arXiv:1508.05813v3 [] 7 Apr 2016

DUAL OF BASS NUMBERS AND DUALIZING MODULES
MOHAMMAD RAHMANI AND ABDOLJAVAD TAHERIZADEH
Abstract. Let R be a Noetherian ring and let C be a semidualizing R-module. In this
paper, we impose various conditions on C to be dualizing. For example, as a generalization of Xu [22, Theorem 3.2], we show that C is dualizing if and only if for an R-module
M , the necessary and sufficient condition for M to be C-injective is that πi (p, M ) = 0
for all p ∈ Spec (R) and all i 6= ht (p), where πi is the invariant dual to the Bass numbers
defined by E.Enochs and J.Xu [8].

1. introduction
Throughout this paper, R is a commutative Noetherian ring with non-zero identity.
A finitely generated R-module C is semidualizing if the natural homothety map R −→
Hom R (C, C) is an isomorphism and Ext iR (C, C) = 0 for all i > 0. Semidualizing modules
have been studied by Foxby [9], Vasconcelos [20] and Golod [10] who used the name suitable for these modules. Dualizing complexes, introduced by A.Grothendieck, is a powerful
tool for investigating cohomology theories in algebraic geometry. A bounded complex of Rmodules D with finitely generated homologies is said to be a dualizing complex for R, if the
natural homothety morphism R → RHom R (D, D) is quasiisomorphism, and id R (D) < ∞.
These notion has been extended to semidualizing complexes by L.W. Christensen [5]. A
bounded complex of R-modules C with finitely generated homologies is semidualizing for
R if the natural homothety morphism R → RHom R (C, C) is quasiisomorphism. He used
these notion to define a new homological dimension for complexes, namely GC -dimension,
which is a generalization of Yassemi’s G-dimension [23]. The following, is the translation of
a part of [5, Proposition 8.4] to the language of modules:
Theorem 1. Let (R, m, k) be a Noetherian local ring and let C be a semidualizing Rmodule. The following are equivalent:
(i) C is dualizing.
(ii) GC -dim R (M ) < ∞ for all finite R-modules M .
(iii) GC -dim R (k) < ∞.
In particular, the above theorem recovers [4, 1.4.9]. Note that k is a Cohen-macaulay Rmodule of type 1. R.Takahashi, in [17, Theorem 2.3], replaced the condition G-dim R (k) < ∞
in [4, 1.4.9] by weaker conditions and obtained a nice characterization for Gorenstein rings.
2000 Mathematics Subject Classification. 13C05, 13D05, 13D07, 13H10.
Key words and phrases. Semidualizing modules, dualizing modules, GC -dimension, Bass numbers, dual
of Bass numbers, minimal flat resolution, local cohomology.
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M. RAHMANI AND A.- J. TAHERIZADEH

Indeed, he showed that R is Gorenstein, provided that either R admits an ideal I of finite Gdimension such that R/I is Gorenstein, or there exists a Cohen-Macaulay R-module of type
1 and of finite G-dimension. The following is the main result of section 3, which generalizes
Theorem 1 as well as [17, Theorem 2.3]. See Theorem 3.4 below.
Theorem 2. Let (R, m) be a Noetherian local ring and let C be a semidualizing R-module.
The following are equivalent:
(i) C is dualizing.
(ii) There exists an ideal a with GC -dim R (aC) < ∞ such that C/aC is dualizing for
R/a.
(iii) There exists a Cohen-Macaulay R-module M with rR (M ) = 1 and GC -dim R (M ) <
∞.
(iv) rR (C) = 1 and there exists a Cohen-Macaulay R-module M with GC -dim R (M ) <
∞.
E.Enochs et al. [1], solved a long standing conjecture about the existence of flat covers.
Indeed, they showed that if R is any ring, then all R-modules have flat covers. E.Enochs
[6], determined the structure of flat cotorsion modules. Also, E.Enochs and J.Xu [8, Definition 1.2], defined a new invariant πi , dual to the Bass numbers, for modules related to
flat resolutions. J.Xu [22], studied the minimal injective resolution of flat R-modules and
minimal flat resolution of injective R-modules. He characterized Gorenstein rings in terms
of vanishing of Buss numbers of flat modules, and vanishing of dual of Bass numbers of
injective modules. More precisely, the following theorem is [22, Theorems 2.1 and 3.2].
Theorem 3. Let R be a Noetherian ring. The following are equivalent:
(i) R is Gorenstein.
(ii) An R-module F is flat if and only if µi (p, F ) = 0 for all p ∈ Spec (R) whenever
i 6= ht (p).
(iii) An R-module E is injective if and only if πi (p, E) = 0 for all p ∈ Spec (R) whenever
i 6= ht (p).
In section 4, we give a generalization of Theorem 3. Indeed, in Theorem 4.3, we prove the
following result.
Theorem 4. Let R be a Noetherian ring and let C be a semidualizing R-module. The
following are equivalent:
(i) C is pointwise dualizing.
(ii) An R-module M is C-injective if and only if πi (p, M ) = 0 for all p ∈ Spec (R)
whenever i 6= ht (p).
(iii) An R-module M is injective if and only if πi (p, Hom R (C, M )) = 0 for all p ∈
Spec (R) whenever i 6= ht (p).
Theorem 4 has several applications. Let (R, m) be a d-dimensional Cohen-Macaulay local
ring possessing a canonical module. In this section, we give the structure of the minimal flat
resolution of Hdm (R), the top local cohomology of R. More precisely, the following theorem
is Corollary 4.7.

DUAL OF BASS NUMBERS AND DUALIZING MODULES

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Theorem 5. Let (R, m) be a d-dimensional Cohen-Macaulay local ring possessing a canonical module. The minimal flat resolution of Hdm (R) is of the form
cm −→ · · · −→ Q Tp −→ Q Tp −→ Hdm (R) −→ 0,
0 −→ R
ht (p)=1
ht (p)=0
in which Tp is the completion of a free Rp -module with respect to pRp -adic topology.
In this section, by using the above resolution, we obtain the following isomorphism for a
d-dimensional Cohen-Macaulay local ring (See Corollary 4.8).
(
Hdm (R)
i = d,
d
d
∼
Tor R
i (Hm (R), Hm (R)) =
0
i=
6 d.

2. preliminaries
In this section, we recall some definitions and facts which are needed throughout this
paper. By an injective cogenerator, we always mean an injective R-module E for which
Hom R (M, E) 6= 0 whenever M is a nonzero R-module. For an R-module M , the injective
hull of M , is always denoted by E(M ).
Definition 2.1. Let X be a class of R-modules and M an R-module. An X -resolution of
M is a complex of R-modules in X of the form
∂X

∂X

1
n
X0 −→ 0
Xn−1 −→ . . . −→ X1 −→
X = . . . −→ Xn −→
∼
such that H0 (X) = M and Hn (X) = 0 for all n ≥ 1. Also the X -projective dimension of M

is the quantity
X -pd R (M ) := inf{sup{n ≥ 0|Xn 6= 0} | X is an X -resolution of M }

.

So that in particular X -pd R (0) = −∞. The modules of X -projective dimension zero are
precisely the non-zero modules in X . The terms of X -coresolution and X -id are defined
dually.
Definition 2.2. A finitely generated R-module C is semidualizing if it satisfies the following
conditions:
(i) The natural homothety map R −→ Hom R (C, C) is an isomorphism.
(ii) Ext iR (C, C) = 0 for all i > 0.
For example a finitely generated projective R-module of rank 1 is semidualizing. If
R is Cohen-Macaulay, then an R-module D is dualizing if it is semidualizing and that
id R (D) < ∞ . For example the canonical module of a Cohen-Macaulay local ring, if exists,
is dualizing.
Definition 2.3. Following [12], let C be a semidualizing R-module. We set
FC (R) = the subcategory of R–modules C ⊗R F where F is a flat R–module.
IC (R) = the subcategory of R–modules Hom R (C, I) where I is an injective R–
module.

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M. RAHMANI AND A.- J. TAHERIZADEH

The R-modules in FC (R) and IC (R) are called C-flat and C-injective, respectively. If
C = R, then it recovers the classes of flat and injective modules, respectively. We use the
notations C-fd and C-id instead of FC -pd and IC -id , respectively.
Proposition 2.4. Let C be a semidualizing R-module. Then we have the following:
(i) Supp (C) = Spec (R), dim (C) = dim (R) and Ass (C) = Ass (R).
(ii) If R → S is a flat ring homomorphism, then C ⊗R S is a semidualizing S-module.
(iii) If x ∈ R is R–regular, then C/xC is a semidualizing R/xR-module.
(iv) If, in addition, R is local, then depth R (C) = depth (R).
Proof. The parts (i), (ii) and (iii) follow from the definition of semidualizing modules. For
(iv), note that an element of R is R-regular if and only if it is C-regular since Ass (C) =
Ass (R). Now an easy induction yields the equality.



Definition 2.5. Let C be a semidualizing R-module. A finitely generated R-module M is
said to be totally C-reflexive if the following conditions are satisfied:
(i) The natural evaluation map M −→ Hom R (Hom R (M, C), C) is an isomorphism.
(ii) Ext iR (M, C) = 0 = Ext iR (Hom R (M, C), C) for all i > 0.
For an R-module M , if there exists an exact sequence 0 → Gn → · · · → G1 → G0 → M → 0,
of R-modules such that each Gi is totally C-reflexive, then we say that M has GC -dimension
at most n, and write GC -dim R (M ) ≤ n. If there is no shorter such sequence, we set GC dim R (M ) = n. Also, if such an integer n does not exist, then we say that M has infinite
GC -dimension, and write GC -dim R (M ) = ∞.
The next proposition collects basic properties of GC -dimension. For the proof, see [10].
Proposition 2.6. Let (R, m) be local, M a finitely generated R-module and let C be a
semidualizing R-module. The following statements hold:
(i) If GC -dim R (M ) < ∞, and x ∈ m is M -regular, then
GC -dim R (M ) = GC -dim R (M/xM ) − 1.
If, also, x is R-regular, then
GC -dim R (M ) = GC/xC -dim R/xR (M/xM ).
(ii) If GC -dim R (M ) < ∞ and x is an R-regular element in Ann R (M ), then
GC -dim R (M ) = GC/xC -dim R/xR (M ) + 1.
(iii) Let 0 → K → L → N → 0 be a short exact sequence of R-modules. If two of L, K, N
are of finite GC -dimension, then so is the third.
(iv) If GC -dim R (M ) < ∞, then
GC -dim R (M ) = sup{i ≥ 0 | Ext iR (M, C) 6= 0}
= depth (R) − depth R (M ).
Definition 2.7. Let C be a semidualizing R-module. The Auslander class with respect to
C is the class AC (R) of R-modules M such that:
i
(i) Tor R
i (C, M ) = 0 = Ext R (C, C ⊗R M ) for all i ≥ 1, and

DUAL OF BASS NUMBERS AND DUALIZING MODULES

5

(ii) The natural map M → Hom R (C, C ⊗R M ) is an isomorphism.
The Bass class with respect to C is the class BC (R) of R-modules M such that:
(i) Ext iR (C, M ) = 0 = Tor R
i (C, Hom R (C, M )) for all i ≥ 1, and
(ii) The natural map C ⊗R Hom R (C, M )) → M is an isomorphism.
The class AC (R) contains all R-modules of finite projective dimension and those of finite Cinjective dimension. Also the class BC (R) contains all R-modules of finite injective dimension
and those of finite C-projective dimension (see [18, Corollary 2.9]). Also, if any two Rmodules in a short exact sequence are in AC (R) (resp. BC (R)), then so is the third (see
[13]).
Proposition 2.8. Let (R, m) be a local ring and let C be a semidualizing R-module.
b is a dualizing R-module
b
.
(i) C is a dualizing R-module if and only if C ⊗R R

(ii) Let x ∈ m be R-regular. Then C is a dualizing R-module if and only if C/xC is a
dualizing R/xR-module.

Proof. Just use the definition of dualizing modules.



Theorem 2.9. Let C be a semidualizing R-module and let M be an R-module.
(i) C-id R (M ) = id R (C ⊗R M ) and id R (M ) = C-id R (Hom R (C, M )).
(ii) C-fd R (M ) = fd R (Hom R (C, M )) and fd R (M ) = C-fd R (C ⊗R M ).
Proof. For (i), see [18, Theorem 2.11] and for (ii), see [19, Proposition 5.2].



Lemma 2.10. Let C be a semidualizing R-module, E be an injective cogenerator and M be
an R-module.
(i) One has C-id R (M ) = C-fd R (Hom R (M, E)).
(ii) One has C-fd R (M ) = C-id R (Hom R (M, E)).
Proof. (i). We have the following equalities
C-id R (M ) = id R (C ⊗R M )
= fd R (Hom R (C ⊗R M, E))
= fd R (Hom R (C, Hom R (M, E))
= C-fd R (Hom R (M, E)),
in which the first equality is from Theorem 2.9(i), and the last one is from Theorem 2.9(ii).
(ii). Is similar to (i).



Remark 2.11. Let (R, m) be a local ring and let M be a finitely generated R-module. We
use νR (M ) to denote the minimal number of generators of M . More precisely, νR (M ) =
vdim R/m (M ⊗R R/m). It is easy to see that if x ∈ m, then νR (M ) = νR/xR (M/xM ). In
particular, if x ∈ Ann R (M ), then νR (M ) = νR/xR (M ). Assume that depth R (M ) = n. The
type of M , denoted by rR (M ), is defined to be vdim R/m (Ext nR (R/m, M )). If x ∈ m, then
rR (M/xM ) = rR/xR (M/xM ) by [2, Exercise 1.2.26]. Also, if x ∈ m is M - and R-regular,
then rR (M ) = rR/xR (M/xM ) by [2, Lemma 3.1.16]. Assuma that C is a semidualizing

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M. RAHMANI AND A.- J. TAHERIZADEH

R-module. Then rR (C) | rR (R). Indeed, by reduction modulo a maximal R-sequence, we
can assume that depth R (C) = 0 = depth (R). Then we have
rR (R) = vdim R/m Hom R (R/m, R)
= vdim R/m Hom R (R/m, Hom R (C, C))
= vdim R/m Hom R (R/m ⊗R C, C)
= vdim R/m Hom R (R/m ⊗R C ⊗R/m R/m, C)
= vdim R/m Hom R/m (R/m ⊗R C, Hom R (R/m, C))
= νR (C)rR (C).
In particular, if rR (R) = 1 (e.g. R is Gorenstein local), then νR (C) = 1 and then C ∼
= R.
Definition 2.12. Let M be an R-module and let X be a class of R-modules . Following
[7], a X -precover of M is a homomorphism ϕ : X → M , with X ∈ X , such that every
homomorphism Y → M with Y ∈ X , factors through φ; i.e., the homomorphism
Hom R (Y, ϕ) : Hom R (Y, X) → Hom R (Y, M )
is surjective for each module Y in X . A X -precover ϕ : X → M is a X -cover if every
ψ ∈ Hom R (X, X) with ϕψ = ϕ is an automorphism.
Definition 2.13. Following [6], an R-module M is called cotorsion if Ext 1R (F, M ) = 0 for
any flat R-module F.
Remark 2.14. In [1], E. Enochs et al. showed that if R is any ring, then every R-module
has a flat cover. It is easy to see that flat cover must be surjective. By [6, Lemma 2.2], the
kernel of a flat cover is always cotorsion. So that if F → M is flat cover and M is cotorsion,
then so is F . Therefore for an R-module M , one can iteratively take flat covers to construct
a flat resolution of M . Since flat cover is unique up to isomorphism, this resolution is unique
up to isomorphism of complexes. Such a resolution is called the minimal flat resolution of M .
Note that the minimal flat resolution of M is a direct summand of any other flat resolution
of M . Assume that
· · · → Fi → · · · → F1 → F0 → M → 0,
is the minimal flat resolution of M . Then Fi is cotorsion for all i ≥ 1. If, in addition, M
is cotorsion, then all the flat modules in the minimal flat resolution of M are cotorsion. E.
Enochs [6], determined the structure of flat cotorsion modules. He showed that if F is flat
∼ QTp where Tp is the completion of a free Rp -module with respect
and cotorsion, then F =
p

to pRp -adic topology. So that we can determine the structure of the minimal flat resolution
of cotorsion modules.
Definition 2.15. Let M be a cotorsion R-module and let
· · · → Fi → · · · → F1 → F0 → M → 0,
be the minimal flat resolution of M . Following [8], for a prime ideal p of R and an integer
i ≥ 0, the invariant πi (p, M ) is defined to be the cardinality of the basis of a free Rp -module
Q
whose completion is Tp in the product Fi ∼
= Tp . By [8, theorem 2.2], for each i ≥ 0,
p
Rp

πi (p, M ) = vdim Rp /pRp Tor i


Rp /pRp , Hom R (Rp , M ) .

DUAL OF BASS NUMBERS AND DUALIZING MODULES

7

Remark 2.16. Let M be a finitely generated R-module. There are isomorphisms
Hom R (M, E(R/p)) ∼
= Hom R (M, E(R/p) ⊗R Rp )
∼
= Hom R (M, E(R/p)) ⊗R Rp
∼
= Hom Rp (Mp , ERp (Rp /pRp )),
where the the first isomorphism holds because E(R/p) ∼
= ERp (Rp /pRp ), and the second
isomorphism is tensor-evaluation [7, Theorem 3.2.14].
3. Finiteness of GC -dimension
Throughout this section, C is a semidualizing R-module. We begin with three lemmas
that are needed for the main result of this section. It is well-known that a local ring over
which there exists a non-zero finitely generated injective module, must be Artinian. Our first
lemma generalizes this fact by replacing the injectivity condition with weaker assumption.
Lemma 3.1. Let (R, m) be local and let M be a finitely generated R-module with depth (M ) =
0. If Ext 1R (R/m, M ) = 0, then R is Artinian. In particular, M is injective.
Proof. We show that dim (R) = 0. Assume, on the contrary, that dim (R) > 0. Note that if
N is an R-module of finite length, then by using a composition series for N in conjunction
with the assumption, we have Ext 1R (N, M ) = 0. Now an easy induction on ℓR (N ) yields the
equality ℓR (Hom R (N, M )) = ℓR (N )ℓR (Hom R (R/m, M )). Next, note that ℓR (R/mi ) < ∞
i
i+1
for any i ≥ 1, and that the sequence {ℓR (R/mi )}∞
for
i=1 is not bounded since m 6= m
2
any i ≥ 1. Hence {ℓR (Hom R (R/mi , M ))}∞
i=1 is not bounded. But 0 :M m ⊆ 0 :M m ⊆ · · ·

is a chain of submodules of M , and hence is eventually stationary. This is a contradiction.
Therefore R is Artinian. Finally, the assumption Ext 1R (R/m, M ) = 0 implies that M is
injective.



Lemma 3.2. Let (R, m) be a local ring and let M be a Cohen-Macaulay R-module with GC dim R (M ) < ∞. Then rR (C) | rR (M ).
Proof. We use induction on n = depth (R). If n = 0, then by Proposition 2.6(iv), we have
∼ Hom R (Hom R (M, C), C), and
GC -dim R (M ) = 0, and hence there is an isomorphism M =
the equalities depth R (C) = 0 = depth R (M ). Hence we have
rR (M ) = vdim R/m Hom R (R/m, M )
= vdim R/m Hom R (R/m, Hom R (Hom R (M, C), C))
= vdim R/m Hom R (R/m ⊗R Hom R (M, C), C)
= vdim R/m Hom R (R/m ⊗R Hom R (M, C) ⊗R/m R/m, C)
= vdim R/m Hom R/m (R/m ⊗R Hom R (M, C), Hom R (R/m, C))
= νR (Hom R (M, C))rR (C).
Therefore rR (C) | rR (M ). Now, assume inductively that n > 0. We consider two cases:
Case 1. If depth R (M ) = 0, then M is of finite length since it is Cohen-Macaulay. Hence
we can take an R-regular element x such that xM = 0. Set (−) = (−) ⊗R R/xR. Then by

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M. RAHMANI AND A.- J. TAHERIZADEH

Proposition 2.6(ii), we have GC -dim R (M ) < ∞. Also, note that M is a Cohen-Macaulay
R-module. Hence by induction hypothesis we have rR (C) | rR (M ). Thus rR (C) | rR (M ).
Case 2. If depth R (M ) > 0, then we can take an element y ∈ m to be M - and R-regular.
Set (−) = (−) ⊗R R/yR. Now M is a Cohen-Macaulay R-module, and that
GC -dim R (M ) = GC -dim R (M ) < ∞,
by Proposition 2.6(i). Therefore, by induction hypothesis, we have rR (C) | rR (M ), whence
rR (C) | rR (M ). This complete the inductive step.



Lemma 3.3. Let (R, m) be local and that rR (C) = 1. If there exists a totally C-reflexive
R-module of finite length, then C is dualizing.
Proof. Assume that M is a finite length C-reflexive R-module. Then depth R (M ) = 0,
and hence depth (R) = GC -dim (M ) = 0 by Proposition 2.6(iv).

Therefore, we have

depth R (C) = 0 by Proposition 2.4(iv). Now assume, on the contrary, that C is not dualizing. Hence, by Lemma 3.1, we have Ext 1R (R/m, C) 6= 0. Let
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M ,
be a composition series for M . Thus the factors are all isomorphic to R/m, and we have
exact sequences
0 → Mi−1 → Mi → R/m → 0,
for all 1 ≤ i ≤ r. Applying the functor Hom R (−, C), we get the exact sequence
0 → Hom R (R/m, C) → Hom R (Mi , C) → Hom R (Mi−1 , C),
for each 1 ≤ i ≤ r−1. Now since depth R (C) = 0 and rR (C) = 1, we have Hom R (R/m, C) ∼
=
R/m. Hence we have the inequality ℓR (Hom R (Mi , C)) ≤ ℓR (Hom R (Mi−1 , C)) + 1 for each
1 ≤ i ≤ r − 1. On the other hand, application of the functor Hom R (−, C) on the exact
sequence 0 → Mr−1 → M → R/m → 0, yields an exact sequence
0 → Hom R (R/m, C) → Hom R (M, C) → Hom R (Mr−1 , C)
→ Ext 1R (R/m, C) → Ext 1R (M, C) = 0.
Therefore ℓR (Hom R (M, C)) = ℓR (Hom R (Mr−1 , C)) + 1 − ℓR (Ext 1R (R/m, C)). But since
ℓR (Ext 1R (R/m, C)) > 0, we have
ℓR (Hom R (M, C))

< ℓR (Hom R (Mr−1 , C)) + 1
≤ ℓR (Hom R (Mr−2 , C)) + 2
≤ ···
≤ ℓR (Hom R (M0 , C)) + r
=r
= ℓR (M ).

Now since Hom R (M, C) is again a totally C-reflexive R-module of finite length, the same

argument shows that ℓR Hom R (Hom R (M, C), C) ≤ ℓR (Hom R (M, C)). But since M is
∼ Hom R (Hom R (M, C), C), which implies that ℓR (M ) <
totally C-reflexive, we have M =
ℓR (M ), a contradiction. Hence C is dualizing.
The following theorem is a generalization of [17, Theorem 2.3].
Theorem 3.4. Let (R, m) be local. The following are equivalent:



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9

(i) C is dualizing.
(ii) There exists an ideal a with GC -dimR (aC) < ∞ such that C/aC is dualizing for
R/a.
(iii) There exists a Cohen-Macaulay R-module M with rR (M ) = 1 and GC -dimR (M ) <
∞.
(iv) rR (C) = 1 and there exists a Cohen-Macaulay R-module M of finite GC -dimension.
Proof. (i)=⇒(ii). Choose a = 0.
(ii)=⇒(iii). We show that C/aC has the desired properties. First, the exact sequence
0 → aC → C → C/aC → 0,
in conjunction with Proposition 2.6(iii), show that GC -dim R (C/aC) < ∞. On the other
hand, C/aC is a Cohen-Macaulay R/aR-module and hence is a Cohen-Macaulay R-module.
Finally, by [2, Exercise 1.2.26], we have rR (C/aC) = rR/a (C/aC) = 1.
(iii)=⇒(iv). By Lemma 3.2, we have rR (C) = 1.
(iv)=⇒(i). Assume that M is a Cohen-Macaulay R-module with GC -dim R (M ) < ∞. We
use induction on m = depth R (M ). If m = 0, then M is of finite length since it is Cohenp
Macaulay. Since Ann R (M ) = m, we can choose a maximal R-sequence from elements

of Ann R (M ), say x. In view of Proposition 2.8(ii) and Proposition 2.6(ii), we can replace
C by C/xC and R by R/xR, and assume that M is totally C-reflexive. In this case, C
is dualizing by Lemma 3.3. Now assume inductively that m > 0. Hence depth (R) > 0
by Proposition 2.6(iv), and we can take an element x ∈ m to be M - and R-regular. Set
(−) = (−)⊗R R/xR. Now M is a Cohen-Macaulay R-module and rR (C) = rR (C) = 1. Also,
by Proposition 2.6(i), we have GC -dim R (M ) =GC -dim R (M ) < ∞. Hence, by induction
hypothesis, C is dualizing for R, whence C is dualizing for R by Proposition 2.8(ii).



It is well-known that the existence of a finitely generated (resp. Cohen-Macaulay) module
of finite injective (resp. projective) dimension implies Cohen-Macaulyness of the ring. But,
in the special case that C is dualizing, the proof is easy, as the following relations show
dim (R) = dim R (C) ≤ id R (C) = depth (R),
where the first equality is from Proposition 2.4(i), and the remaining parts are from [2,
Theorem 3.1.17]. Therefore, in view of Theorem 3.4, we can state the following corollary.
Corollary 3.5. Let (R, m) be local. If there exists a Cohen-Macaulay R-module of type 1
and of finite GC -dimension, then R is Cohen-Macaulay.
4. C-injective modules
In this section, our aim is to extend two nice results of J.Xu [22]. It is well-known that
a Noetherian ring R is Gorenstein if and only if µi (p, R) = δi,ht(p) (the Kronecker δ). As a
generalization, J.Xu [22, Theorem 2.1], showed that R is Gorenstein if and only if for any
R-module F , the necessary and sufficient condition for F to be flat is that µi (p, F ) = 0 for all
p ∈ Spec (R) and all i 6= ht (p). Next, in [22, Theorem 3.2], he proved a dual for this theorem.
Indeed, he proved that R is Gorenstein if and only if for any R-module E, the necessary
and sufficient condition for E to be injective is that πi (p, E) = 0 for all p ∈ Spec (R) and

10

M. RAHMANI AND A.- J. TAHERIZADEH

all i 6= ht (p). In the present section, first we generalize the mentioned results. Next, we use
our new results to determine the minimal flat resolution of some top local cohomology of a
Cohen-Macaulay local rings and their torsion products.
Lemma 4.1. The followings are equivalent:
(i) C is pointwise dualizing.
(ii) C-fd R (E(R/m)) = ht (m) for any m ∈ Max (R).
(iii) C-fd R (E(R/m)) < ∞ for any m ∈ Max (R).
(iv) C-fd R (E(R/p)) = ht (p) for any p ∈ Spec (R).
(v) C-fd R (E(R/p)) < ∞ for any p ∈ Spec (R).
(vi) C-id R (Tm ) = ht (m) for any m ∈ Max (R).
(vii) C-id R (Tm ) < ∞ for any m ∈ Max (R).
(viii) C-id R (Tp ) = ht (p) for any p ∈ Spec (R).
(ix) C-id R (Tp ) < ∞ for any p ∈ Spec (R).
Proof. (i)=⇒(ii). Assume that m ∈ Max (R). There are equalities
C-fd R (E(R/m)) = fd R (Hom R (C, E(R/m)))
= fd Rm (Hom Rm (Cm , ERm (Rm /mRm ))
= id Rm (Cm )
= dim (Rm )
= ht (m),
in which the first equality is from Theorem 2.9(ii), and the second one is from Remark 2.16.
(ii)=⇒(iii). Is clear.
(iii)=⇒(i). We can assume that (R, m) is local. Now one can use Theorem 2.9(ii), to see
that
id R (C) = fd R (Hom R (C, E(R/m)))
= C-fd R (E(R/m)) < ∞,
whence C is dualizing.
(i)=⇒(iv). Let p be a prime ideal of R. Note that E(R/p)q 6= 0 if and only if q ⊆ p. Now
as in (i)=⇒(ii), we have C-fd R (E(R/p)) = dim (Rp ) = ht (p).
(iv)=⇒(v). Is clear.
(v)=⇒(i). Again, we can assume that R is local. Now the proof is similar to that of
(iii)=⇒(i).


(ii)⇐⇒(vi) and (iii)⇐⇒(vii). Note that Tm = Hom R E(R/m), E(R/m)(X) for some set

X. Now we have the equalities

C-id R (Tm ) = id R (C ⊗R Tm )
= id R C ⊗R Hom R E(R/m), E(R/m)(X)



= id R Hom R Hom R (C, E(R/m)), E(R/m)(X)
= fd R (Hom R (C, E(R/m)))
= C-fd R (E(R/m)),



DUAL OF BASS NUMBERS AND DUALIZING MODULES

11

in which the first equality is from Theorem 2.9(i), the fourth equality is from Remark 2.16
and the fact that E(R/m)(X) is an injective cogenerator in the category of Rm -modules, and
the last one is from Theorem 2.9(ii).
(iv)⇐⇒(viii) and (v)⇐⇒(ix). Are similar to (ii)⇐⇒(vi).



The following theorem is a generalization of [21, theorem 2.1].
Theorem 4.2. The following are equivalent:
(i) C is pointwise dualizing.
(ii) An R-module M is C-flat if and only if µi (p, M ) = 0 for all p ∈ Spec (R) whenever
i 6= ht (p).
(iii) An R-module M is flat if and only if µi (p, C ⊗R M ) = 0 for all p ∈ Spec (R)
whenever i 6= ht (p).
Proof. (i)=⇒(ii). First assume that M is C-flat. Set M = C ⊗R F , where F is a flat
R-module. Since C is pointwise dualizing, we have µi (p, C) = 0 for all p ∈ Spec (R) with
i 6= ht (p). Assume that
0 → C → E 0 (C) → E 1 (C) → ... → E i (C) → ...
is the minimal injective resolution of C. By applying the exact functor − ⊗R F to this
resolution, we find an exact complex
0 → M = C ⊗R F → E 0 (C) ⊗R F → E 1 (C) ⊗R F → ... → E i (C) ⊗R F → ..., (∗)
which is an injective resolution for M . By [7, Theorem 3.3.12] the injective R-module
E(R/p) ⊗R F is a direct sum of copies of E(R/p) for each p ∈ Spec (R). Now, since the
minimal injective resolution of M is a direct summand of the complex (∗), we get the result.
Conversely, suppose that M is an R-module such that µi (p, M ) = 0 for all p ∈ Spec (R)
whenever i 6= ht (p). In order to show that M is C-flat, it is enough to prove that Mm is Cm flat Rm -module for all m ∈ Max (R). For if Mm is Cm -flat Rm -module for all m ∈ Max (R),
then Hom R (C, M )m ∼
= Hom Rm (Cm , Mm ) is flat as an Rm -module for all m ∈ Max (R) by
Theorem 2.9(ii). Hence Hom R (C, M ) is a flat R-module and thus M is C-flat by Theorem
2.9(ii). Hence, replacing R by Rm , we can assume that (R, m) is local. Clearly we may
assume that M 6= 0. In this case we have id R (M ) < ∞ since by assumption µi (p, M ) = 0
for all p ∈ Spec (R) and all i > dim (R) . Hence the assumption in conjunction with Lemma
4.1, imply that M has a bounded injective resolution all of whose terms have finite C-flat
dimensions. More precisely, by Lemma 4.1, if E i is the i-th term in the minimal injective
resolution of M , then C-fdR (E i ) = i for all 0 ≤ i ≤ id (M ). Breaking up this resolution to
short exact sequences and using [19, Corollary 5.7], we can conclude that C-fdR (M ) = 0.
Hence M is C-flat, as wanted.
(ii)=⇒(iii). Assume that M is a flat R-module. Then C ⊗R M ∈ FC and µi (p, C ⊗R
M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p) by assumption. Conversely, suppose that
µi (p, C ⊗R M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Then, by assumption, C ⊗R M
is C-flat. Set C ⊗R M = C ⊗R F , where F is flat. Therefore C ⊗R M ∈ BC (R), whence

12

M. RAHMANI AND A.- J. TAHERIZADEH

M ∈ AC (R) by [18, Theorem 2.8(b)]. Thus we have the isomorphisms
M

∼
= Hom R (C, C ⊗R M )
∼
= Hom R (C, C ⊗R F )
∼
= F,

where the first and the last isomorphism hold since both M and F are in AC (R).
(iii)=⇒(i). Note that R is a flat R-module. Hence by assumption, if m ∈ Max (R), then
i

µ (m, C ⊗R R) = 0 for all i > ht (m). Thus id Rm (Cm ) < ∞, as wanted.



Theorem 4.3. The following are equivalent:
(i) C is pointwise dualizing.
(ii) An R-module M is C-injective if and only if πi (p, M ) = 0 for all p ∈ Spec (R)
whenever i 6= ht (p).
(iii) An R-module M is injective if and only if πi (p, Hom R (C, M )) = 0 for all p ∈
Spec (R) whenever i 6= ht (p).
Proof. (i)=⇒(ii) . Assume that M is a nonzero C-injective R-module. Set M = Hom R (C, E)
with E is injective. First, we show that M is cotorsion. Assume that F is a flat R-module.
Then, by [7, Theorem 3.2.1], we have Ext 1 (F, Hom R (C, E)) ∼
= Hom R (Tor R (F, C), E) = 0,
R

1

and hence M is cotorsion. Fix a prime ideal p of R and set k(p) = Rp /pRp . Note
∼ ⊕ E(R/q), where
that Hom R (Rp , E) is an injective R-module and that Hom R (Rp , E) =
q∈X

X ⊆ Ass R (E) and each element of X is a subset of p. There are isomorphisms


R
R
Tor i p k(p), Hom R (Rp , Hom R (C, E)) ∼
= Tor i p k(p), Hom R (Cp , E)

R
∼
= Tor i p k(p), Hom Rp (Cp , Hom R (Rp , E)

R
∼
= Tor i p k(p), Hom Rp (Cp , ⊕ E(R/q)
q∈X

∼
= Hom Rp Ext iRp (k(p), Cp ), ⊕ E(R/q) ,
q∈X

where the last isomorphism is from [7, Theorem 3.2.13]. Now since Cp is dualizing for Rp ,

we have Ext iRp (k(p), Cp ) = 0 for all i 6= ht (p). Therefore πi (p, M ) = 0 for all i 6= ht (p).
Conversely, assume that M is a non-zero R-module with πi (p, M ) = 0 for all i 6= ht (p). By
assumption, the minimal flat resolution of M is of the form
Q

· · · −→ Fi −→ · · · −→ F1 −→ F0 −→ M −→ 0,

in which Fi =
Tp for all i ≥ 1. Also, in view of [22, Lemma 3.1], we have
ht (p)=i
Q
Tp . Hence the minimal flat resolution of M is of the form
F0 =
ht (p)=0
Q
Q
Q
· · · −→
Tp −→ · · · −→
Tp −→
Tp −→ M −→ 0. (∗)
ht (p)=i
ht (p)=1
ht (p)=0
Let E be an injective cogenerator. According to Lemma 2.10(i), it is enough to show
that Hom R (M, E) is C-flat.

In fact, by Theorem 4.2, we need only to show that

i

µ (p, Hom R (M, E)) = 0 for all i 6= ht (p) and all i ≥ 0. Applying the exact functor
Hom R (−, E) on (∗), we get an injective resolution 

Q
0 −→ Hom R (M, E) −→ Hom R
Tp , E −→
ht (p)=0

DUAL OF BASS NUMBERS AND DUALIZING MODULES

Hom R



Q



Tp , E −→ · · · −→ Hom R

ht (p)=1

for Hom R (M, E). Note that Hom R


Q





Q

Tp , E
ht (p)=i



Q

ht (p)=i

13


Tp , E −→ · · · ,

is an injective R-module for all i ≥ 0.

∼ ⊕E(R/q). We show that ht (q) = i. Since C is pointwise
Tp , E =
ht (p)=i
dualizing, by Lemma 4.1, we have C-fd R (E(R/q)) = ht (q). On the other hand, we have
Set Hom R

the equalities
C-fd R (E(R/q)) = C-fd R 
(⊕E(R/q))



Q
= C-fd R Hom R
Tp , E
ht(p)=i
 Q
= C-id R
Tp
ht (p)=i
= i,

in which the third equality is from Lemma 2.10(i), and the last one is from Lemma 4.1.
Hence µi (p, Hom R (M, E)) = 0 for all i ≥ 0 with i 6= ht (p), as wanted.
(ii)=⇒(iii). Assume that M is an injective R-module. Then Hom R (C, M ) ∈ IC and
i

µ (p, Hom R (C, M )) = 0 for all p ∈ Spec (R) whenever i 6= ht (p) by assumption. Conversely,
suppose that µi (p, Hom R (C, M )) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Then,
by assumption, Hom R (C, M ) is C-injective. Set Hom R (C, M ) = Hom R (C, I), where I is
injective. Therefore Hom R (C, M ) ∈ AC (R), whence M ∈ BC (R) by [18, Theorem 2.8(a)].
Thus we have the isomorphisms
M

∼
= C ⊗R Hom R (C, M )
∼
= C ⊗R Hom R (C, I)
∼
= I,

where the first and the last isomorphism hold since both M and I are in BC (R).
(iii)=⇒(i). Assume that m is a maximal ideal of R. Set k(m) = Rm /mRm . Since E(R/m)

is injective, by assumption, we have πi m, Hom R (C, E(R/m)) = 0 fo all i 6= ht (m). On the

other hand, there are isomorphisms


Hom Rm Ext iRm (k(m), Cm ), E(k(m))


m
∼
k(m), Hom Rm (Cm , E(k(m))
= Tor R
i

∼ Tor Rm k(m), Hom R (Cm ⊗R Rm , E(k(m))
=
m
m
i

m
∼
k(m), Hom R (Rm , Hom Rm (Cm , E(k(m))
= Tor R
i

∼
= Tor Rm k(m), Hom R (Rm , Hom R (C, E(R/m)) ,
i

where the first isomorphism is from [7, Theorem 3.2.13], and the last one is from Remark

2.16. From this isomorphisms, it follows that Hom Rm Ext iRm (k(m), Cm ), E(k(m)) = 0 for

all i 6= ht (m), from which we conclude that Ext iRm (k(m), Cm ) = 0 for all i 6= ht (m), since

E(k(m)) is an injective cogenerator in the category of Rm -modules. Thus Cm is dualizing
for Rm , as required.



Corollary 4.4. Let C be pointwise dualizing. Then flat cover of any C-injective R-module
is C-injective.

14

M. RAHMANI AND A.- J. TAHERIZADEH

Proof. By Lemma 4.1, C-id R (Tp ) = 0 for any prime ideal p with ht (p) = 0. Hence Tp is
C-injective for any prime ideal p with ht (p) = 0. Assume that M is a C-injective R-module.
Q
By Theorem 4.3, we have F (M ) =
Tp . Now the result follows since the class IC
ht (p)=0
closed under arbitrary direct product.

Corollary 4.5. The R-module C is pointwise dualizing if and only if for any prime ideal p
of R,

πi p, Hom R (C, E(R/p)) =

(

1

i = ht (p),

0

i 6= ht (p).

Proof. Assume that p ∈ Spec (R). Set k(p) = Rp /pRp . We have the following equalities


R
πi p, Hom R (C, E(R/p)) = vdim k(p) Tor i p k(p), Hom R (Rp , Hom R (C, E(R/p))

R
= vdim k(p) Tor i p k(p), Hom R (Cp , Hom Rp (Rp , E(R/p))

R
= vdim k(p) Tor i p k(p), Hom R (Cp , E(R/p))

= vdim k(p) Hom Rp Ext iRp (k(p), Cp ), E(R/p) ,

where the second equality is from Remark 2.16, and the last equality is from [7, Theorem
3.2.13]. Now, C is pointwise dualizing if and only if Cp is the dualizing module of Rp for all
p ∈ Spec (R), and this is the case if and only if
(
k(p)
Ext iRp (k(p), Cp ) ∼
=
0

i = ht (p),
i 6= ht (p).

for all p ∈ Spec (R).

Thus we are done by the above equalities and the fact that
∼

Hom Rp (k(p), E(R/p)) = k(p).
In the following corollaries, we are concerned with the local cohomology. For an R-

module M , the i-th local cohomology module of M with respect to an ideal a of R, denoted
by Hia (M ), is defined to be
Hia (M ) = limExt iR (R/an , M ).
−→

n≥1

For the basic properties of local cohomology modules, please see the textbook [3].
Corollary 4.6. Let (R, m) be a Cohen-Macaulay local ring with dim (R) = d possesing


a canonical module ωR . Then πi m, Hdm (R) = δi,d , and πi q, Hdm (R) = 0 for any nonmaximal prime ideal q whenever i 6= ht (q).

Proof. By [3, Theorem 11.2.8], we have Hdm (R) ∼
= Hom R (ωR , E(R/m)), and hence Hdm (R) is
ωR -injective. Assume that q is a non-maximal prime ideal of R. Then by the Theorem 4.3, we


have πi q, Hdm (R) = 0 for all i 6= ht (q). Finally, by corollary 4.5, we have πi m, Hdm (R) = 0


for all i 6= d and that πd m, Hdm (R) = 1, as wanted.
If (R, m) is a Cohen-Macaulay local ring with dim (R) = d, then by [3, Corollary 6.2.9] the

only non-vanishing local cohomology of R with respect to m is Hdm (R). Also, if R admits a

DUAL OF BASS NUMBERS AND DUALIZING MODULES

15

canonical module, then by [7, Proposition 9.5.22], we have fd R (Hdm (R)) = d. The following
corollary describes the structure of the minimal flat resolution of Hdm (R).
Corollary 4.7. Let (R, m) be a d-dimensional Cohen-Macaulay local ring possessing a
canonical module. The minimal flat resolution of Hdm (R) is of the form
cm −→ · · · −→ Q Tp −→ Q Tp −→ Hdm (R) −→ 0.
0 −→ R
ht (p)=1

ht (p)=0

In the following corollary, we give another proof of [16, Corollary 3.7]. Our approach is

direct, and uses the well-known fact that the homology functor Tor can be computed by a
flat resolution.
Corollary 4.8. Let (R, m) be a d-dimensional Cohen-Macaulay local ring. Then
(
Hdm (R)
i = d,
d
R
d
∼
Tor i (Hm (R), Hm (R)) =
0
i 6= d.

b is a d-dimensional complete Cohen-Macaulay local ring, and hence
Proof. Note that R

admits a canonical module ωRb . The R-module Hdm (R) is Artinian by [3, Theorem 7.1.6], and
b
thus naturally has a R-module
structure by [3, Remark 10.2.9]. Hence Tor R (Hd (R), Hd (R))
i

m

m

is Artinian for all i ≥ 0 by [14, Corollary 3.2]. Thus there are isomorphisms
d
d
Tor R
i (Hm (R), Hm (R))

d
d
∼
b
= Tor R
i (Hm (R), Hm (R)) ⊗R R
b
d
∼
b
b d
= Tor R
i (Hm (R) ⊗R R, Hm (R) ⊗R R)

b
d
d
∼
b ,
b H b (R)
= Tor R H b (R),
i

mR

mR

in which the second isomorphism is from [7, Theorem 2.1.11], and the last one is flat base
change [3, Theorem 4.3.2]. Also, we have the isomorphisms
b
Hdm (R) ∼
= Hdm (R) ⊗R R
d
∼
b
= H b (R)

mR

∼
b R)
b ,
= Hom Rb ωRb , ERb (R/m

in which the first isomorphism holds because Hdm (R) is Artinian, the second isomorphism is
the flat base change, and the last one is local duality [3, Theorem 11.2.8]. Thus Hdm (R) is
b
a ωRb -injective R-module.
Hence, by Corollary 4.7, the minimal flat resolution of Hdm (R), as
b
an R-module,
is of the form
[
b b −→ · · · −→ Q TQ −→ Q TQ −→ Hdm (R) −→ 0,
0 −→ R
mR
ht (Q)=1
ht (Q)=0
b
bQ -adic topology,
in which TQ is the completion of a free RQ -module with respect to QR
b Observe that the above resolution is a flat resolution of Hdm (R) as an
for Q ∈ Spec (R).

R-module since the modules in the above resolution are all flat R-modules. Therefore, we
b and assume that R is complete. So that, the minimal flat resolution of
can replace R by R,
Hdm (R) is of the form
cm −→ · · · −→
0 −→ R

Q

Tp −→

ht (p)=1

Q

Tp −→ Hdm (R) −→ 0,
ht (p)=0

16

M. RAHMANI AND A.- J. TAHERIZADEH

in which Tp is the completion of a free Rp -module with respect to pRp -adic topology, for p 
∈ Spec(R). Next, note that for each prime ideal p with p 6= m, we have
Q
d
Tp = 0. Indeed, we can write Hdm (R) = limMα , where Mα is a finitely
Hm (R) ⊗R
−→

generated submodule of

Hdm (R).

α∈I

Also Tp = Hom R E(R/p), E(R/p)(X) for some set X.

Now since Mα is of finite length by [3, Theorem 7.1.3], we can take an element x ∈ m r q

such that xMα = 0. But multiplication of x induces an automorphism
on E(R/p) and hence
Q 
Q
Tp is both an isomorphism and
on Tp . Consequently, multiplication of x on Mα ⊗R
Q 
Q 
Tp = 0 since
Tp = 0, from which we conclude that Hdm (R) ⊗R
zero. Hence Mα ⊗R
d
d
tensor commutes with direct limit. Thus Tor R
i (Hm (R), Hm (R)) = 0 for i 6= d. Finally, we

have
d
d
Tor R
d (Hm (R), Hm (R))

∼
cm ⊗R Hd (R)
=R
m
∼
cm )
(R
= HdmR
d
m
∼
c
c
= Hom d (ωd
Rm , E d (Rm /mRm ))
Rm

Rm

∼
cm
= Hom Rm (ωRm , ERm (Rm /mRm )) ⊗Rm R
∼ Hom R (ωR , ER (Rm /mRm ))
=
m
m
m
∼
= Hom R (ωR , E(R/m)) ⊗R Rm
∼
= Hom R (ωR , E(R/m) ⊗R Rm )
∼
= Hom R (ωR , E(R/m))
∼
= Hd (R),
m

in which the second isomorphism is the flat base change [3, Theorem 4.3.2], the third isomorphism is local duality [3, Theorem 11.2.8], and the fifth one is from [3, Remark 10.2.9], since
Hom Rm (ωRm , ERm (Rm /mRm )) is an Artinian Rm -module and hence has a natural structure
cm -module.
as an R

The following theorem is a slight generalization of [22, Theorem 3.3].

Theorem 4.9. The following are equivalent:
(i) C is pointwise dualizing.
(ii) If M is a cotorsion R-module such that C-id R (M ) = n < ∞, then M admits
a minimal flat resolution such that πi (p, M ) = 0 for all p ∈ Spec (R) whenever
ht (p) ∈
/ {i, ..., i + n}.
Proof. (i) =⇒ (ii). We use induction on n. If n = 0, then we are done by Theorem 4.3.
Now assume inductively that n > 0 and the case n ie settled. Fix a prime ideal p of R.
Assume that M is a cotorsion R-module with C-id R (M ) = n + 1. Hence M ∈ AC (R), and
so the IC -preenvelope of M is injective by [18, Corollary 2.4(b)]. Thus there exists an exact
sequence
0 → M → Hom R (C, I) → L → 0, (∗)
in which I is injective, and L = Coker (M → Hom R (C, I)). Note that L is cotorsion
since both M and Hom R (C, I) are cotorsion. Also, since both M and Hom R (C, I) are
in AC (R), we have L ∈ AC (R), and therefore Tor R
1 (C, L) = 0.

On the other hand

DUAL OF BASS NUMBERS AND DUALIZING MODULES

17

C ⊗R Hom R (C, I) ∼
= I, by [7, Theorem 3.2.11]. Hence application of C ⊗R − on (∗) yields
an exact sequence
0 → C ⊗R M → I → C ⊗R L → 0.
By Theorem 2.9(i), we have id R (C ⊗R M ) = n + 1. Therefore id R (C ⊗R L) = n, whence
C-id R (L) = n.

Now induction hypothesis applied to Hom R (C, I) and L yields that

/ {i, ..., i + n}.
πi p, Hom R (C, I) = 0 for all i 6= ht (p), and that πi (p, L) = 0 fo all ht (p) ∈

Note that Ext 1R (Rp , M ) = 0 since M is cotorsion. Hence the exact sequence (∗) yields an

exact sequence
0 → Hom R (Rp , M ) → Hom R (Rp , Hom R (C, I)) → Hom R (Rp , L) → 0,
and the later exact sequence, by applying k(p) ⊗Rp −, yields the long exact sequence


Rp
Rp
k(p), Hom R (Rp , L) →
k(p), Hom R (Rp , Hom R (C, E)) → Tor i+1
· · · → Tor i+1


R
R
Tor i p k(p), Hom R (Rp , M ) → Tor i p k(p), Hom R (Rp , Hom R (C, E)) → · · · .

R
From the above long exact sequence, it follows that Tor i p k(p), Hom R (Rp , M ) = 0 for all
ht (p) ∈
/ {i, ..., i + n + 1}, as wanted. This completes the inductive step.

(ii) =⇒ (i). Let m be a maximal ideal of R. Now Hom R (C, E(R/m)) is C-injective and

hence by assumption πi m, Hom R (C, E(R/m)) = 0 for all i 6= ht (m). Now by the same

argument as in the proof of Theorem 4.3, we have Ext iRm (k(m), Cm ) = 0 for all i 6= ht (m),
whence Cm is dualizing for Rm .



Corollary 4.10. The following statements hold true:
(i) If C is pointwise dualizing, then C-id R (F (M )) ≤ C-id R (M ) for any cotorsion Rmodule M .
(ii) If C-id R (F (M )) ≤ C-id R (M ) for any R-module M , then C is pointwise dualizing.
Proof. (i). Assume that M is a cotorsion R-module. If C-id R (M ) = ∞, then we are done.
Q
So assume that C-id R (M ) = n < ∞. Then by Theorem 4.9, we have F (M ) = Tp where

0 ≤ ht (p) ≤ n. Now the result follows by Lemma 4.1.

(ii). Assume that m is a maximal ideal of R. We have to show that Cm is dualizing
for Rm . Assume that x is a maximal R-sequence in m. Then fd R (R/xR) < ∞, and
Ass R (C/xC) = {m} since x is also a maximal C-sequence. Hence we have the equalities
C-fd (C/xC)

= fd R (Hom R (C, C/xC))
= fd R (Hom R (C, C ⊗R R/xR))
= fd R (R/xR)
< ∞,

in which the first equality is from Theorem 2.9(ii), and the third one holds because R/xR ∈
AC (R). Assume that E is an injective cogenerator. Set (−)∨ = Hom R (−, E). Then Cid R ((C/xC)∨ ) < ∞ by Lemma 2.10(ii). Now if F is the flat cover of (C/xC)∨ , then by
assumption, we have C-id R (F ) < ∞. Therefore, we have C-fd R (F ∨ ) < ∞ by Lemma
2.10(i). Next, note that we have
C/xC ֒→ (C/xC)∨∨ ֒→ F ∨ .
Hence, the injective envelope of C/xC is a direct summand of F ∨ . Thus, in fact, E(R/m)

18

M. RAHMANI AND A.- J. TAHERIZADEH

is a direct summand of F ∨ , since R/m ֒→ C/xC. It follows that C-fd R (E(R/m)) < ∞, and
hence we are done by Lemma 4.1, since m was arbitrary.



Acknowledgments. We thank the referee for very careful reading of the manuscript
and also for his/her useful suggestions.

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DUAL OF BASS NUMBERS AND DUALIZING MODULES

Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.
E-mail address: m.rahmani.math@gmail.com
E-mail address: taheri@khu.ac.ir

19

