1, Remark 3.3]) For each sequence i = {i1 , · · · , ij } ⊆ [n],
the R-module Bi is a semidualizing.
(2) ([16, Lemma 4.6] and [1, Remark 3.3]) If i, s ⊆ [n] are two sequences with s ⊆ i,
then [Bi ] E [Bs ] and HomR (Bs , Bi ) ∼
= Bi\s .

(3) ([16, Theorem 3.3] and [1, Remark 3.3]) |G0 (R)| > | [Ci ] | i ⊆ [n] | = 2n .


(4) ([16, Theorem 4.7] and [1, Remark 3.3]) [Bu ] | u ⊆ [n] = [Ci ] | i ⊆ [n] .
For an R-module M , the ith Bass number of M is the integer µiR (M ) = rankk (ExtiR (k, M )),
M (t) = Σ
i
i
and the Bass series of M is the formal Laurent series IR
i∈Z µR (M )t . The ith

Betti number of M is the integer βiR (M ) = rankk (ExtiR (M, k)) = rankk (TorR
i (k, M )), and
R (t) = Σ
R
i
the Poincaré series of M is the formal Laurent series PM
i∈Z βi (M )t .

3. Results
First we investigate suitable chains modulo regular sequences.
Proposition 3.1. Assume that x = x1 , · · · , xd is an R-regular sequence and [Cn ] ⊳ · · · ⊳
[C1 ] ⊳ [C0 ] is a suitable chain in G0 (R) of length n. Then [C n ] ⊳ · · · ⊳ [C 1 ] ⊳ [C 0 ]
is also a suitable chain in G0 (R) of length n, where R = R/xR and C i = R ⊗R Ci for
i = 0, 1, · · · , n.

4

E. AMANZADEH

Proof. By [4, Theorem 5.1], C i is a semidualizing R-module for all i = 0, · · · , n. Also, by
[4, Theorem 5.10] and [17, Proposition 4.2.18], [C i ] ⊳ [C i−1 ] for i = 1, · · · , n. Therefore
[C n ] ⊳ · · · ⊳ [C 1 ] ⊳ [C 0 ] is a chain in G0 (R) of length n. Now we show that this chain is
suitable.
For a semidualizing R-module C, set (−)†C = HomR (−, C). For each sequence of
†C

integers i = {i1 , · · · , ij } such that j > 0 and 1 6 i1 < · · · < ij 6 n, set Ci = C 0

···†C
†
i1 C i2
ij

.

(When j = 0, set Ci = C∅ = C 0 = R).
By induction on j we prove that
†C

Ci = C 0

†
···†C
ij
i1 C i2

†Ci †Ci ···†Ci

∼
= C0

1

2

†Ci †Ci ···†Ci

j

2

1

= R ⊗ R C0

j

= Ci

and Ci is totally C t -reflexive for all i and t with ij 6 t 6 n.
†

Ci
When j = 0, there is nothing to prove. If j = 1, then Ci = HomR (C 0 , C i1 ) ∼
= C0 1 . As

†Ci

[Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a suitable chain, C0

1

is totally Ct -reflexive for all i1 6 t 6 n.

Hence Ci is totally C t -reflexive for all i1 6 t 6 n, by [4, Theorem 5.10].
†C

Let j > 1. We have Ci = (C 0
†C

C0

···†C
†
i1 C i2
ij−1

···†C
†
i1 C i2
ij−1

†C

)

ij

and by induction

†Ci †Ci ···†Ci

∼
= C0

1

2

j−1

†Ci †Ci ···†Ci

is totally C t -reflexive for all ij−1 6 t 6 n. As C0

1

2

j−1

is totally Cij -reflexive, we

obtain the isomorphism
†Ci †Ci ···†Ci

HomR (C0

1

2

j−1

†Ci †Ci ···†Ci

, C ij ) ∼
= HomR (C0

1

2

j−1

, C ij ) .

†C †C ···†C

ij
i2
i1
Therefore Ci ∼
= C i and Ci is totally C t -reflexive for all ij 6 t 6 n, by [4,
= C0

Theorem 5.10]. Thus [C n ] ⊳ · · · ⊳ [C 1 ] ⊳ [C 0 ] is a suitable chain in G0 (R).

For the remaining part of this paper we assume that (R, m, k) is an Artinian local ring
and that all modules are finite.
Definition 3.2. [8, Definitions 4.1 and 4.2] The modules K1 , K2 , · · · , Kn are said to be
weakly Tor-independent if amp(⊗L
16i6n Ki ) = 0. These modules are said to be strongly
Tor-independent if for any subset Λ ⊆ [n] we have amp(⊗L
i∈Λ Ki ) = 0.
In the case n = 2 both notions are equivalent to the classical Tor-independence, i.e., to
the condition that TorR
>0 (K1 , K2 ) = 0.
Theorem 3.3. [8, Theorem 4.5] If the modules K1 , K2 , · · · , Kn are non-free and strongly
Tor-independent, then mn 6= 0. If, under the same conditions, mn+1 = 0, then the Poincaré
Q
series of k has the form 1/ ni=1 (1 − di t) for some positive integers di .

CHAINS OF SEMIDUALIZING MODULES

5

Conjecture 3.4. [8, Conjecture 4.6] If [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a chain in G0 (R) of
length n, then mn 6= 0. If, under the same conditions, mn+1 = 0, then the Poincaré series
Q
of k has the form 1/ ni=1 (1 − di t) for some positive integers di .
In the next result we prove the conjecture for suitable chains.
Theorem 3.5. Let [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] be a suitable chain in G0 (R) of length n,
then mn 6= 0. If, under the same conditions, mn+1 = 0, then the Poincaré series of k has
Q
the form 1/ ni=1 (1 − di t) for some positive integers di .
Proof. The non-free semidualizing modules B1 , B2 , · · · , Bn are strongly Tor-independent,
where Bi is as in Notation 2.3, for each i ∈ [n]. Indeed, by Proposition 2.6, for each
sequence of integers i = {i1 , · · · , ij } with j > 1 and 1 6 i1 < · · · < ij 6 n, Bi =
L
Bi1 ⊗R · · ·⊗R Bij is a semidualizing R-module. Thus amp(Bi1 ⊗L
R · · ·⊗R Bij ) = amp(Bi ) = 0.

Therefore the assertion concludes by Theorem 3.3.



Note that Theorem 3.5 implies [8, Theorem 4.8]. Indeed, for n 6 3 any chain of the
form [D] = [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] = [R], where D is dualizing, is also a suitable chain of
length n. At this moment we do not know whether a chain for which Conjecture 3.4 holds
true, is a suitable chain. However we guess that, if [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a chain
in G0 (R) such that the semidualizing modules B1 , · · · , Bn are strongly Tor-independent,
then this chain is suitable.
Remark 3.6. When (S, n) is a Cohen-Macaulay local ring with dimension d and [Cn ] ⊳
· · · ⊳ [C1 ] ⊳ [C0 ] is a suitable chain in G0 (S) of length n, then, by Proposition 3.1,
[C n ] ⊳ · · · ⊳ [C 1 ] ⊳ [C 0 ] is also a suitable chain in G0 (S) of length n, where S =
S/xS, C i = S ⊗S Ci and x = x1 , · · · , xd is an S-regular sequence. If nn+1 = 0, then by
Qn
Theorem 3.5, the Poincaré series of S/n ∼
= S/n has the form 1/ i=1 (1 − di t) for some
positive integers di .

Also, one may see that the set G0 (S) can not contain suitable chains of arbitrary length.
Indeed, the length of a suitable chain in G0 (S) is less than the number l = min { i > 0 | ni =
0 }. In [15], some other upper bounds for the length of chains of semidualizing modules
are given.
Definition 3.7. [8, Definition 4.9] An Artinian ring R is called SD(n)-full if the following
conditions are satisfied.
(i) mn+1 = 0.
(ii) There are strongly Tor-independent non-free semidualizing modules
K1 , K2 , · · · , Kn such that for any subset Λ ⊆ [n] the module ⊗i∈Λ Ki
is semidualizing.

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E. AMANZADEH

Theorem 3.8. An Artinian ring R is SD(n)-full if and only if there is a suitable chain
in G0 (R) of length n and mn+1 = 0.
Proof. First we assume that mn+1 = 0 and [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a suitable chain
in G0 (R) of length n. Then, by Proposition 2.6, the non-free semidualizing modules
B1 , B2 , · · · , Bn are strongly Tor-independent and, for each subset i = {i1 , · · · , ij } ⊆ [n],
Bi = Bi1 ⊗R · · · ⊗R Bij is a semidualizing R-module. Therefore R is SD(n)-full.
For the converse, assume that K1 , K2 , · · · , Kn are strongly Tor-independent non-free
semidualizing modules such that for any subset Λ ⊆ [n] the module ⊗i∈Λ Ki is semidualizing. Set C0 = R and Cj = ⊗16i6j Ki for each j ∈ [n]. By [8, Proposition 3.5], it can be
seen that the semidualizing modules C0 , C1 , · · · , Cn form a chain in G0 (R) and then the
sequence
(3.1)

[Cn ] ⊳ [Cn−1 ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ]

is a chain in G0 (R) of length n. We show that (3.1) is a suitable chain. For each sequence
i = {i1 , · · · , ij } ⊆ [n] such that j > 0 and 1 6 i1 < · · · < ij 6 n, we have Ci =
†Ci †Ci ···†Ci

C0

2

1

j

. (When j = 0, set Ci = C∅ = C0 ). Note that for each t, ij 6 t 6 n,

Ct = ⊗16l6t Kl = (⊗16l6i1 Kl ) ⊗R (⊗i1 <l6i2 Kl ) ⊗R · · · ⊗R (⊗ij−1 <l6ij Kl ) ⊗R (⊗ij <l6t Kl ).
On the other hand, by [8, Proposition 3.5], we have
†Ci †Ci

C0

1

2

= HomR (⊗16l6i1 Kl , ⊗16l6i2 Kl ) ∼
= ⊗i1 <l6i2 Kl

and so
†Ci †Ci †Ci

C0

1

2

3

= HomR (⊗i1 <l6i2 Kl , ⊗16l6i3 Kl ) ∼
= (⊗16l6i1 Kl ) ⊗R (⊗i2 <l6i3 Kl ).

By proceeding in this way and using [8, Proposition 3.5], one may see that Ci is totally
Ct -reflexive. Thus the sequence (3.1) is a suitable chain.



Example 3.9. A ring with mn+1 = 0 is SD(n)-full if and only if there is a suitable chain
in G0 (R) of length n.
Let F be a field. Set Si = F ⋉F ai for all 1 6 i 6 n, where ai > 1. Then Si is an Artinian
ring with dualizing module Di = HomF (Si , F ). As type Si = ai 6= 1, the ring Si is not
Gorenstein. Set S = ⊗F16i6n Si . By [8, Example 4.11], S is SD(n)-full. Now, we construct a
suitable chain of length n in G0 (S). If Ki and Mi are Si -modules such that Ki is finite, then
there is S-module isomorphism HomS (⊗16i6n Ki , ⊗16i6n Mi ) ∼
= ⊗16i6n HomS (Ki , Mi ), by
F

F

F

i

[17, Proposition A.1.5]. When Ki is semidualizing, by [17, Proposition 2.3.6], ⊗F16i6n Ki
is also a semidualizing S-module, and ⊗F16i6n Mi is totally ⊗F16i6n Ki -reflexive by [17,
Proposition 5.3.3], whenever Mi is totally Ki -reflexive. Note that if we take, for each
i, a module Ki ∈ {Si , Di }, then the module ⊗F16i6n Ki is semidualizing S-module. Set
C0 = S and Cj = (⊗F16i6j Di ) ⊗F (⊗Fj<i6n Si ) for all 1 6 j 6 n. Therefore we obtain the

CHAINS OF SEMIDUALIZING MODULES

7

chain [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] in G0 (S) of length n. We show that this chain is suitable.
Assume that u = {u1 , · · · , uj } is a sequence of integers such that 1 6 u1 < · · · < uj 6 n
†Cu ···†Cu
1

and Cu = C0
†Cu †Cu

C0

1

2

1

†Cu

. We have C0

1

∼
= Cu 1 ,

∼
= HomS (Cu1 , Cu2 ) ∼
= (⊗F16i6u1 Si ) ⊗F (⊗Fu1 <i6u2 Di ) ⊗F (⊗uF2 <i6n Si ) , and

†Cu †Cu †Cu

C0

j

2

3

†Cu †Cu

= HomS (C0 1 2 , Cu3 )
∼
= (⊗F16i6u1 Di ) ⊗F (⊗uF1 <i6u2 Si ) ⊗F (⊗Fu2 <i6u3 Di ) ⊗F (⊗uF3 <i6n Si ).

By proceeding in this way one obtains the following isomorphism
Cu ∼
=

 16i6u1
u <i6n
u
<i6uj
u1 <i6u2
Di ) ⊗F (⊗Fj
Si )
Di ) ⊗F · · · ⊗F (⊗Fj−1
Si ) ⊗F (⊗F

(⊗F

 16i6u1
u <i6n
u
<i6uj
u1 <i6u2
Di ) ⊗F (⊗Fj
Si )
Si ) ⊗F · · · ⊗F (⊗Fj−1
Di ) ⊗F (⊗F
(⊗F

if j is even,
if j is odd.

As Ct = (⊗F16i6t Di ) ⊗F (⊗t<i6n
Si ), one can see that Cu is totally Ct -reflexive for all
F
uj 6 t 6 n. Hence [Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a suitable chain in G0 (S).
Note that Bj = HomS (Cj−1 , Cj ) ∼
= S1 ⊗F · · · ⊗F Sj−1 ⊗F Dj ⊗F Sj+1 ⊗F · · · ⊗F Sn for
all 1 6 j 6 n. The semidualizing modules B1 , · · · , Bn are strongly Tor-independent and
for any subset Λ ⊆ [n] the module ⊗i∈Λ Bi is semidualizing.
The next result follows from [8, Proposition 5.4] and the proof of Theorem 3.8.
Lemma 3.10. Let R be an Artinian ring with mn+1 = 0. Assume that [Cn ] ⊳ · · · ⊳ [C1 ] ⊳
[C0 ] is a suitable chain in G0 (R) of length n. Then for each i ∈ [n] the Poincaré series
B

of Bi is PBRi (t) = (β0R (Bi ) − t)/(1 − β0R (Bi )t) and the Bass series of B[n]\i is IR [n]\i (t) =
(µ0R (B[n]\i ) − t)/(1 − µ0R (B[n]\i )t).
Proposition 3.11. Let R be an Artinian ring with mn+1 = 0. Assume that [Cn ] ⊳ · · · ⊳
[C1 ] ⊳ [C0 ] is a suitable chain in G0 (R) of length n. Then for each i ∈ [n] the Poincaré
series of Ci is
PCRi (t)

=

Qi

R
j=1 (β0 (Bj ) − t)
Qi
R
j=1 (1 − β0 (Bj )t)

,

Cn
(t) = 1, and for i 6= n, the Bass series of Ci is
IR
Qn
Qn
R
0
j=i+1 (β0 (Bj ) − t)
j=i+1 (µR (B[n]\j ) − t)
Ci
Q
=
IR (t) = Qn
.
n
0
R
j=i+1 (1 − µR (B[n]\j )t)
j=i+1 (1 − β0 (Bj )t)

Proof. As mn+1 = 0, this suitable chain is of maximum length, by Theorem 3.5. Thus the
Cn
(t) = 1. For i ∈
module Cn is the dualizing R-module. Hence the Bass series of Cn is IR
R
R
[n], we have Ci ∼
= B1 ⊗R · · · ⊗R Bi , by Fact 2.4. Therefore P (t) = P (t)P R (t) · · · P R (t),
Ci

B1

by [2, Lemma 1.5.3]. Hence, by Lemma 3.10, one gets
PCRi (t) = (

β0R (Bi ) − t
β0R (B1 ) − t β0R (B2 ) − t
)(
)
·
·
·
(
).
1 − β0R (B1 )t 1 − β0R (B2 )t
1 − β0R (Bi )t

B2

Bi

8

E. AMANZADEH

Also, we have Ci ∼
= B1 ⊗ R · · · ⊗ R Bi ∼
= HomR (B{i+1,··· ,n} , B[n] ), by Proposition 2.6. Thus,
by [2, Lemma 1.5.3], one gets
B

B

Ci
IR
(t) = PBR{i+1,··· ,n} (t)IR [n] (t) = PBRi+1 (t) · · · PBRn (t)IR [n] (t) .
B[n]
Cn
(t) = 1. Therefore, by Lemma 3.10,
As B[n] = B1 ⊗R · · ·⊗R Bn ∼
= Cn , one has IR (t) = IR
Ci
IR
(t) = PBRi+1 (t) · · · PBRn (t) = (

β0R (Bi+1 ) − t
β0R (Bn ) − t
)
·
·
·
(
).
1 − β0R (Bi+1 )t
1 − β0R (Bn )t

B
B[n]\j
(t) = PBRj (t)IR [n] (t) = PBRj (t).
By Proposition 2.6, B[n]\j ∼
= HomR (Bj , B[n] ) and so IR
Note that µ0 (B[n]\j ) = β0 (Bj ) and then the final equality follows.


From Proposition 3.11, the Bass series of R is
IR (t) =

Hom (C ,C )
IR R 1 1 (t)

=

C1
PCR1 (t)IR
(t)

IR (t) =

=

PCRn (t)

B

B
C1
IR [n]\1 (t)IR
(t)

Qn

0
j=1 (µR (B[n]\j ) − t)
= Qn
, and
0
j=1 (1 − µR (B[n]\j )t)

Qn

R
j=1 (β0 (Bj ) − t)
R
j=1 (1 − β0 (Bj )t)

= Qn

.

For each i ∈ [n], IR [n]\i (t) = αi + (α2i − 1)t + αi (α2i − 1)t2 + α2i (α2i − 1)t3 + · · · , where αi =
µ0R (B[n]\i ) (see [8, Propositions 5.1, 5.2 and 5.4]). Thus {µjR (B[n]\i )} is strictly increasing
(indeed, it has exponential growth, since µjR (B[n]\i ) > αji for all j > 1). Therefore {µjR (R)}
B

B

B

is also strictly increasing whenever n > 1, since IR (t) = IR [n]\1 (t)IR [n]\2 (t) · · · IR [n]\n (t).
Also, similar to the proof of [15, Theorem 3.5], it is easy to see that {µjR (R)} is bounded
below by a polynomial in j of degree n − 1.
Remark 3.12. Assume that (S, n) is a Cohen-Macaulay local ring with dimension d
and x = x1 , · · · , xd is an S-regular sequence. Set S = S/xS. As Exti (S/n, S) ∼
=
S

ExtSi−d (S/n, S) for all i
IS (t) = td IS (t). Now, if
n+1

and n

> d, we have

µiS (S)

=

(S),
µi−d
S

for all i > d. Thus we get

[Cn ] ⊳ · · · ⊳ [C1 ] ⊳ [C0 ] is a suitable chain in G0 (S) of length n

= 0, then, by Proposition 3.1 and the above discussion, IS (t) has a very specific

form and {µiS (S)} is strictly increasing.
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School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box:
19395-5746, Tehran, Iran.
E-mail address: en.amanzadeh@gmail.com

