Math. J. Okayama Univ. 51 (2009), 71–81

LINKAGE AND DUALITY OF MODULES
Kenji NISHIDA

Abstract. Martsinkovsky and Strooker [13] recently introduced module theoretic linkage using syzygy and transpose. This generalization
brings possibility of much application of linkage, especially, to homological theory of modules. In the present paper, we connect linkage ofmodules to certain duality of modules. We deal with Gorenstein dimension, Cohen-Macaulay modules over a Gorenstein local ring using
linkage and generalize the results to non-commutative algebras.

1. Introduction
The theory of linkage, devised by Peskine and Szpiro[16], is recently
generalized to module theoretic version by Martsinkovsky and Strooker in[13]. They generalize the theory for wide class of rings, including noncommutative semiperfect Noetherian rings. The use of composition of two
sort of functors, syzygy and transpose, enables them to extend the definition of linkage. These functors are fundamental for homological theory of
Noetherian rings. There appears a relation to the duality theory introduced
and studied by Auslander and Bridger[1] and Iyama[11], that is, vanishingof certain extension group is equivalent to existence of linkage for a dual ofa module.
In section two, we shall apply this fact to Gorenstein dimension and
Cohen-Macaulayness of modues, mainly over commutative Gorenstein local ring, and obtain characterization of these notions by linkage. A duality
studied by Iyama[11] can be applicable to the full subcategory of CohenMacaulay modules of codimension k > 0 over a Gorenstein local ring. We
decide the image of this subcategory and give a characterization of a CohenMacaulay module of codimension k > 0 (Theorem 2.6, Corollary 2.7).
In section three, we make an attempt to gerenalize the results of section
two to non-commutative Noetherian ring. It is desirable that a candidate
satisfies the equations (2.1) and (2.2) in section two. A module-finite Gorenstein algebra studied in [8] satisfies them. Therefore, we treat such algebrasand show that these equations hold. We find that similar proof works formodule-finite Gorenstein algebras and get the same results as in section
two. The equation (2.1), called Auslander-Bridger Formula, is proved in2000 Mathematics Subject Classification. 13C14, 13C40, 16E65
keywords: Linkage, Duality, Gorenstein dimension
Research of the author is supported by Grant-in-Aid for Scientific Researches C(2) in
Japan
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many ways. Among others, we use the original version of [1] to prove it.
To do so, we dissolve a question, proposed by [14], about the proof of thisformula in [1] (see Remark 3.4.1).
2. Linkage and duality
Let Λ be a left and right Noetherian ring. Let modΛ (respectively,modΛop ) be the category of all finitely generated left (respectively, right) Λmodules. Throughout the paper, all modules are finitely generated and left
modules (if the ring is non-commutative) and right modules are consideredas Λop -modules. We denote the stable category by modΛ, the syzygy functorby Ω : modΛ → modΛ, and the transpose functor by Tr : modΛ → modΛop .
Recall the definition of the functor Tr ([1], Chapter two, section 1 where itf

is denoted by D, or [2], Chapter IV, section 1). Let P1 → P0 → M → 0
be a projective resolution of a module M ∈ modΛ. Then the transpose of
M , TrM ∈ modΛop , is equal to Cokf ∗ , where (−)∗ : modΛ → modΛop is
defined by M ∗ := HomΛ (M, Λ).
In order to relate linkage and duality, we combine the above functors.
One is the functor Tk := TrΩk−1 for k > 0 which was firstly introduced by
Auslander and Bridger [1] to define and study the (homological) torsionfreeness and reflexivity, and recently used by Iyama [11] to define a duality
between some full subcategories given by grade. An another tool, an operator λ := ΩTr, is introduced in the nice article by Martsinkovsky andStrooker [13]. Using the operator λ, they defined the notion of linkage ofmodules.
Let Λ be a semiperfect ring and M, N are finitely generated Λ-modules
having no nonzero projective direct summands. Then M ∼= N in modΛ∼
if and only if M = N in modΛ (e.g., [6], 25.1.5 Corollary). Then λM isdetermined up to isomorphism for a finitely gererated Λ-module M .
Throughout the paper, we assume that the underlying rings are always
semiperfect. It is well-known that a commutative local ring ([17], Theorem
4.46) and a module-finite algebra over a commutative Noetherian complete
local ring ([5], p.132) are semiperfect.
Following [13], we define
2.1. Definition. A finitely generated Λ-module M and a Λop -module N
are said to be horizontally linked if M ∼= λM , in other words,= λN and N ∼∼M is horizontally linked (to λM ) if and only if M = λ2 M .
A rather different definition of linkage of modules is proposed by Yoshino
and Isogawa [18]. However, both definitions coincide for Cohen-Macaulay
modules over a commutative Gorenstein ring (see [13], section 3).

LINKAGE AND DUALITY OF MODULES73

It is noted in [13], that a projective module is horizontally linked if andonly if it is isomorphic to the zero module (the statement after [13], Definition 3). We also note that, for a finitely generated Λ-module M having
no nonzero projective direct summands, if the projective dimension of TrM
equals one, i.e. M ∗ = 0, then M is horizontally linked if and only if M is
isomorphic to the zero module.
Let us start with the following simple observation which connect duality
with linkage.
2.2. Proposition. Let k > 0. Then Tk M is horizontally linked if andonly if ExtkΛ (M, Λ) = 0.
Proof. This is a direct consequence of the definition of linkage and thefollowing exact sequence:
0 → ExtkΛ (M, Λ) → Tk M → λ2 Tk M → 0,
which is given in [11], section 2 or [10], the proof of Lemma 2.1. We prepare the facts about Gorenstein dimension from [1] and [4], Chapter 1. A Λ-module M is said to have G-dimension zero, denoted by GdimΛ M = 0, if M ∗∗ ∼= M and ExtkΛ (M, Λ) = ExtkΛop (M ∗ , Λ) = 0 for
k > 0. This is equivalent to ‘ExtkΛ (M, Λ) = ExtkΛop (TrM, Λ) = 0 for
k > 0’ ([1], Proposition 3.8). For a positive integer k, we say that M
has G-dimension less than or equal to k, denoted by G-dimΛ M ≤ k, if
there exists an exact sequence 0 → Gk → · · · → G0 → M → 0 with
G-dimΛ Gi = 0 for (0 ≤ i ≤ k). It follows from [1], Theorem 3.13 that
G-dimΛ M ≤ k if and only if G-dimΛ Ωk M = 0. If G-dimΛ M < ∞, thenG-dimΛ M = sup{k : ExtkΛ (M, Λ) 6= 0}([1], p. 95 or [4], (1.2.7)).
Invariance of G-dimension under linkage is studied in [13].
2.3. Theorem([13], Theorem 1). Let Λ be a semiperfect right and left
Noetherian ring and M a Λ-module having no nonzero projective direct
summands. Then the following conditions are equivalent.
(1) G-dimΛ M = 0,
(2) G-dimΛop λM = 0 and M is horizontally linked.
In the rest of this section, we consider a commutative ring case and apply
the above results to Cohen-Macaulay modules over a commutative Gorenstein local ring. See [3] for Cohen-Macaulay rings and modules and Gorenstein rings.

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KENJI NISHIDA

Let R be a (commutative) Gorenstein local ring and M a finitely generated
R-module. Then there are the following useful equalities:(2.1) G-dimR M + depthM = dim R(2.2) gradeR M + dim M = dim R,
where gradeR M := inf{k ≥ 0 : ExtkR (M, R) 6= 0}. The first equality is dueto [1], Theorems 4.13 and 4.20 and the second one to [15], Lemma 4.8. (Seealso [8], Proposition 4.11.)
The combination of linkage and duality produces the following characterization of a maximal Cohen-Macaulay module which improves [13], Proposition 8.
2.4. Theorem. Let R be a Gorenstein local ring and M a finitely generated R-module having no nonzero projective direct summands. Then thefollowing are equivalent(1) M is a maximal Cohen-Macaulay module,(2) Tk M is horizontally linked for k > 0,(3) λM is a maximal Cohen-Macaulay module and M is horizontally
linked.
Proof. (1) ⇔ (2): Since R is Gorenstein, M is maximal Cohen-Macaulay
if and only if G-dimR M = 0. This is equivalent to ExtkR (M, R) = 0 for
k 6= 0 by [1], Theorem 4.20. Thus (1) ⇔ (2) follows from Proposition 2.2.
(1) ⇔ (3):This is a direct consequence of Theorem 2.3. 
By the above theorem, G-dimension is described using linkage.
2.5. Proposition. Let R be a Gorenstein local ring and M a finitely
generated module. Then G-dimR M ≤ k if and only if Ti+k M is horizontally
linked for i > 0.
Proof. Ti+k M is horizontally linked for i > 0 ⇔ Ti Ωk M is horizontally
linked for i > 0 ⇔ G-dimR Ωk M = 0 ⇔ G-dimR M ≤ k. 
We apply duality theory on a non-commutative Noetherian ring due toIyama [11] to the category of Cohen-Macaulay modules. Suppose that Λ
is a right and left Noetherian ring. Then the functor Tk gives a duality
between the categories {X ∈ modΛ : gradeΛ X ≥ k} and {Y ∈ modΛop :rgradeΛop Y ≥ k and pdΛop Y ≤ k} [11], 2.1.(1), where rgradeΛop Y := {k >0 : ExtkΛop (Y, Λ) 6= 0} stands for a reduced grade of Y [9]. Returning to ourcase, we consider a commutative Noetherian local ring R and a finitely generated R-module M . Then it holds that G-dimR M ≥ gradeR M , in general.

LINKAGE AND DUALITY OF MODULES

75

Moreover, if G-dimR M < ∞, then M is a Cohen-Macaulay module if andonly if G-dimR M = gradeR M by the equations (2.1) and (2.2). Thus wecan apply the above duality to the category of Cohen-Macaulay R-modules.
A finitely generated module M over a Cohen-Macaulay local ring R iscalled a Cohen-Macaulay module of codimension k, if depthM = dim M =dim R − k. Put the full subcategory of modR
Ck := {M ∈ modR : M is a Cohen-Macaulay R-module of codimension k}.
In order to give a duality, we need a counterpart of the category Ck . We
put the full subcategory of modRrgradeR N = pdR N = k and′,Ck := N ∈ modR : 2
λ N is a maximal Cohen-Macaulay module
where pdR N stands for a projective dimension of N . Let C k , respectively
C ′k , be the full subcategory of modR induced from Ck , respectively Ck′ .
2.6. Theorem. Let R be a Gorenstein local ring. Let k > 0. Then thefunctor Tk gives a duality between full subcategories C k and C ′k such thatTk ◦ Tk is isomorphic to the identity funtor.
Proof. It follows from [11], 2.1. (1) that
M ∈ Ck ⇒ rgradeR Tk M = pdR Tk M = k and Tk Tk M ∼= M,
N ∈ Ck′ ⇒ gradeR Tk N ≥ k and Tk Tk N ∼= N.
It suffices to prove that Tk M ∈ Ck′ (respectively, Ck ) whenever M ∈ Ck
(respectively, Ck′ ). Let M ∈ Ck and · · · → Pi → · · · → P0 → M → 0
a minimal free resolution of M . Then we have an exact sequence 0 →∗→ TrΩk M → 0, which gives an exact sequence(Ωk M )∗ → Pk∗ → Pk+1
0 → (Ωk M )∗ → Pk∗ → λ2 Tk M → 0,
by definition of Tk . From an exact sequence 0 → Ωk+1 M → Pk → Ωk M → 0
and the assumption that G-dimR M = k, we have an exact sequence
0 → (Ωk M )∗ → Pk∗ → (Ωk+1 M )∗ → 0.
Hence λ2 Tk M ∼= (Ωk+1 M )∗ . Since G-dimR M = k, we have G-dimR Ωk+1 M =0, so that G-dimR (Ωk+1 M )∗ = 0, hence G-dimR λ2 Tk M = 0 which implies
that λ2 Tk M is a maximal Cohen-Macaulay module.
Conversely, let N ∈ Ck′ . It suffices to prove that G-dimR Tk N ≤ k. SinceΩk Tk N is a syzygy module, the canonical homomorphism ϕ : Ωk Tk N →(Ωk Tk N )∗∗ is monic. From Auslander formula [1], Proposition 2.6, Cokϕ =Ext2R (TrΩk Tk N, R), whose right hand side is equal to Ext1R (λ2 Tk Tk N, R) =Ext1R (λ2 N, R) = 0 by assumption. Hence Ωk Tk N ∼= (Ωk Tk N )∗∗ . From a76

KENJI NISHIDA

minimal free resolution · · · → Qi → · · · → Q0 → Tk N → 0, we get an exact
sequence
0 → (Ωk Tk N )∗ → Q∗k → λ2 N → 0.
Since λ2 N is a maximal Cohen-Macaulay module, (Ωk Tk N )∗ is also a maximal Cohen-Macaulay module, so that (Ωk Tk N )∗∗ ∼= Ωk Tk N is a maximal
Cohen-Macaulay module. This implies that G-dimR Ωk Tk N = 0, and thenG-dimR Tk N ≤ k. 
2.7. Corollary. Let M be a finitely generated R-module of gradeR M =k > 0. Then the following are equivalent(1) M is a Cohen-Macaulay module of codimension k,(2) rgradeR Tk M =pdR Tk M = k and λ2 Tk M is a maximal Cohen-Macaulay
module.
3. A generalization to non-commutative Gorenstein algebras
We shall generalize the results of section two concerning Gorenstein rings
to module finite (non-commutative) Gorenstein algebras over a commutative
Noetherian complete local ring. Such algebras are extensively studied in [8].
Throughout this section, let (R, m) be a commutative Noetherian complete local ring. Let Λ be a module-finite R-algebra, that is, Λ is an Ralgebra which is finitely generated as an R-module. As is noted in section
2, Λ is semiperfect. We define a Gorenstein algebra (cf. [8], Lemma 4.7).
3.1. Definition. Let Λ be a module-finite R-algebra. We call Λ a
Gorenstein R-algebra, if Λ is a Cohen-Macaulay R-module with idΛ Λ =dimR Λ, where idΛ X stands for an injective dimension of a Λ-module X.
It follows from [8], Corollary 4.8 that this definition is left-right symmetric. In order to characterize a Gorenstein algebra, we need the following‘homogeneity condition’ studied in [7].
(h) ExttΛ (S, Λ) 6= 0 for every simple Λ-module S, where t = depthR Λ.
We denote by (hop ), when we consider Λop -modules. Then we state a theorem which is essential for generalizing the results in §2 to non-commutative
algebras.
3.2. Theorem. (cf. [1], Theorem 4.20) Let Λ be a module-finite Ralgebra and t = depthR Λ. Then the following are equivalent.
(1) Λ is a Gorenstein R-algebra,(2) Λ satisfies the condition (h) and idΛ Λ < ∞,

LINKAGE AND DUALITY OF MODULES

77(2 ′ ) Λ satisfies the condition (hop ) and idΛop Λ < ∞,(3) Λ satisfies the condition (h) and G-dimΛ M < ∞ for any finitely
generated Λ-module M ,(3 ′ ) Λ satisfies the condition (hop ) and G-dimΛop M < ∞ for any finitely
generated Λop -module M
When this is the case, it holds that idΛ Λ = idΛop Λ = dimR Λ = t.
The proof is done after generalizing several facts in [1], Chapter four toour case.
3.3. Proposition. (cf. [1], Proposition 4.12) Assume that Λ satisfies thecondition (h) and M is a finitely generated Λ-module with G-dimΛ M < ∞.
Then the following are equivalent(1) G-dimΛ M = 0,(2) depthR M ≥ depthR Λ,(3) depthR M = depthR Λ,
Proof. Let depthR Λ = t.
(1) ⇒ (2): Let x1 , · · · , xi be a Λ-sequence in m. We show that x1 , · · · , xi
is an M -sequence by induction on i. Let i = 1. Since M ∼= M ∗∗ and x1 isΛ-regular, x1 is M -regular.
Suppose that i > 1 and the assertion holds for i − 1. Then x1 , · · · , xi−1
is an M -sequence. Put I = (x1 , · · · , xi−1 ), R = R/I, Λ = Λ/IΛ, M =M/IM . Then (R, m/I) is a commutative Noetherian complete local ring.
By [1], Lemma 4.9 or [4], Corollary 1.4.6, G-dimΛ M = G-dimΛ M = 0.
Since xi ∈ R is a Λ-regular element, xi is M -regular, hence x1 , · · · , xi is an
M -sequence. Therefore, depthM ≥ depthΛ.
(2) ⇒ (3): Assume that t = 0. We prove that if G-dimM < ∞, then GdimM = 0. The condition (h) implies HomΛ (S, Λ) 6= 0 for every simple
Λ-module S. Hence, for a finitely generated Λ-module M , if M ∗ = 0 thenM = 0.
Suppose that G-dimΛ M ≤ 1. We have an exact sequence 0 → L1 → L0 →M → 0 with G-dimΛ Li = 0 (i = 0, 1). Hence we have an exact sequence
0 → M ∗ → L∗0 → L∗1 → Ext1Λ (M, Λ) → 0
and ExtiΛ (M, Λ) = 0 for i > 1. By this sequence, we have an exact sequence
0 → Ext1Λ (M, Λ)∗ → L1 → L0 ,where L1 → L0 is monic. Thus Ext1Λ (M, Λ)∗ = 0, and so Ext1Λ (M, Λ) = 0 by
the previous paragraph. Since G-dimΛ M < ∞, this proves G-dimΛ M = 0.

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KENJI NISHIDA
fn

f1

Suppose that G-dimΛ M ≤ n. Let 0 → Ln → · · · → L0 → M → 0 be
exact with G-dimΛ Li = 0 (0 ≤ i ≤ n). Since G-dimΛ (Imfn−1 ) ≤ 1, we haveG-dimΛ (Imfn−1 ) = 0 by the above argument. Repeating this process, we
get G-dimΛ M = 0.
Next, we prove that depthR M = 0. We only need the reflexivity of M .
By the condition (h), there exists an exact sequence 0 → C → Λ, where C
is a direct sum of all non-isomorphic simple Λ-modules. This sequence gives
an exact sequence
0 → HomΛ (M ∗ , C) → M ∗∗ ∼= M.
Since M ∗ 6= 0, we have HomΛ (M ∗ , C) 6= 0. Since mHomR (M ∗ , C) = 0, we
see that m has no M -regular element, so that depthM = 0. This proves (2)⇒ (3) for the case that depthR Λ = 0.
Let t = depthΛ > 0. We have depthM ≥ depthΛ ≥ 1, so that there isan element x ∈ m which is Λ and M -regular. By [1], Lemma 4.9, we haveG-dimΛ/xΛ M/xM < ∞. It follows from [3], Proposition 1.2.10 (d) that
depthR/xR M/xM = depthR M − 1 ≥ depthR Λ − 1 = depthR/xR Λ/xΛ.
Hence, by induction on t, we have depthR/xR M/xM = depthR/xR Λ/xΛ,and then depthM = depthΛ. Therefore, (2) ⇒ (3) holds.
(3) ⇒ (1): By assumption, it suffices to prove that ExtiΛ (M, Λ) = 0 for
i > 0. We show the assertion by induction on t.
Let t = 0. Then G-dimΛ M = 0 by the proof (2) ⇒ (3).
Let t > 0. Then depthM = depthΛ ≥ 1. We take an element x ∈ mwhich is Λ and M -regular. Then, by [3], Proposition 1.2.10 (d),depthR/xR M/xM = depthR M − 1 = depthR Λ − 1 = depthR/xR Λ/xΛ.
Hence we have ExtiΛ/xΛ (M/xM, Λ/xΛ) = 0 for i > 0 by induction. This
gives ExtiΛ (M, Λ/xΛ) = 0 for i > 0 (cf. [12], p.155). From an exact sequencex
0 → Λ → Λ → Λ/xΛ → 0, we get an exact sequencexExtiΛ (M, Λ) → ExtiΛ (M, Λ) → ExtiΛ (M, Λ/xΛ) = 0.
By Nakayama’s Lemma, it holds that ExtiΛ (M, Λ) = 0 for i > 0 
3.4. Proposition. (cf. [1], Theorem 4.13) Assume that Λ satisfies thecondition (h). Let M ∈ modΛ with G-dimΛ M < ∞. Then we have an
equality
G-dimΛ M + depthR M = depthR Λ
Proof. If G-dimM = 0, we are done by the previous proposition. Supposethat G-dimM = n > 0 and the equation holds for n − 1. Let 0 → K → P →LINKAGE AND DUALITY OF MODULES

79

M → 0 be exact with P finitely generated projective. Since G-dimK =n − 1, we have G-dimK + depthK = depthΛ by induction. Suppose that
depthM ≥ depthP . Then depthM ≥ depthΛ. Hence G-dimM = 0 holdsby the previous proposition. This contradicts to G-dimM > 0. Hence
depthM < depthP , so depthK = depthM + 1 by, e.g., [3], Proposition
1.2.9. Therefore, n + depthM = depthΛ. 
3.4.1. Remark. Recently, it is pointed out in [14] that the proof of [1],Theorem 4.13 (b) includes a ‘serious mistake’. Although we think that there
is no ‘serious mistake’, we might say that the proof needs some supplementary explanation as above to use induction on G-dimM .
Proof of Theorem 3.2. The equivalences (1) ⇔ (2) and (1) ⇔ (2′ )are showed by [8], Theorem 3.7 and Corollary 4.10. Then idΛ Λ = idΛop Λ =dimR Λ = t is a direct consequence of [8], Proposition 4.11.(5).
(2) ⇒ (3): Since idΛ Λ = t, we have ExtiΛ (Ωt M, Λ) ∼= Exti+tΛ (M, Λ) = 0 forti > 0. Let · · · → Pk → · · · → P0 → Ω M → 0 be a projective resolution ofΩt M . Then we have an exact sequence
0 → (Ωt M )∗ → P0∗ → · · · → Pk∗ → · · · .
∗ ). Then there is an exact sequence
Put C := Cok(Pt∗ → Pt+1∗
0 → TrΩt M → P2∗ → · · · → Pt∗ → Pt+1
→ C → 0.

Since idΛop Λ = t, we have
Exti op (TrΩt M, Λ) ∼= Exti+t (C, Λ) = 0, for i > 0.
= Exti op (Ωt C, Λ) ∼ΛΛt
Ω M = 0, and then G-dim

Λ

Thus G-dimΛΛ M ≤ t.
(3) ⇒ (2): Suppose that ExtiΛ (M, Λ) 6= 0 holds. Then G-dimΛ M ≥ i. By
Proposition 3.4, we see that t − depthR M ≥ i, so t ≥ i. Therefore, if t < i,then ExtiΛ (M, Λ) = 0 for all M ∈ modΛ. This implies idΛ Λ ≤ t. 
We assume that Λ is a Gorenstein R-algebra. Let M be a finitely generated Λ-module. Then the following equalities hold true by [8], Proposition
4.11(3.1) G-dimΛ M + depthR M = dimR Λ(3.2) gradeΛ M + dimR M = dimR Λ.
We consider the functors Ω, Tr, Tk over modΛ or modΛop , and an operator
λ on modΛ or modΛop then we can define linkage and duality of modules
over Λ.
A finitely generated Λ-module is called a maximal Cohen-Macaulay Λmodule, respectively a Cohen-Macaulay Λ-module of codimension k if it isa maximal Cohen-Macaulay R-module, respectively a Cohen-Macaulay Rmodule of codimension k. It follows from Theorem 3.2 and equalities (3.1),

80

KENJI NISHIDA(3.2) that M ∈ modΛ is a maximal Cohen-Macaulay Λ-module if and only
if G-dimΛ M = 0.
Using above arrangement, a generalization of 2.4 - 2.7 in section two areroutin task. The proofs are almost the same as in section two. Here, we
provide non-commutative version of main theorems, Theorems 2.4 and 2.6.
3.5. Theorem (a non-commutative version of Theorem 2.4) Let Λ bea Gorenstein R-algebra and M a finitely generated Λ-module having nononzero projective direct summands. Then the following are equivalent(1) M is a maximal Cohen-Macaulay Λ-module,(2) Tk M is horizontally linked for k > 0,(3) λM is a maximal Cohen-Macaulay Λ-module and M is horizontally
linked.
Proof. This theorem is proved by the similar way to Theorem 2.4. Let
Ck (Λ) := {M ∈ modΛ : M is a Cohen-Macaulay Λ-module of codimension k},
rgradeΛ N = pdΛ N = k and λ2 N is′
Ck (Λ) := N ∈ modΛ :a maximal Cohen-Macaulay Λ-module

and C k (Λ), C ′k (Λ) be the full subcategories of modΛ induced from Ck (Λ),
Ck′ (Λ).
3.6. Theorem. (a non-commutative version of Theorem 2.6) Let Λ
be a Gorenstein R-algebra. Let k > 0. Then the functor Tk gives a duality
between full subcategories C k (Λ) and C ′k (Λop ) such that Tk ◦Tk is isomorphic
to the identity functor.
Proof. It follows from [8], Theorem 4.12 that if L is a maximal CohenMacaulay Λ-module, i.e., L ∈ C0 (Λ), then ExtiΛ (L, Λ) = 0 for i > 0,
HomΛ (L, Λ) ∈ C0 (Λop ), and L ∼= HomΛop (HomΛ (L, Λ), Λ). Then letting∗(−) := HomΛ (−, Λ), the proof is done by the similar way to Theorem 2.6.

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Department of Mathematical Sciences, Shinshu University, Matsumoto,390-8621, Japane-mail address: kenisida@math.shinshu-u.ac.jp(Received January 13, 2006 )