arXiv:1510.01845v2 [] 7 Apr 2016

CHARACTERIZING LOCAL RINGS VIA PERFECT AND COPERFECT
MODULES
MOHAMMAD RAHMANI AND ABDOLJAVAD TAHERIZADEH
Abstract. Let R be a Noetherian ring and let C be a semidualizing R-module. In this
paper, by using the classes PC and IC , we extend the notions of perfect and coperfect
modules introduced by D.Rees [14] and O.Jenda [12]. First, we study the basic properties
of these modules and relations between them. Next, we characterize local rings in terms
of the existence of special perfect (resp. coperfect) modules.

1. introduction
Throughout this paper, R is a commutative Noetherian ring with non-zero identity. Recall that the grade of M , defined by D. Rees [14], is the least integer i ≥ 0 such that
Ext iR (M, R) 6= 0. He showed that if M is finitely generated, then grade R (M ) is the length
of a maximal R-sequence contained in Ann R (M ). An R-module M is called perfect (resp.
n-perfect) if one has grade R (M ) = pd R (M ) (resp. grade R (M ) = n = pd R (M )) where n
is a non-negative integer. He, also, showed that grade R (M ) is the least integer i ≥ 0 such
that Ext iR (M, P ) 6= 0 for some projective R-module P . As a dual notion of grade, O. Jenda
[12] introduced an invariant, namely E-dimension. For an R-module M , E-dimension of
M , denoted by E-dim R (M ), is the least integer i ≥ 0 such that Ext iR (I, M ) 6= 0 for some
injective R-module I. Also, M is called a coperfect (resp. n-coperfect) R-module if one
has E-dim R (M ) = id R (M ) (resp. E-dim R (M ) = n = id R (M )). 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 [6], Vasconcelos [17] and Golod [8]. For a semidualizing R-module C, the class
of C-projectives (resp. C-injectives), denoted PC (resp. IC ), consists of those R-modules
of the form C ⊗R P (resp. Hom R (C, I)) for some projective (resp. injective) R-module P
(resp. I). In [11], H. Holm and D. White showed that for an R-module M , there exists a
projective resolution (resp. injective coresolution) by modules from PC (resp. IC ). Despite
the fact that these (co)resolutions may not be exact, they still have good lifting properties.
By using these (co)resolutions, R. Takahashi and D. White [16] defined two new homological
dimensions, namely C-pd and C-id . Recently, B. Kubik [13], introduced the dual notion
of semidualizing modules, namely quasidualizing modules. Over a local ring (R, m), an Arb −→ Hom R (T, T ) is
tinian R-module T is quasidualizing if the natural homothety map R

an isomorphism and Ext iR (T, T ) = 0 for all i > 0. In section 2, for an R-module M , we
2010 Mathematics Subject Classification. 13C05, 13H10, 13D07, 13D05.
Key words and phrases. Semidualizing modules, quasidualizing modules, dualizing modules, perfect modules, coperfect modules.
1

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M. RAHMANI AND A.- J. TAHERIZADEH

introduce two invariant for modules, namely PC -grade R (M ) and IC -grade R (M ) as generalization of the classical invariants grade R (M ) and E-dim R (M ), respectively. We study the
basic properties of these new invariants and the relations of them with the relative homological dimensions C-pd R (M ) and C-id R (M ). For instance, the following result, proved in
Theorem 3.11, is a generalization of [12, Theorem 3.1].
Theorem A. Let M be a finitely generated R-module and let I be an injective R-module
for which Hom R (M, I) 6= 0. Then
PC -grade R (M ) ≤ IC -grade R (Hom R (M, I)) ≤ C-pd R (M ).
Next, we define the notions of C-perfect (resp. n-C-perfect) and C-coperfect (resp. nC-coperfect) modules, and use them to impose some conditions on C to be dualizing. The
next result is Theorem 3.19.
Theorem B. Let (R, m) be a local ring with dim (R) = n an let C be a semidualizing
R-module. The following are equivalent:
(i) C is dualizing.
(ii) E(R/m) is n-C-perfect.
b is n-C-coperfect.
(iii) R

We, also, use n-perfect (resp. n-coperfect) modules to characterize local rings. We find
some special n-perfect (resp. n-coperfect) modules whose existence imply the Regularity
(resp. Gorensteinness) of the ring. For instance, the following result is Theorem 3.20.
Theorem C. Let (R, m) be a local ring with dim (R) = n. The following are equivalent:
(i) R is Gorenstein.
(ii) any quasidualizing R-module is n-perfect.
(iii) there exists an n-perfect quasidualizing R-module with 1-dimensional socle.
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:

CHARACTERIZING LOCAL RINGS

3

(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. An Rmodule D is dualizing if it is semidualizing and that id R (D) < ∞. Moreover, D is pointwise
dualizing if Dm is dualizing Rm -module for any m ∈ Max (R). For example the canonical
module of a Cohen-Macaulay local ring, if exists, is dualizing.
Assume that (R, m) is local. Following [13], an Artinian R-module T is called quasidualizing if it satisfies the following conditions:
b −→ Hom R (T, T ) is an isomorphism.
(i) The natural homothety map R
(ii) Ext iR (T, T ) = 0 for all i > 0.

For example E(R/m) is a quasidualizing R-module.
Definition 2.3. Following [10], let C be a semidualizing R-module. We set
PC (R) = the subcategory of R–modules C ⊗R P where P is a projective R-module.
IC (R) = the subcategory of R–modules Hom R (C, I) where I is an injective Rmodule.
The R-modules in PC (R) and IC (R) are called C-projective and C-injective, respectively.
If C = R, then it recovers the classes of projective and injective modules, respectively. We
use the notations C-pd and C-id instead of PC -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) 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. 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

(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 [16, 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 [11,
Corollary 6.3]).

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M. RAHMANI AND A.- J. TAHERIZADEH

Definition 2.6. Let M be a non-zero R-module. Following [14], the grade of M , denoted
by grade R (M ), is defined to be
grade R (M ) = inf{i ≥ 0 | Ext iR (M, R) 6= 0}.
Also, M is called a perfect R-module precisely when grade R (M ) = pd R (M ). Following
[12], the E-dimension of M , denoted by E-dim R (M ), is defined to be
E-dim R (M ) = inf{i ≥ 0 | Ext iR (E, M ) 6= 0 for some injective R-module E }.
Also, M is called a coperfect R-module precisely when E-dim R (M ) = id R (M ).
Theorem 2.7. 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-pd R (M ) = pd R (Hom R (C, M )) and pd R (M ) = C-pd R (C ⊗R M ).
Proof. See [16, Theorem 2.11].



Lemma 2.8. Let C be a semidualizing R-module and let M be a finitely generated R-module.
Let E be an injective cogenerator and F be a faithfully flat R-module.
(i) One has C-pd R (M ) = C-id R (Hom R (M, E)).
(ii) One has C-id R (M ) = C-id R (M ⊗R F ).
Proof. (i). We have the following equalities
C-id R (Hom R (M, E))

= id R (C ⊗R Hom R (M, E))
= id R (Hom R (Hom R (C, M ), E))
= fd R ((Hom R (C, M ))
= pd R (Hom R (C, M ))
= C-pd R (M ),

in which the first equality is from Theorem 2.7(i), the second equality is from [5, Theorem
3.2.11], the fourth equality holds since Hom R (C, M ) is finitely generated, and the last one
is from Theorem 2.7(ii).
(ii). One has
C-id R (M ) = id R (C ⊗R M )
= id R (C ⊗R M ) ⊗R F




= id R C ⊗R (M ⊗R F )
= C-id R (M ⊗R F ),

in which the first and the last equalities are from Theorem 2.7(i).



3. main results
Throughout this section, C is a semidualizing R-module. In the rest of the paper, we
use the notations P-grade R (M ) and I-grade R (M ) instead of grade R (M ) and E-dim R (M ),
respectively.
Definition 3.1. Let M be an R-module. The PC -grade of M , denoted by PC -gradeR (M ),
is defined to be
PC -gradeR(M ) = inf{i ≥ 0 | Ext iR (M, C ⊗R P ) 6= 0 for some projective P }.

CHARACTERIZING LOCAL RINGS

5

Also, the IC -grade of M , denoted by IC -gradeR (M ), is defined to be
IC -gradeR (M ) = inf{i ≥ 0 | Ext iR (Hom R (C, E), M ) 6= 0 for some injective E}.
Remark 3.2. One should note that if C = R, then the above definition recovers the
notions of [14] and [12], respectively. It is clear that PC -grade R (M ) ≤ pd R (M ) and IC grade R (M ) ≤ id R (M ). Also, note that if M is finitely generated, then
PC -grade R (M ) = inf{i ≥ 0 | Ext iR (M, C) 6= 0}.
For if PC -grade R (M ) = n and Ext nR (M, C ⊗R P ) 6= 0 for some projective R-module P , then
Ext n (M, C)⊗R P ∼
= Ext n (M, C ⊗R P ) 6= 0 by [5, Theorem 3.2.15], and so Ext n (M, C) 6= 0.
R

R

R

Therefore, we have
PC -grade R (M ) = inf{i ≥ 0 | Ext iR (M, C) 6= 0}
= inf{depth Rp (Cp ) | p ∈ Supp R (M )}
= inf{depth (Rp ) | p ∈ Supp R (M )}
= P-grade R (M ).
Lemma 3.3. Assume that M is an R-module. The following statements hold true.
(i) One has PC -gradeR (M ) ≤ C-pd R (M ).
(ii) One has IC -gradeR (M ) ≤ C-id R (M ).
Proof. (i). We may assume that C-pd R (M ) = n < ∞, since otherwise we have nothing to
prove. In this case, we have M ∈ BC (R). By Theorem 2.7(ii), we have pd R (Hom R (C, M )) =
n. For any projective R-module P , we have the isomorphisms
Ext iR (M, C ⊗R P ) ∼
= Ext iR (C ⊗R Hom R (C, M ), C ⊗R P )
∼
= Ext i (Hom R (C, M ), P )
IC

∼
= Ext iR (Hom R (C, M ), P ),
in which the first isomorphism holds since M ∈ BC (R), the second isomorphism is from
[16, Theorem 4.1], and the third isomorphism is from [16, Corollary 4.2(b)].
Ext iR (M, PC )

Hence

= 0 for all i > n, whence PC -grade R (M ) ≤ n.

(ii). Is similar to the part (i).



Definition 3.4. Let n be a non-negative integer. An R-module M is said to be C-perfect
(resp. n-C-perfect ) if PC -grade R (M ) = C-pd R (M ) (resp. PC -grade R (M ) = C-pd R (M ) =
n). Also, M is said to be C-coperfect (resp. n-C-coperfect ) if IC -grade R (M ) = C-id R (M )
(resp. IC -grade R (M ) = C-id R (M ) = n).
Note that if C = R, then the above definition recovers the notions of [14] and [12],
respectively.
Proposition 3.5. Let M be an R-module.
(i) If M ∈ BC (R), then PC -gradeR (M ) = P-grade R (Hom R (C, M )).
(ii) If M ∈ AC (R), then IC -gradeR (M ) = I-grade R (C ⊗R M ).

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M. RAHMANI AND A.- J. TAHERIZADEH

Proof. (i). For any projective R-module P , we have the isomorphisms
Ext iR (M, C ⊗R P ) ∼
= Ext iR (C ⊗R Hom R (C, M ), C ⊗R P )
∼ Ext i (Hom R (C, M ), P )
=
IC

∼
= Ext iR (Hom R (C, M ), P ),
in which the first isomorphism holds since M ∈ BC (R), the second isomorphism is from [16,
Theorem 4.1], and the third isomorphism is from [16, Corollary 4.2(b)]. Hence the result
follows.
(ii). Is similar to the part (i).



Lemma 3.6. Let M be an R-module with C-id R (M ) < ∞ and let n be a non-negative
integer. The following are equivalent:
(i) C-id R (M ) ≤ n.
(ii) Ext iR (IC , M ) = 0 for all i > n.
Proof. (i)=⇒(ii). By Theorem 2.7(i), we have id R (C ⊗R M ) ≤ n. Also, by [16, Theorem
2.8(b)], C ⊗R M ∈ BC (R) since M ∈ AC (R). Now, for any injective R-module I, we have
the isomorphisms
Ext iR (Hom R (C, I), M )

∼
= Ext iR (Hom R (C, I), Hom R (C, C ⊗R M )
∼
= Ext iPC (I, C ⊗R M )
∼
= Ext i (I, C ⊗R M ),
R

in which the first isomorphism holds because M ∈ AC (R), the second isomorphism is from
[16, Theorem 4.1], and the last one is from [16, Corollary 4.2(a)] Now, the result follows.
(ii)=⇒(i). Assume, on the contrary, that C-id R (M ) = t > n. By [16, Corollary 2.4(b)],
there is an exact sequence
0 → M → Hom R (C, I 0 ) → Hom R (C, I 1 ) → · · · → Hom R (C, I t ) → 0,
i

in which I is an injective R-module. Let K i be the image of the i-th boundary map of the
above coresolution. Note that Ext jR (IC , IC ) = 0 for all j ≥ 1, by [5, Theorem 3.2.1] since
Tor R
j (C, IC ) = 0 for all j ≥ 1. Therefore a dimension shifting argument yields
Ext 1 (IC , K t−1 ) ∼
= Ext t (IC , M ) = 0.
R

Thus, in particular, the exact sequence 0 → K
splits. Consequently, K

t−1

R
t−1

→ Hom R (C, I t−1 ) → Hom R (C, I t ) → 0

is C-injective and then C-id R (M ) = t − 1, a contradiction.



Lemma 3.7. Let M be an R-module with C-pd R (M ) < ∞ and let n be a non-negative
integer. The following are equivalent:
(i) C-pd R (M ) ≤ n
(ii) Ext iR (M, PC ) = 0 for all i > n.
Proof. Is dual to the proof of Lemma 3.6.



Proposition 3.8. Let M be an R-module and let n be a non-negative integer. The following
statements hold true.
(i) Assume that C-pd R (M ) < ∞. Then M is n-C-perfect if and only if Ext iR (M, PC ) = 0
for all i 6= n and that Ext nR (M, C ⊗R P ) 6= 0 for some projective R-module P .

CHARACTERIZING LOCAL RINGS

7

(ii) Assume that C-id R (M ) < ∞. Then M is n-C-coperfect if and only if Ext iR (IC , M ) = 0
for all i 6= n and that Ext nR (Hom R (C, I), M ) 6= 0 for some injective R-module I.
Proof. (i). Assume that M is n-C-perfect. Then we have PC -gradeR(M ) = C-pd (M ) = n,
by definition. Thus Ext iR (M, PC ) = 0 for all i < n. On the other hand, by Lemma 3.7, we
have Ext iR (M, PC ) = 0 for all i > n. The remaining part follows from definition. For the
converse, note that by assumption, we have PC -gradeR (M ) = n. Now, since M ∈ BC (R),
in view of Lemma 3.3(i) and Lemma 3.7, we have
n = PC -gradeR (M ) ≤ C-pd R (M ) ≤ n,
whence M is n-C-perfect.
(ii). Is dual of part (i).



Theorem 3.9. Let M and N be R-modules. The following statements hold true.
(i) Assume that IC -gradeR (M ) = n and C-id R (N ) = m < n. Then Ext iR (N, M ) = 0
for all i < n − m.
(ii) Assume that PC -gradeR (N ) = n and C-pd R (M ) = m < n. Then Ext iR (N, M ) = 0
for all i < n − m.
Proof. (i). Induct on C-id R (N ). The case C-id R (N ) = 0 is just definition. Assume,
inductively, that C-id R (N ) = m. Then, by [16, Corollary 2.4(b)], there exists an exact
sequence
0 → N → L → K → 0,
such that L ∈ IC and K = Coker (N → L). since both N and L are in AC (R), we have
K ∈ AC (R), and therefore Tor R
1 (C, K) = 0. On the other hand C ⊗R L ∈ I, by [5, Theorem
3.2.11]. Hence application of C ⊗R − on above exact sequence yields an exact sequence
0 → C ⊗R N → C ⊗R L → C ⊗R K → 0.
By Theorem 2.7(i), we have id R (C ⊗R N ) = m. Therefore id R (C ⊗R K) = m − 1, whence
C-id R (K) = m−1. Now, induction hypothesis applied to L and K yields Ext i<n
R (L, M ) = 0
i<m−(n−1)

and Ext R

(K, M ) = 0, respectively. Therefore, the long exact sequence

· · · → Ext iR (L, M ) → Ext iR (N, M ) → Ext i+1
R (K, M ) → · · ·
completes the inductive step.
(ii). Is similar to (i).



Corollary 3.10. Let a and b be two ideals of R. The following statements hold true.
(i) If IC -gradeR (R/a) > C-id R (R/b), then b : a = b.
(ii) If PC -gradeR(R/a) > C-pd R (R/b), then b : a = b.
(iii) One has IC -gradeR(R/a) ≤ C-id R (R/p), for all p ∈ ass R (a).
(iv) One has PC -gradeR (R/a) ≤ C-pd R (R/p), for all p ∈ ass R (a).
Proof. For (i) and (iii) use Theorem 3.9(i), and for (ii) and (iv) use Theorem 3.9(ii).



Theorem 3.11. Let M be a finitely generated R-module and let I be an injective R-module
for which Hom R (M, I) 6= 0. Then
PC -gradeR (M ) ≤ IC -gradeR(Hom R (M, I)) ≤ C-pd R (M ).

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M. RAHMANI AND A.- J. TAHERIZADEH

Proof. By Remark 3.2, we have PC -gradeR (M ) = inf{i ≥ 0 | Ext iR (M, C) 6= 0}. Now,
if Ext iR (M, C) = 0, then for any injective R-module E, we have the isomorphism
∼ Hom R (Ext i (M, C), E) = 0, by [5, Theorem 3.2.13]. Hence,
Tor R (Hom R (C, E), M ) =
i

R

we have
Ext iR (Hom R (C, E), Hom R (M, I)) ∼
= Hom R (Tor R
i (Hom R (C, E), M ), I) = 0.
Therefore the inequality PC -gradeR (M ) ≤ IC -gradeR (Hom R (M, I)) follows.

Next, if

C-pd R (M ) = ∞, then we are done. So, we can assume that C-pd R (M ) < ∞. Thus
M ∈ BC (R), and then Hom R (M, I) ∈ AC (R) by [11, Proposition 7.2(b)]. Therefore, we
have the following (in)equalities
IC -grade R (Hom R (M, I))

≤ C-id (Hom R (M, I))
= id R (C ⊗R Hom R (M, I))
= id R (Hom R (Hom R (C, M ), I))
≤ fd (Hom R (C, M ))
= pd (Hom R (C, M )),
= C-pd (M ),

in which the first inequality is from Lemma 3.3(ii), the second equality is from [5, Theorem
3.2.11], the third equality holds because Hom R (C, M ) is finitely generated, and the other
equalities are from Theorem 2.7. This completes the proof.



Corollary 3.12. Let M be a finitely generated R-module and let E be an injective cogenerator. Then
PC -gradeR (M ) = IC -gradeR(Hom R (M, E)).
Proof. in view of Theorem 3.11, we need only to prove the inequality ’≥’. Assume that i
is a non-negative integer for which Ext iR (Hom R (C, I), Hom R (M, E)) = 0 for any injective
R-module I. Thus, in particular, Ext iR (Hom R (C, E), Hom R (M, E)) = 0. Set (−)∨ =
Hom R (−, E). Then we have
Ext iR (Hom R (C, E), Hom R (M, E))

= Ext iR (C ∨ , M ∨ )
∼
= Tor R (C ∨ , M )∨
i

∼
= Ext iR (M, C)∨∨ ,
in which the first isomorphism is from [5, Theorem 3.2.1], and the second isomorphism is
from [5, Theorem 3.2.13]. Now, since Ext iR (M, C) ֒→ Ext iR (M, C)∨∨ , the result follows. 
Corollary 3.13. Let (R, m) be local with dim (R) = n. The following are equivalent:
(i) R is Cohen-Macaulay.
(ii) One has PC -grade R (R/m) = n.
(iii) One has IC -grade R (R/m) = n.
Proof. (i)=⇒(ii). In this case, by Proposition 2.4(iv), we have depth R (C) = n. Hence,
according to Remark 3.2, we have
PC -grade R (R/m) = inf{i ≥ 0 | Ext iR (R/m, C) 6= 0} = n.

CHARACTERIZING LOCAL RINGS

9

(ii)=⇒(iii). Use Corollary 3.12, the isomorphism Hom R (R/m, E(R/m)) ∼
= R/m, and the
fact that E(R/m) is an injective cogenerator.
(iii)=⇒(i). Set (−)∨ = Hom R (−, E(R/m)). By assumption, Ext iR (C ∨ , R/m) = 0 for all
0 ≤ i < n. There are isomorphisms
Ext iR (C ∨ , R/m) ∼
= Ext iR (C ∨ , (R/m)∨ )
∼
= Tor R (C ∨ , R/m)∨
i

∼
= Ext iR (R/m, C)∨∨ ,
where the second isomorphism is from [5, Theorem 3.2.1], and the third isomorphism is from
[5, Theorem 3.2.13]. Hence Ext iR (R/m, C) = 0 for all 0 ≤ i < n, from which we conclude
that
n = IC -grade R (R/m) ≤ depth R (C) = depth (R) ≤ n.
Therefore R is Cohen-Macaulay.



Theorem 3.14. Let E be an injective cogenerator. The following are equivalent:
(i) C is pointwise dualizing.
(ii) One has PC -grade R (Ext iR (M, C)) ≥ i for all finitely generated R-modules M and
all i ≥ 0.
(iii) One has IC -grade R (Tor iR (M, Hom R (C, E))) ≥ i for all finitely generated Rmodules M and all i ≥ 0.
Proof. (i)=⇒(ii). In this case, R is Cohen-Macaulay. For any p ∈ Spec (R) with ht (p) < i,
∼ Ext i (Mp , Cp ) = 0, since id R (Cp ) = ht (p). Hence, if we set
we have Ext iR (M, C)p =
p
Rp
a = Ann R (Ext iR (M, C)), then we have
grade (a, C) = ht C (a) = ht (a) ≥ i.
Now, since

Ext iR (M, C)

is finitely generated, we have

PC -grade R (Ext iR (M, C))

= inf{j ≥ 0 | Ext jR (Ext iR (M, C), C) 6= 0}
= inf{j ≥ 0 | Ext jR (R/a, C) 6= 0}
≥ i.

(ii)⇐⇒(iii). By using Corollary 3.12 and the isomorphism [5, Theorem 3.2.13], we have

PC -grade R (Ext iR (M, C)) = IC -grade R Hom R (Ext iR (M, C), E

= IC -grade R Tor R
i (M, Hom R (C, E) .
(ii)=⇒(i). Assume that m is a maximal ideal of R. For any finitely generated R-module
M and any i 6= 0, we set aiM = Ann R (Ext iR (M, C)). Note that grade Rm (aiM Rm , Cm ) < ∞
since Rm is Noetherian. On the other hand, by assumption, we have
inf{j ≥ 0 | Ext jR (R/aiM , C) 6= 0} ≥ i,
for any finitely generated R-module M and for any i ≥ 0. Hence we have
inf{j ≥ 0 | Ext jRm (Rm /aiM Rm , Cm ) 6= 0} ≥ i,
for any finitely generated R-module M and for any i ≥ 0. It follows that aiM Rm = Rm for
∼ Ext i (M, C)m = 0 for all i > ht (m). Therefore,
all i > ht (m), whence Ext iRm (Mm , Cm ) =
R
since M was arbitrary, we get id Rm (Cm ) < ∞, as wanted.



10

M. RAHMANI AND A.- J. TAHERIZADEH

Assume that (R, m) is local. Recall that the depth of a (not necessarily finitely generated)
R-module M is
depth R (M ) = inf{i ≥ 0 | Ext iR (R/m, M ) 6= 0}.
Also the width of M is defined to be
width R (M ) = inf{i ≥ 0 | Tor R
i (R/m, M ) 6= 0}.
Theorem 3.15. Let (R, m) be local and M be a non-zero R-module.
(i) One has depth (R) ≤ depth R (M ) + C-pd R (M ).
(ii) One has depth (R) ≤ width R (M ) + C-id R (M ).
Proof. (i). We can assume that C-pd R (M ) < ∞, since otherwise we have nothing to prove.
We have PC -grade R (R/m) = P-grade R (R/m) = depth (R) by Remark 3.2. Hence we have
Ext iR (R/m, M ) = 0 for all i < depth (R) − C-pd R (M ) by Theorem 3.9(ii). It follows that
depth (R) ≤ depth R (M ) + C-pd R (M ).
(ii). We can assume that C-id R (M ) < ∞, since otherwise we have nothing to prove. We
have IC -grade R (R/m) = PC -grade R (R/m) = depth (R) by Corollary 3.12. Hence we have
Ext iR (M, R/m) = 0 for all i < depth (R) − C-id R (M ) by Theorem 3.9(i). But there are
isomorphisms
Ext iR (M, R/m) ∼
= Ext iR (M, Hom R (R/m, E(R/m)))
∼ Hom R (Tor R (M, R/m), E(R/m)),
=
i

in which the second equality is from [5, Theorem 3.2.1]. Hence Tor R
i (M, R/m) = 0 for all
i < depth (R) − C-id R (M ), whence depth (R) ≤ width R (M ) + C-id R (M ).



Lemma 3.16. Let M be a finitely generated R-module and let I be an injective R-module
for which Hom R (M, I) 6= 0.
(i) If M is n-C-perfect, then Hom R (M, I) is n-C-coperfect.
(ii) If I is an injective cogenerator, then the converse of (i) holds true.
Proof. (i). Assume that M is n-C-perfect. Then C-pd R (M ) = n, and hence M ∈ BC (R).
Therefore, by [11, Proposition 7.2(b)], we have Hom R (M, I) ∈ AC (R). Hence we have the
(in)equalities
PC -grade R (M ) ≤ IC -grade R (Hom R (M, I))
≤ C-id R (Hom R (M, I))
= id R (C ⊗R Hom R (M, I))
= id R (Hom R (Hom R (C, M ), I))
≤ fd (Hom R (C, M ))
= pd (Hom R (C, M ))
= C-pd (M ),
in which the first inequality is from Theorem 3.11, the second inequality is from Lemma
3.3(ii), and the first and the last equalities are from Theorem 2.7. Hence Hom R (M, I) is
n-C-coperfect.
(ii). Use Lemma 2.8(i) and Corollary 3.12.



CHARACTERIZING LOCAL RINGS

11

Corollary 3.17. Let (R, m) be local with dim (R) = n. The following are equivalent:
(i) R is regular.
(ii) R/m is n-C-perfect.
(iii) R/m is n-C-coperfect.
Proof. (i)=⇒(ii). In this case, by [3, Corollary 8.6], we have C = R, and then
P-grade R (R/m) = inf{i ≥ 0 | Ext iR (R/m, R) 6= 0} = n = pd R (R/m).
∼ R/m, and the
(ii)=⇒(iii). Use Lemma 3.16, the isomorphism Hom R (R/m, E(R/m)) =
fact that E(R/m) is an injective cogenerator.
(iii)=⇒(i). Note that C ⊗R R/m is a finite dimensional vector space over R/m. Therefore,
by Theorem 2.7(i) and the assumption, we have
id R (R/m) = id R (C ⊗R R/m) = C-id R (R/m) = n.
Hence R is regular by [1, Exercise 3.1.26].



Theorem 3.18. Let (R, m) be a Cohen-Macaulay local ring. Let M be a finitely generated
R-module with C-pd R (M ) < ∞ and let n = depth (R) − depth (M ). The following are
equivalent:
(i) M is Cohen-Macaulay.
(ii) M is n-C-perfect.
(iii) Hom R (M, E(R/m)) is n-C-coperfect.
Proof. (i)=⇒(ii). Since C-pd R (M ) < ∞, we have M ∈ BC (R) and hence Ext iR (C, M ) = 0
for all i ≥ 0. Thus depth R (Hom R (C, M )) = depth R (M ) by [4, Corollary 5.2.7]. Therefore
we have the equalities
C-pd R (M ) = pd R (Hom R (C, M )
= depth (R) − depth R (Hom R (C, M ))
= depth (R) − depth R (M )
= n,
in which the first equality is from Theorem 2.7(ii), and the second equality is AuslanderBuchsbaum formula. Hence, in view of Lemma 3.7, we have Ext iR (M, PC ) = 0 for all i > n.
Next, since R is Cohen-Macaulay, there are equalities
inf{i ≥ 0 | Ext iR (M, C) 6= 0} = inf{i ≥ 0 | Ext iR (R/Ann R (M ), C) 6= 0}
= grade (Ann R (M ), C)
= grade (Ann R (M ), R)
= dim (R) − dim R (M )
= n.
Thus, since M is finitely generated, we have Ext iR (M, PC ) = 0 for all i < n by [5, Theorem
3.2.15]. Hence M is n-C-perfect by Proposition 3.8(i).
(ii)=⇒(iii). Use Lemma 3.16 and the fact that E(R/m) is an injective cogenerator.

12

M. RAHMANI AND A.- J. TAHERIZADEH

(iii)=⇒(i). We have the equalities
dim R (M ) = dim (R) − P-grade R (M )
= dim (R) − PC -grade R (M )

= dim (R) − IC -grade R Hom R (M, E(R/m))
= dim (R) − n

= depth (M ),
in which the first equality holds since R is Cohen-Macaulay, the second equality holds by
Remark 3.3, the third equality holds by Corollary 3.12, and the fourth equality holds by
assumption. Hence, M is Cohen-Macaulay.



Theorem 3.19. Let (R, m) be local with dim (R) = n. The following are equivalent:
(i) C is dualizing.
(ii) E(R/m) is n-C-perfect.
(iii) Hom R (C, E(R/m)) is n-perfect.
b is n-C-coperfect.
(iv) R
b is n-coperfect.
(v) C

Proof. (i)=⇒(ii). First, note that fd R (Hom R (C, E(R/m))) = n. Hence by [7, Corollary 3.4],
we have pd R (Hom R (C, E(R/m))) ≤ n, from which we conclude that C-pd R (E(R/m)) =
pd R (Hom R (C, E(R/m))) = n by Theorem 2.7(ii). Next, note that the minimal injective
resolution of C is of the form
L
L
0→C →
E(R/p) →
E(R/p) → · · · → E(R/m) → 0. (†)
ht (p)=0
ht (p)=1
Let P be a projective R-module. Applying the exact functor − ⊗R P on the (†), we get the
following exact complex
L
L
E(R/p) ⊗R P → · · · → E(R/m) ⊗R P → 0.
E(R/p) ⊗R P →
0 → C ⊗R P →
ht (p)=1
ht (p)=0
which is an injective resolution for C ⊗R P . By [5, Theorem 3.3.12], the injective R-module
E(R/p) ⊗R P is a direct sum of copies of E(R/p) for each p ∈ Spec (R). Therefore, since
Hom R (E(R/m), E(R/p)) = 0 for any prime ideal p 6= m, we have Ext iR (E(R/m), PC ) = 0
for all i 6= n. But, if P = R, then
Ext nR (E(R/m), C) = Hom R (E(R/m), E(R/m))
∼
b
=R
6= 0.

Hence, E(R/m) is n-C-perfect by proposition 3.8(i), as wanted.
(ii)⇐⇒(iii). Note that Hom R (C, E(R/m)) ∈ AC (R). Now, E(R/m) is n-C-perfect if
and only if PC -grade R (E(R/m)) = C-pd R (E(R/m)) = n, and this is the case if and only
if P-grade R (Hom R (C, E(R/m))) = pd R (Hom R (C, E(R/m))) = n by Theorem 2.7(ii) and
Lemma 3.3(i).
(i)=⇒(iv). First, note that there are equalities
b
n = id R (C) = C-id R (R) = C-id R (R)

in which the second equality is from Theorem 2.7(i) and the last one is from Lemma 2.8(ii).

CHARACTERIZING LOCAL RINGS

13

b = 0 for all i > n by Lemma 3.6. Next, we show that n is the
Hence, we have Ext iR (IC , R)
b 6= 0
b 6= 0. To do this, first we show that if Ext i (IC , R)
least integer for which Ext n (IC , R)
R

R

b 6= 0. Assume that I is an injective Rfor some integer i, then Ext iR (Hom R (C, E(R/m)), R)
i
b
module for which Ext R (Hom R (C, I), R) 6= 0. Suppose, on the contrary, that m ∈
/ Ass R (I).
Without loss of generality, we can assume that I = E(R/p) with p 6= m. We have the

isomorphisms

b ∼
Ext iR (Hom R (C, I), R)
= Ext iR Hom R (C, I), Hom R (E(R/m), E(R/m))

∼
= Hom iR Tor R
i (Hom R (C, I), E(R/m)), E(R/m) ,
b 6=
in which the second isomorphism is from [5, Theorem 3.2.1]. Hence Ext iR (Hom R (C, I), R)
0 if and only if Tor R
i (Hom R (C, I), E(R/m)) 6= 0. Choose an element x ∈ m r p. Then

multiplication of x induces an isomorphism on I and hence on Hom R (C, I). It follows that
the homomorphism
x.

R
Tor R
i (Hom R (C, I), E(R/m)) −→ Tor i (Hom R (C, I), E(R/m))

is both an isomorphism and locally nilpotent, whence Tor R
i (Hom R (C, I), E(R/m)) = 0, a
contradiction. Consequently, we need only to show that n is the least integer for which
b 6= 0 or equivalently Tor R (Hom R (C, E(R/m)), E(R/m)) 6= 0.
Ext nR (Hom R (C, E(R/m)), R)
n
Applying the exact functor Hom R (−, E(R/m)) on (†) above, we get a flat resolution

0→
Fn → · · · → F1 → F0 → Hom
 R (C, E(R/m)) → 0,
L
in which Fi = Hom R
E(R/p), E(R/m) . Now, by applying − ⊗R E(R/m), on
ht (p)=i
this flat resolution, we can compute Tor R
i (Hom R (C, E(R/m)), E(R/m)). By the above
argument, Tor R
i (Hom R (C, E(R/m)), E(R/m)) = 0 for all 0 ≤ i < n. On the other hand,
we have
Tor R
n (Hom R (C, E(R/m)), E(R/m)) = Fn ⊗R E(R/m)
= Hom R (E(R/m), E(R/m)) ⊗R E(R/m)
b ⊗R E(R/m)
=R
∼
= E(R/m)
6= 0.
b = 0 for all i 6= 0 and Ext n (Hom R (C, E(R/m)), R)
b 6= 0. Consequently,
Thus Ext iR (IC , R)
R
b is n-C-coperfect by Proposition 3.8(ii).
R

∼ C ⊗R R
b ∈ BC (R) by [16, Theorem 2.8(b)].
b ∈ AC (R), and so C
b=
(iv)⇐⇒(v). Note that R
b
b
b = n, and this is the case
Now, R is n-C-coperfect if and only if IC -grade R (R) = C-id R (R)
b = id R (C)
b = n by Theorem 2.7(i) and Lemma 3.3(ii).
if and only if I-grade R (C)
(iv)=⇒(i). One has

b =n
id R (C) = C-id R (R) = C-id R (R)

where the first equality is from Theorem 2.7(i), the second equality is from Lemma 2.8(ii),
and the last equality is the assumption. Hence C is dualizing.



14

M. RAHMANI AND A.- J. TAHERIZADEH

Recall that 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

Theorem 3.20. Let (R, m) be local with dim (R) = n. The following are equivalent:
(i) R is Gorenstein.
(ii) any quasidualizing R-module is n-perfect.
(iii) there exists an n-perfect quasidualizing R-module with 1-dimensional socle.
Proof. (i)=⇒(ii). Assume that T is a quasidualizing R-module. By [13, Proposition 2.1], T
b
b is a complete, by [13, Theorem 3.1], there exists
is a quasidualizing R-module.
Also, since R

∼ Hom b (T, E(R/m)). But, as R
b
b is Gorenstein,
a semidualizing R-module
K for which K =
R
∼
b
we have K = R by [3, Corollary 8.6]. Therefore we have the isomorphisms
T

∼
= Hom Rb (Hom Rb (T, E(R/m)), E(R/m))
∼
b E(R/m))
= Hom Rb (R,
∼
= E(R/m),

b Now, if P is a
in which the first isomorphism holds because T is Matlis reflexive over R.

projective R-module, then it is free and hence by using the minimal injective resolution of
R, we have µi (p, P ) = 0 for all i 6= ht (p). Therefore Ext iR (E(R/m), P ) = 0 for all i 6= n,
and that
n
Ext nR (E(R/m), P ) ∼
= Hom R (E(R/m), E(R/m)µ (m,P ) ) 6= 0.

Thus, since pd R (E(R/m)) = n, the R-module T = E(R/m) is n-perfect by Proposition
3.8(i).
(ii)=⇒(iii). The desired module is E(R/m).
(iii)=⇒(i). Assume that T is an n-perfect quasidualizing R-module with 1-dimensional
b
socle. Observe that, by [13, Proposition 2.1], T is a quasidualizing R-module.
Also, we have
∼
b
pd b (T ) = n since T ⊗R R = T . Next, a similar argument as in [5, Lemma 9.1.4] yields the
R

isomorphisms

b
Ext iR (T, R) ∼
= Ext iRb (T, R),

Ext iR (T, F )

for all i ≥ 0. Hence, if
= 0 for any free R-module F , then Ext iRb (T, X) = 0
b
for any free R-module
X. Finally, if vdim R/m (Soc R (T )) = 1, then Hom R (R/m, T ) ∼
= R/m
∼
and therefore Hom b (R/m, T ) = R/m, whence vdim R/m (Soc b (T )) = 1. Consequently, T
R

R

b
is an n-perfect R-module
with 1-dimensional socle. On the other hand, R is Gorenstein if
b
b and assume that R is complete. In
and only if R is so. Hence, we can replace R by R

the new case, by [13, Theorem 3.1], there exists a semidualizing R-module C for which
C∼
= Hom R (T, E(R/m)). Thus we have id R (C) ≤ pd R (T ) = n, whence C is dualizing, and
hence the (in)equalities
dim (R) = dim R (C) ≤ id R (C) = depth (R),
show that R is Cohen-Macaulay. Now, we can use local duality [2, Theorem 11.2.8] to get
the isomorphisms
Hnm (R) = Hom R (C, E(R/m)) ∼
= T,

CHARACTERIZING LOCAL RINGS

15

from which we conclude that vdim R/m (Hnm (R)) = 1. Consider the following Grothendieck’s
third quadrant spectral sequence [15, theorem 10.47]
p
p+q
q
Ep,q
2 = Ext R (R/m, Hm (R)) =⇒ Ext R (R/m, R).
p

Since R is Cohen-Macaulay, we have Hqm (R) = 0 for all q 6= n by [2, Corollary 6.2.9], and
hence Ep,q
2 collapses on the n-th column, whence
Ext n (R/m, R) ∼
= Hom R (R/m, Hn (R)) ∼
= R/m.
R

m

It follows, from [5, Theorem 9.2.27], that R is Gorenstein.



By using the same argument as in the proof of the above theorem, we have the following
corollary which is a characterization of Cohen-Macaulay local rings.
Corollary 3.21. Let (R, m) be a complete local ring with dim (R) = n. The following are
equivalent:
(i) R is Cohen-Macaulay.
(ii) there exists an n-perfect quasidualizing R-module.
Proof. (i)=⇒(ii). By assumption, R has a dualizing module, say D. Now, by Theorem 3.19
and [13, theorem 3.1], Hom R (D, E(R/m)) is an n-perfect quasidualizing R-module.
(ii)=⇒(i). Suppose that T is an n-perfect quasidualizing R-module. Then by [13, theorem
3.1], there exists a semidualizing module C for which T = Hom R (C, E(R/m)). Now, we
have id R (C) ≤ pd (T ) = n, whence C is dualizing and R is Cohen-Macaulay.



Question 3.22. Can we omit the condition ’1-dimensional socle’ in the Theorem 3.20?
More precisely, if there exists an n-perfect quasidualizing R-module, then is R Gorenstein?
Acknowledgment. The authors are grateful to the referee for his/her invaluable comments.
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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

