Having the material developed in Sections 2 and 3 at our disposal, we are fully prepared to prove the celebrated Greenlees-May Duality Theorem.

Theorem 22. Let a be an ideal of R, and X, Y ∈ D(R). Then there is a natural isomorphism

RHom_{R}(RΓa(X), Y)≃RHom_{R}(X,LΛ^{a}(Y))
in D(R).

Proof. Using Corollary21 and the Adjointness Isomorphism, we have RHomR(RΓa(X), Y)≃RHomR

(RΓa(R)⊗^{L}_{R}X, Y)

≃RHomR(X,RHomR(RΓa(R), X))

≃RHom_{R}(X,LΛ^{a}(Y)).

Corollary 23. Let a be an ideal of R, and X, Y ∈ D(R). Then there are natural isomorphisms:

LΛ^{a}(RHom_{R}(X, Y))≃RHom_{R}(LΛ^{a}(X),LΛ^{a}(Y))

≃RHomR(X,LΛ^{a}(Y))

≃RHomR(RΓa(X),LΛ^{a}(Y))

≃RHom_{R}(RΓ_{a}(X), Y)

≃RHom_{R}(RΓ_{a}(X),RΓ_{a}(Y)).

Proof. By Corollary21, Adjointness Isomorphism, and Theorem22, we have
LΛ^{a}(RHom_{R}(X, Y))≃RHom_{R}(RΓ_{a}(R),RHom_{R}(X, Y))

≃RHomR

(RΓa(R)⊗^{L}_{R}X, Y)

≃RHom_{R}(RΓ_{a}(X), Y)

≃RHom_{R}(X,LΛ^{a}(Y)).

(4.1)

Further, by Theorem 22, [2, Corollary on Page 6], and [11, Proposition 3.2.2], we have

RHom_{R}(RΓ_{a}(X),LΛ^{a}(Y))≃RHom_{R}(RΓ_{a}(RΓ_{a}(X)), Y)

≃RHom_{R}(RΓ_{a}(X), Y)

≃RHomR(RΓa(X),RΓa(Y)).

(4.2)

Moreover, by Theorem22 and [2, Corollary on Page 6], we have
RHomR(LΛ^{a}(X),LΛ^{a}(Y))≃RHomR(RΓa(LΛ^{a}(X)), Y)

≃RHom_{R}(RΓ_{a}(X), Y). (4.3)
Combining the isomorphisms (4.1), (4.2), and (4.3), we get all the desired isomorphisms.

Now we turn our attention to the Grothendieck’s Local Duality, and demonstrate how to derive it from the Greenlees-May Duality.

We need the definition of a dualizing complex.

Definition 24. A dualizing complex forRis anR-complexD∈ D_{}^{f}(R)that satisfies
the following conditions:

(i) The homothety morphism χ^{D}_{R} : R → RHom_{R}(D, D) is an isomorphism in
D(R).

(ii) idR(D)<∞.

Moreover, if R is local, then a dualizing complex D is said to be normalized if sup(D) = dim(R).

It is clear that if D is a dualizing complex for R, then so is Σ^{s}D for every
s ∈ Z, which accounts for the non-uniqueness of dualizing complexes. Further,
Σdim(R)−sup(D)D is a normalized dualizing complex.

Example 25. Let (R,m, k) be a local ring with a normalized dualizing complex D.

Then RΓ_{m}(D)≃E_{R}(k). For a proof, refer to [8, Proposition 6.1].

The next theorem determines precisely when a ring enjoys a dualizing complex.

Theorem 26. The the following assertions are equivalent:

(i) R has a dualizing complex.

(ii) R is a homomorphic image of a Gorenstein ring of finite Krull dimension.

Proof. See [8, Page 299] and [10, Corollary 1.4].

Now we prove the Local Duality Theorem for complexes. We recall that given a
local ring (R,m, k), we let (−)^{∨} := Hom_{R}(−, E_{R}(k)), whereE_{R}(k) is the injective
envelope ofk.

Theorem 27. Let(R,m)be a local ring with a dualizing complexD, andX ∈ D^{f}_{}(R).

Then

H_{m}^{i}(X)∼= Extdim(R)−i−sup(D)

R (X, D)^{∨}

for everyi∈Z.

Proof. Clearly, we have

Extdim(R)−i−sup(D)

R (X, D)∼= Ext^{−i}_{R}
(

X,Σdim(R)−sup(D)D)

for every i ∈ Z, and Σdim(R)−sup(D)D is a normalized dualizing for R. Hence by
replacing D with Σdim(R)−sup(D)D, it suffices to assume that D is a normalized
dualizing complex and prove the isomorphism H_{m}^{i}(X) ∼= Ext^{−i}_{R}(X, D)^{∨} for every
i∈Z. By Theorem 22, we have

RHom_{R}(RΓ_{m}(X), E_{R}(k))≃RHom_{R}(X,LΛ^{m}(E_{R}(k))). (4.4)

But sinceE_{R}(k) is injective, it provides a semi-injective resolution of itself, so we
have

RHomR(RΓm(X), ER(k))≃HomR(RΓm(X), ER(k)). (4.5) Besides, by Example25, [2, Corollary on Page 6], and [6, Proposition 2.7], we have

LΛ^{m}(E_{R}(k))≃LΛ^{m}(RΓ_{m}(D))

≃LΛ^{m}(D)

≃D⊗^{L}_{R}Rˆ^{m}

≃D⊗_{R}Rˆ^{m}.

(4.6)

Combining (4.4), (4.5), and (4.6), we get

Hom_{R}(RΓ_{m}(X), E_{R}(k))≃RHom_{R}(
artinian Rˆ^{m}-module by [9, Proposition 2.1], and thus Matlis reflexive for everyi∈Z.
Moreover,D⊗_{R}Rˆ^{m} is a normalized dualizing complex forRˆ^{m}. Therefore, using the
isomorphism (4.7) over them-adically complete ring Rˆ^{m}, we obtain

H_{m}^{i}(X)∼=H_{m}^{i}(X)⊗_{R}Rˆ^{m}
generatedR-module for everyi∈Z. It follows that

HomRˆ^{m}

for everyi∈Z. Combining (4.8) and (4.9), we obtain
H_{m}^{i}(X)∼= HomR

(Ext^{−i}_{R}(X, D), ER(k))
for everyi∈Zas desired.

Our next goal is to obtain the Local Duality Theorem for modules. But first we need the definition of a dualizing module.

Definition 28. Let (R,m) be a local ring. A dualizing module for R is a finitely generatedR-moduleω that satisfies the following conditions:

(i) The homothety map χ^{ω}_{R} :R →Hom_{R}(ω, ω), given by χ^{ω}_{R}(a) =a1^{ω} for every
a∈R, is an isomorphism.

(ii) Ext^{i}_{R}(ω, ω) = 0 for everyi≥1.

(iii) idR(ω)<∞.

The next theorem determines precisely when a ring enjoys a dualizing module.

Theorem 29. Let(R,m)be a local ring. Then the following assertions are equivalent:

(i) R has a dualizing module.

(ii) Ris a Cohen-Macaulay local ring which is a homomorphic image of a Gorenstein local ring.

Moreover in this case, the dualizing module is unique up to isomorphism.

Proof. See [20, Corollary 2.2.13] and [3, Theorem 3.3.6].

Since the dualizing module for R is unique whenever it exists, we denote a choice
of the dualizing module by ωR. It can be seen thatR is Gorenstein if and only if
ω_{R}∼=R.

Proposition 30. Let (R,m) be a Cohen-Macaulay local ring, and ω a finitely generatedR-module. Then the following assertions are equivalent:

(i) ω is a dualizing module for R.

(ii) ω^{∨}∼=H_{m}^{dim(R)}(R).

Proof. See [4, Definition 12.1.2, Exercises 12.1.23 and 12.1.25, and Remark 12.1.26], and [3, Definition 3.3.1].

We can now derive the Local Duality Theorem for modules.

Theorem 31. Let (R,m) be a local ring with a dualizing module ω_{R}, and M a
finitely generated R-module. Then

H_{m}^{i}(M)∼= Ext^{dim(R)−i}_{R} (M, ω_{R})^{∨}
for everyi≥0.

Proof. By Theorem29,R is a Cohen-Macaulay local ring which is a homomorphic
image of a Gorenstein local ringS. SinceS is local, we have dim(S)<∞. Hence
Theorem 26implies that R has a dualizing complexD. SinceR is Cohen-Macaulay,
we have H_{m}^{i}(R) = 0 for everyi̸= dim(R). On the other hand, by Theorem27, we
have

H_{m}^{i}(R)∼= Extdim(R)−i−sup(D)

R (R, D)^{∨}

∼=H_{−}dim(R)+i+sup(D)(RHom_{R}(R, D))^{∨}

∼=H_{−}dim(R)+i+sup(D)(D)^{∨}.

(4.10)

It follows from the display (4.10) that H_{−}dim(R)+i+sup(D)(D) = 0 for every i ̸=

dim(R), i.e. Hi(D) = 0 for everyi̸= sup(D). Therefore, we have
D≃Σ^{sup(D)}H_{sup(D)}(D).

In addition, lettingi= dim(R) in the display (4.10), we getH_{m}^{dim(R)}(R)∼=H_{sup(D)}(D)^{∨},
which implies that ωR ∼= H_{sup(D)}(D) by Proposition 30. It follows that D ≃
Σ^{sup(D)}ω_{R}.

Now let M be a finitely generated R-module. Then by Theorem27, we have
H_{m}^{i}(M)∼= Extdim(R)−i−sup(D)

R (M, D)^{∨}

∼=H_{−}dim(R)+i+sup(D)(RHom_{R}(M, D))^{∨}

∼=H_{−}dim(R)+i+sup(D)

(

RHom_{R}(

M,Σ^{sup(D)}ω_{R}))∨

∼=H_{−}_{dim(R)+i}(RHomR(M, ωR))^{∨}

∼= Ext^{dim(R)−i}_{R} (M, ωR)^{∨}.

Acknowledgment. The paper was received by editors on April 21, 2018, and it was accepted for publication on March 12, 2019.

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Hossein Faridian,

School of Mathematical and Statistical Sciences, Clemson University, SC 29634, USA.

E-mail: hfaridi@g.clemson.edu, h.faridian@yahoo.com

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