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Left-Derived Functors of the Generalized I-AdicCompletion and Generalized Local HomologyTran Tuan Nam aa Department of Mathematics-Informatics , Ho Chi Minh University of Pedagogy , Ho Chi MinhCity, VietnamPublished online: 18 Feb 2010.

To cite this article: Tran Tuan Nam (2010) Left-Derived Functors of the Generalized I-Adic Completion and Generalized LocalHomology, Communications in Algebra, 38:2, 440-453, DOI: 10.1080/00927870802578043

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Communications in Algebra®, 38: 440–453, 2010Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802578043

LEFT-DERIVED FUNCTORS OF THE GENERALIZED I-ADICCOMPLETION AND GENERALIZED LOCAL HOMOLOGY

Tran Tuan NamDepartment of Mathematics-Informatics, Ho Chi MinhUniversity of Pedagogy, Ho Chi Minh City, Vietnam

We study left-derived functors of the generalized I-adic completion and compute thesefunctors by generalized local homology functors.

Key Words: Linearly compact module; Local cohomology; Local homology.

2000 Mathematics Subject Classification: 13D45; 16E30; 13J99.

1. INTRODUCTION

Let R be a commutative ring and I an ideal of R. There are two importantfunctors in commutative algebra and algebraic geometry which are the I-torsionfunctor �I and the I-adic completion functor �I defined by �I�N� =

⋃t>0�0 �N It�

and �I�N� = lim←−t

N/ItN for each R-module N .

It should be noted that the I-torsion functor �I is left exact and its ith rightderived functor Hi

I is called the ith local cohomology functor with respect to I byGrothendieck [3, 9]. However, the I-adic completion functor �I is neither right norleft exact, so computing its left-derived functors is in general difficult. Some authorstried to compute these left-derived functors by local homology functors in [1, 5, 7,8, 20]. Naturally, they studied the generalized I-torsion functor �I�M�−� definedby �I�M�N� = lim−→

t

HomR�M/ItM�N� for each pair of R-modules M�N . The functor

�I�M�−� is also left exact, and its ith right derived functor HiI�M�−� is called the

ith generalized local cohomology functor [10].In [17, 18] we studied the generalized local homology for artinian R-modules

and linearly compact modules which are in some sense dual to generalized localcohomology and in fact a generalizition of the usual local homology. We know thatthe class of linearly compact modules is big, it contains important classes of modulesin algebra. Even its subclass of semidiscrete linearly compact modules containsartinian modules; moreover, it contains also finitely generated modules over acomplete ring. From the properties of (generalized) local homology, we have got

Received August 12, 2008; Revised October 19, 2008. Communicated by A. Singh.Address correspondence to Tran Tuan Nam, Department of Mathematics-Informatics, Ho Chi

Minh University of Pedagogy, 280 An Duong Vuong, District 5, Ho Chi Minh City, Vietnam; E-mail:namtuantran@gmail.com

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LEFT-DERIVED FUNCTORS 441

new properties by duality and extend some well-known properties of (generalized)local cohomology modules of Grothendieck to linearly compact modules.

In this article, we study the generalized I-adic completion functor �I�M�−�and its left-derived functors. The generalized I-adic completion �I�M�N� of M�N isdefined by �I�M�N� = lim←−

t

�R/It ⊗R M ⊗R N��Unfortunately, the functor �I�M�−� is

in general neither right nor left exact, thus computing its left-derived functors is alsonot easy.We can give only partial answers. The organization of the article is as follows.

In Section 2 we study some basic properties of the generalized I-adiccompletion functor �I�M�−� and its left-derived functors Li�I�M�−�. As thefunctor �I�M�−� is not right exact, in general, L0�I�M�−� �= �I�M�−�. Denoteby ��I

�R� the category of all R-modules M (and homomorphisms) such thatL0�I�M�N� � �I�M�N� for all R-modules N . For each R-module M , let ��I

�M�R�denote the category of all R-modules N (and homomorphisms) such thatL0�I�M�N� � �I�M�N�. We show in Theorem 2.3 that if M is an R-modulesuch that the system �ItMt is stationary, then M ∈ ��I

�R�, and if N is an R-modulesuch that the system �ItNt is stationary, then N ∈ ��I

�M�R� for each R-module M .This section ends with Theorem 2.9 for the Mayer–Vietoris sequence of the left-derived functors.

The last section is devoted to study the relations between the left-derivedfunctors Li�I�M�−� and the generalized local homology functors HI

i �M�−�. Undersome conditions on the modules the functors Li�I�M�−� can be computed by thethe generalized local homology functors HI

i �M�−�.If M ∈ ��I

�R�� then Li�I�M�N� � HIi �M�N� for all i ≥ 0 and all R-modules N

(Theorem 3.2). Let S be a multiplicative set of R, the co-localization of an R-moduleM with respect to S is the module SM = Hom�RS�M�. Let M be a finitely R-moduleand N a linearly compact R-module, then S�H

Ii �M�N�� � H

IRSi �MS� SN� for all

i≥ 0 (Proposition 3.13). As a consequence, we have CosR�HIi �M�N�� ⊆ SuppR�M� ∩

CosR�N� ∩ V�I� for all i ≥ 0 (Corollary 3.14). We also have Li�I�M�N� � HIi �M�N�

for all i ≥ 0 (Theorem 3.6).Finally, we study the vanishing of generalized local homology and

cohomology. Let �R��� be a local ring with the �-adic topology, M a finitelyR-module and N a linearly compact R-module, then H�

i �M�N� = 0 for all i > dimR(Theorem 3.16). By duality, we extend the vanishing of local cohomology to linearlycompact modules (Corollary 3.17).

2. THE GENERALIZED I-ADIC COMPLETIONAND LEFT-DERIVED FUNCTORS

Let I be an ideal of a commutative ring R and M�N R-modules. The I-adiccompletion of N is the module �I�N� = lim←−

t

N/ItN . This suggests the followingdefinition.

Definition 2.1. The generalized I-adic completion �I�M�N� of the R-modulesM�N is defined by

�I�M�N� = lim←−t

�R/It ⊗R M ⊗R N��

It is clear that �I�M�N� � �I�M ⊗R N� the I-adic completion of M ⊗R N .

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Let ��R� be the category of all R-modules and homomorphisms. We have abi-functor

�I�−�−� � ��R�×��R� −→ ��R��

For each R-module M� there is a covariant functor

�I�M�−� � ��R� −→ ��R��

Note that, in general, inverse limits are left exact but not right exact, and tensorproducts are not left exact. Therefore, the functor �I�M�−� is neither left norright exact. Let Li�I�M�−� be the ith left-derived functor of �I�M�−�. In general,L0�I�M�−� �= �I�M�−�; however, L0�I�M�−� is right exact.

Remark 2.2.

(i) �I�M�N� � �I�N�M�.(ii) When M = R� �I�M�N� = �I�N� the I-adic completion of N and Li�I�R�−� =

Li�I�−� the ith left-derived functor of �I�−�.

Denote by ��I�R� the category of all R-modules M (and homomorphisms)

such that L0�I�M�N� � �I�M�N� for all R-modules N , equivalently, the functor�I�M�−� is right exact. For each R-module M , let ��I

�M�R� denote the categoryof all R-modules N (and homomorphisms) such that L0�I�M�N� � �I�M�N�.It is clear that ��I

�R� and ��I�M�R� are the full subcategories of the category

of R-modules ��R� and preserve finite direct sums of R-modules. Moreover,if M ∈ ��I

�R�, then ��I�M�R� = ��R�. If P is a projective R-module, then

P ∈ ��I�M�R�.

The following theorem shows that the categories ��I�R� and ��I

�M�R� arebig, and they contain all artinian R-modules.

Theorem 2.3.

(i) Let M be an R-module such that the system �ItMt is stationary (i.e., there is apositive integer n such that ItM = InM for all t ≥ n), then M ∈ ��I

�R�.(ii) If N is an R-module such that the system �ItNt is stationary, then N ∈ ��I

�M�R�for each R-module M .

To prove Theorem 2.3, we need the following lemmas.

Lemma 2.4. Let f � G −→ N be a homomorphism of R-modules. If f is surjective,then the homomorphism �I�M� f� � �I�M�G� −→ �I�M�N� is also surjective.

Proof. The epi-morphism f � G −→ N induces an epi-morphism M ⊗R f � M ⊗R

G −→ M ⊗R N . Set L = KerM ⊗R f , we have M ⊗R N � �M ⊗R G�/L. Then theinduced homomorphism of inverse systems

�R/It ⊗R M ⊗R Gt −→ �R/It ⊗R �M ⊗R G�/Lt

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is surjective. It induces a short exact sequence

0 −→ �L/�L ∩ It�M ⊗R G��t −→ ��M ⊗R G�/It�M ⊗R G�t

−→ ��M ⊗R G�/�It�M ⊗R G�+ L�t −→ 0�

As the inverse system �L/�L ∩ It�M ⊗R G��t is a surjective system, pass to inverselimits, we get the homomorphism �I�M� f� which is surjective by [2, 10.2]. �

Replacing M with R, we have an immediate consequence.

Corollary 2.5. Let f � G −→ N be a homomorphism of R-modules. If f is surjective,then the homomorphism �I�f� � �I�G� −→ �I�N� is also surjective.

Lemma 2.6. Let G −→ Lf−→ N −→ 0 be an exact sequence of R-modules such that

the system �ItN is stationary. Then the following induced sequence

�I�G� −→ �I�L� −→ �I�N� −→ 0

is exact.

Proof. Set K = Ker f ; we have induced exact sequences

G −→ K −→ 0

0 −→ K −→ L −→ N −→ 0�

From 2.5, the first exact sequence induces an exact sequence

�I�G� −→ �I�K� −→ 0�

As in the proof of [5, 2.3], the second exact sequence gives an exact sequence

�I�K� −→ �I�L� −→ �I�N� −→ 0�

Therefore, the induced sequence

�I�G� −→ �I�L� −→ �I�N� −→ 0

is exact. �

We now can prove Theorem 2.3.

Proof of Theorem 2.3. (i) Let 0 −→ Kf−→ P

g−→ N −→ 0 be a short exactsequence with P projective. It induces a right exact sequence

M ⊗R K −→ M ⊗R P −→ M ⊗R N −→ 0�

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Then we have an induced sequence

�I�M�K��I �M�f�−→ �I�M� P�

�I �M�g�−→ �I�M�N� −→ 0 (∗)

in which Im�I�M� f� ⊆ Ker�I�M� g� and �I�M� g� is surjective by 2.4. So

L0�I�M�N� � �I�M� P�/Im�I�M� f�� �I�M�N� � �I�M� P�/Ker�I�M� g��

Thus the proof will be complete if we show that Im�I�M� f� = Ker�I�M� g�, whichmeans the sequence �∗� is exact. It should be noted that �I�M�K� � �I�M ⊗R K� forall R-modules M�K� From 2.6, we only need to prove that the system �It�M ⊗R N�is stationary. By the hypothesis, there is a positive integer n such that ItM = InM forall t ≥ n. For each n ∈ In� a ∈ M�b ∈ N , we have na ∈ InM = ItM . Then there ist ∈ It such that na = tc, for some c ∈ M . It follows n�a⊗ b� = na⊗ b = tc ⊗b = t�c ⊗ b�. Thus In�M ⊗R N� = It�M ⊗R N� for all t ≥ n, and part (i) is proved.

(ii) The argument is similar to that in the proof of part (i).�

From Theorem 2.3, we have the following immediate consequences.

Corollary 2.7.

(i) If M is an artinian R-module, then M ∈ ��I�R�.

(ii) If N is an artinian R-module, then N ∈ ��I�M�R� for each R-module M .

Corollary 2.8. Let M�N be R-modules. There is an epi-morphism

��M�N� � L0�I�M�N� −→ �I�M�N��

Proof. It follows from the proof of Theorem 2.3. �

We have the Mayer–Vietoris sequence for the left-derived functors in thefollowing theorem.

Theorem 2.9. Let R be a noetherian ring and M�N R-modules. If N is a finitelygenerated R-module, then there is an exact sequence

� � � Li�I∩J �M�N�→Li�I�M�N�⊕ Li�J�M�N�→Li�I+J �M�N�→Li−1�I∩J �M�N� � � � �

Proof. There is a short exact sequence of inverse sequences of R-modules

0 −→ �M/�ItM ∩ J tM�t�ftt−→�M/ItM ⊕M/JtMt

�gtt−→�M/�ItM + J tM�t −→ 0

defined by ft�m+ ItM ∩ J tM� = �m+ ItM�m+ J tM�� gt�m+ ItM� n+ J tM� = m−n+ �ItM + J tM� for all m�n ∈ M and all t > 0. Let F• be a free resolution of N with

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LEFT-DERIVED FUNCTORS 445

finitely generated free modules Fi �i = 0� 1� � � � �. As the functor F•⊗R- is exact, wehave an induced short exact sequence

0 → �M/�ItM ∩ J tM�⊗ F•t → ��M/ItM ⊕M/JtM�⊗ F•t

→ �M/�ItM + J tM�⊗ F•t → 0�

Since �M/�ItM ∩ J tM�⊗R F•t is a surjective system, it follows from [2, 10.2] a shortexact sequence

0 → lim←−t

�M/�ItM ∩ J tM�⊗ F•� → lim←−t

(�M/ItM ⊕M/JtM�⊗ F•

)→ lim←−

t

�M/�ItM + J tM�⊗R F•� → 0�

It is clear that

lim←−t

(�M/ItM ⊕M/JtM�⊗R F•

) � �I�M�F•�⊕�J�M�F•��

For all t > 0, we have

�I + J�2tM ⊆ ItM + J tM ⊆ �I + J�tM�

So lim←−t

�M/�ItM + J tM�� � lim←−t

�M/�I + J�tM�. Since inverse limits commute with

direct products and the free modules Fi are finitely generated, we get

lim←−t

�M/�ItM + J tM�⊗R F•� � lim←−t

�M/�ItM + J tM��⊗R F•

� lim←−t

�M/�I + J�tM�⊗R F•

� lim←−t

�M/�I + J�tM ⊗R F•� � �I+J �M�F•��

By the Artin–Rees lemma, for each t > 0, there is a positive integer c such thatIt+cM ∩ J tM = It�IcM ∩ J tM�. Then

It+cM ∩ J t+cM ⊆ It+cM ∩ J tM = It�IcM ∩ J tM� ⊆ ItJ tM ⊆ �I ∩ J�tM�

It follows

It+cM ∩ J t+cM ⊆ �I ∩ J�tM ⊆ ItM ∩ J tM�

Thus

lim←−t

�M/�ItM ∩ J tM�⊗R F•� � lim←−t

�M/�I ∩ J�tM ⊗R F•� � �I∩J �M�F•��

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Finally, we have a short exact sequence of complexes

0 −→ �I∩J �M�F•� −→ �I�M�F•�⊕�J�M�F•� −→ �I+J �M�F•� −→ 0�

which induces the exact sequence in the theorem. �

3. GENERALIZED LOCAL HOMOLOGY

The ith generalized local cohomology module HiI�M�N� of M�N with respect

to I was introduced by Herzog [10] defined by HiI�M�N� = lim−→

t

ExtiR�M/ItM�N�� In

[17], the ith generalized local homology module HIi �M�N� of M�N with respect to I

is defined by

HIi �M�N� = lim←−

t

TorRi �M/ItM�N��

This definition is in some sense dual to the definition of generalized localcohomology modules of Herzog. When i = 0� HI

0�M�N� � �I�M�N�.

Lemma 3.1 ([17, 2.3]). Let M�N be R-modules. Then:

(i) The generalized local homology module HIi �M�N� is I-separated for all i ≥ 0� i.e.,⋂

t>0

ItHIi �M�N� = 0�

(ii) Assume that �R��� is a local ring and M is a finite R-module. Then for all i ≥ 0,

HIi �M�D�N�� � D�Hi

I�M�N���

where D�N� = HomR�N� E�R/��� is the Matlis dual of N and E�R/�� is theinjective envelope of the residue ring R/�.

From 2.8 we have an epi-morphism ��M�N� � L0�I�M�N� −→ HI0�M�N��

More general, we have the relation between the modules Li�I�M�N� and HIi �M�N��

Theorem 3.2. Let M�N be R-modules. There is an epi-morphism

�i�M�N� � Li�I�M�N� −→ HIi �M�N�

for all i ≥ 0. Moreover, if M ∈ ��I�R�, then �i�M�N� is an isomorphism.

Proof. When i = 0, we have �0�M�N� = ��M�N� is the epi-morphism. Supposethat i > 0. Let

0 −→ K −→ Pi−1 −→ · · · −→ P0 −→ N −→ 0

be an exact sequence with Pj projective (j = 0� 1� � � � � i− 1�. By the basic propertyof left-derived functors, there is a left exact sequence of inverse systems

0 −→ �TorRi �M/ItM�N�t −→ �M/ItM ⊗R Kt −→ �M/ItM ⊗R Pi−1t�

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LEFT-DERIVED FUNCTORS 447

As inverse limits are left exact, it induces an exact sequence

0 −→ HIi �M�N� −→ �I�M�K� −→ �I�M� Pi−1��

Again, by the basic property of left-derived functors, there is a commutativediagram with the exact rows

Note that ��M�K� is surjective and ��M�Pi−1� is an isomorphism as Pi−1 isprojective. It follows the induced homomorphism �i�M�N� is surjective.

Moreover, if M ∈ ��I�R�� then ��M�K� is an isomorphism. Therefore,

�i�M�N� is also an isomorphism. �

From Theorem 3.2, we have an immediate consequence.

Corollary 3.3. Let 0 −→ N ′ −→ N −→ N ′′ −→ 0 be a short exact sequence ofR-modules and M ∈ ��I

�R�. Then there is a long exact sequence of generalizedhomology modules

· · · −→ HIi+1�M�N ′′� −→ HI

i �M�N ′� −→ HIi �M�N� −→ HI

i �M�N ′′� −→ � � � �

Corollary 3.4. Let M�N be R-modules. If M ∈ ��I�R�, then the module Li�I�M�N�

is I-separated (i.e.,⋂

t>0 ItLi�I�M�N� = 0� for all i ≥ 0.

Proof. By [17, 2.3], HIi �M�N� is I-separated. Thus the result follows from 3.2. �

Proposition 3.5. Let M be an R-module and 0 −→ Kf−→ G

g−→ N −→ 0 a shortexact sequence in which N ∈ ��I

�M�R�. We have the following statements:

(i) There is an exact sequence

�I�M�K� −→ �I�M�G� −→ �I�M�N� −→ 0�

(ii) If G is projective, then there is an exact sequence

0 −→ HI1�M�N� −→ �I�M�K� −→ �I�M�G� −→ �I�M�N� −→ 0�

Proof. (i) There is a commutative diagram

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in which the first row is exact. By the hypothesis, ��M�N� is an isomorphism,and ��M�G� is surjective. We have to prove the second row is also exact.Indeed, we have Im�I�M� f� ⊆ Ker�I�M� g� and �I�M� g� is surjective by 2.4. Letx ∈ Ker�I�M� g�, there is x′ ∈ L0�I�M�G� such that ��M�G��x′� = x and then��M�N�L0�I�M� g��x′� = �I�M� g���M�G��x′� = 0. It follows L0�I�M� g��x′� = 0�so x′ ∈ Im L0�I�M� f�. Then there is y ∈ L0�I�M�K� and L0�I�M� f��y�= x′.We now have �I�M� f��i�M�K��y� = ��M�G�L0�I�M� f��y� = x, that means�I�M� f���i�M�K��y�� = x. Therefore, x ∈ Im�I�M� f�, and then Im�I�M� f� =Ker�I�M� g�.

(ii) As G is projective, Li�I�M�G� = 0 for all i > 0 and ��M�G� �L0�I�M�G� −→ �I�M�G� is an isomorphism. Then there is a commutative diagramwith the exact first row

From part (i), �I�M� g� is surjective and Im�I�M� f� = Ker�I�M� g�� By the leftexactness of inverse limits, is injective. Moreover, analysis similar to that in theproof of part (i), we have Im = Ker�I�M� f�� The proof is complete.

�

From now on R is a noetherian commutative ring and has a topologicalstructure. Let us recall the concept of linearly compact modules by terminologyof Macdonald [13]. Let M be a topological R-module. A nucleus of M is aneighbourhood of the zero element of M� and a nuclear base of M is a base forthe nuclei of M� M is Hausdorff if and only if the intersection of all the nuclei ofM is 0. M is said to be linearly topologized if M has a nuclear base � consistingof submodules. A Hausdorff linearly topologized R-module M is said to be linearlycompact if M has the following property: if � is a family of closed cosets (i.e., cosetsof closed submodules) in M which has the finite intersection property, then thecosets in � have a non-empty intersection. It should be noted that an artinianR-module is linearly compact with the discrete topology [13, 3.10].

A Hausdorff linearly topologized R-module M is called semidiscrete if everysubmodule of M is closed. Thus a discrete R-module is semidiscrete. The classof semidiscrete linearly compact modules contains all artinian modules. Moreover,it also contains all finitely generated modules in case R is a complete localring [13, 7.3].

Theorem 3.6. Let M be a finitely generated R-module and N a linearly compactR-module; then

Li�I�M�N� � HIi �M�N�

for all i ≥ 0.

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LEFT-DERIVED FUNCTORS 449

Proof. Let F• be a free resolution of N with finitely generated free modulesFi �i= 0� 1� � � � �. There are complexes M/ItM ⊗R F• for all t > 0. By [23, 3.5.8],we get a short exact sequence

0 −→ 1lim←−t

TorRi+1�M/ItM�N� −→ Li�I�M�N� −→ HIi �M�N� −→ 0�

On the other hand, it follows from [7, 2.6] that �TorRi+1�M/ItM�N�t forms an inversesystem of linearly compact modules with continuous homomorphisms. Then

1lim←−t

TorRi+1�R/It�M� = 0

by [12, 7.1]. Thus we get the isomorphism in the theorem. �

We have the immediate consequence of 3.6 which is useful to study generalizedlocal homology functors.

Corollary 3.7. Let M be a finitely generated R-module. If

0 −→ N ′ −→ N −→ N ′′ −→ 0

is a short exact sequence of linearly compact modules, then we have a long exactsequence of generalized local homology modules

· · · −→ HIi �M�N ′� −→ HI

i �M�N� −→ HIi �M�N ′′� −→

· · · −→ HI0�M�N ′� −→ HI

0�M�N� −→ HI0�M�N ′′� −→ 0�

Corollary 3.8. Let M be a finitely generated R-module and N a linearly compactR-module, then N ∈ ��I

�M�R�.

Corollary 3.9. Let M be a finitely generated R-module and N a semi-discrete linearlycompact R-module. Then Li���M�N� is a noetherian R̂-module for all i ≥ 0.

Proof. By [18, 3.8], HIi �M�N� is a noetherian R̂-module for all i ≥ 0. Thus the result

follows from 3.6. �

Let S be a multiplicative set of R� Following [15] the co-localization of anR-module M with respect to S is the module SM = Hom�RS�M�. In case M is anArtinian R-module, SM may not Artinian [15, 4], however, SM is a linearly compactR-module [6, 3.2(i)]. Let � be a prime of R and S = R− ��, then instead of SMwe write �M . For an R-module M , Co-support of M to be the set CosR�M� = �� ∈Spec�R� �M �= 0.

Lemma 3.10 ([6, 3.3]). Let S be a multiplicative set and I an ideal of R such thatS ∩ I �= ∅. If M is a I-separated R-module (i.e.,

⋂t>0 I

tM = 0), then SM = 0.

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450 NAM

Corollary 3.11. Let S be a multiplicative set and I an ideal of R such that S ∩ I �= ∅.Then

S�HIi �M�N�� = 0

for all R-modules M�N and i ≥ 0.

Proof. It follows from 3.10 and 3.1(i). �

The following lemma is used to prove Proposition 3.13.

Lemma 3.12 ([16, 3.5(ii)]). Let M be a finitely R-module and N a linearly compactR-module. Then

S�TorRi �M�N�� � TorRS

i �MS� SN��

The following result is the proposition for co-localization of generalized localhomology modules.

Proposition 3.13. Let M be a finitely R-module and N a linearly compact R-module.Then

S�HIi �M�N�� � H

IRSi �MS� SN�

for all i ≥ 0.

Proof. From [22, 2.25], the co-localization functor S�−� preserves inverse limits,then

S�HIi �M�N�� = S

(lim←−t

TorRi �M/ItM�N�

)

� lim←−t

S�TorRi �M/ItM�N���

We now have by 3.12,

S�HIi �M�N�� � lim←−

t

TorRSi �MS/�IRS�

tMS� SN�

= HIRSi �MS� SN�

as required. �

Let V�I� be the set of all primes of R containing the ideal I� we have thefollowing corollary.

Corollary 3.14. Let M be a finitely R-module and N a linearly compact R-module.Then

CosR�HIi �M�N�� ⊆ SuppR�M� ∩ CosR�N� ∩ V�I�

for all i ≥ 0�

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LEFT-DERIVED FUNCTORS 451

Proof. Let � ∈ CosR�HIi �M�N��, we have an isomorphism for all i ≥ 0 by 3.13,

��HIi �M�N�� � H

IR�i �M�� �N��

It follows HIR�i ��M� �N� �= 0, hence M� �= 0, �N �= 0. Moreover, by 3.11, � ∈ V�I�.

Therefore, � ∈ SuppR�M� ∩ CosR�N� ∩ V�I�� �

To prove Theorem 3.16, we need the following lemma. Note that if N is anartinian module over a local ring R� then N has a natural structure as an artinianmodule over R̂ [19, 1.11]. Denote by M̂ the �-adic completion of M�

Lemma 3.15. Let �R��� be a local ring with the �-adic topology, M a finitelygenerated R-module, and N an artinian R-module. Then

HIR̂i

(M̂� N

) � HIi �M�N�

for all i ≥ 0.

Proof. It follows from [22, 11.53] that

TorR̂i(M/ItM ⊗R R̂� N

) � TorRi(M/ItM�N ⊗R̂ R̂

)for all i ≥ 0. By [14, 8.7, 8.11] we have M/ItM ⊗R R̂ � M̂/�IR̂�tM̂ . Then

TorR̂i(M̂/�IR̂�tM̂� N

) � TorRi �M/ItM�N��

Pass to inverse limits, we get the isomorphism in the lemma. �

The following result is the vanishing of generalized local homology modules.

Theorem 3.16. Let �R��� be a local ring with the �-adic topology, M a finitelyR-module, and N a linearly compact R-module. Then

H�i �M�N� = 0

for all i > dimR.

Proof. We first prove in the special case N is an artinian R-module. From 3.15,we may assume in this case that �R��� is a complete ring. By the Matlis duality,DD�N� � N and D�N� is a finitely generated R-module. In virtue of 3.1(ii), wehave H�

i �M�N� � D�Hi��M�D�N���. By [11, 3.2], Hi

��M�D�N�� = 0 for all i > dimR.Hence H�

i �M�N� = 0 for all i > dimR.Let N be a linearly compact R-module. Denote by � a nuclear base of N .

It follows from [13, 4.7] that N = lim←−U ∈�

�N/U�, where the modules N/U�U ∈ ��

are artinian R-modules. In virtue of [18, 3.6], H�i �M�N� � lim←−

U∈�H�

i �M�N/U�. By

our claim above H�i �M�N/U� = 0 for all i > dimR and all U ∈ �. Therefore,

H�i �M�N� = 0 for all i > dimR. �

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452 NAM

Let �R��� be a local ring with the �-adic topology. A Hausdorff linearlytopologized R-module is �-primary if each element of M is annihilated by a powerof � [13, 5]. A Hausdorff linearly topologized R-module M is linearly discrete ifevery �-primary quotient of M is discrete. It should be noted that If M is linearlydiscrete, then M is semidiscrete. In case �R��� is a complete ring, the class oflinearly discrete R-modules contain all finitely generated R-modules.

Recall that in [11, 3.2], Herzog and Zamani proved that generalized localcohomology modules Hi

��M�N� of finitely generated R-modules M�N are equal tozero for all i > dimR. From Theorem 3.16, we have the vanishing of generalizedlocal cohomology modules Hi

��M�N� which is a slight extension of [11, 3.2].

Corollary 3.17. Let �R��� be a local ring with the �-adic topology, M a finitelyR-module, and N a linearly discrete R-module. Then

Hi��M�N� = 0

for all i > dimR.

Proof. Let us first prove in the case �R��� is a complete ring. It follows from[13, 6.2, 9.13] that D�N� is a linearly compact R-module. From 3.16, we haveH�

i �M�D�N�� = 0 for all i > dimR. By 3.1(ii), D�Hi��M�N�� = 0 for all i > dimR.

Then Hi��M�N� = 0 for all i > dimR.

Now let �R��� be a local ring. From [21, 1.3,1.5,1.6], we have an isomorphismDR̂�H

i��M�N�� � DR̂�H

i�̂�M̂� N̂ ��. By our claim above DR̂�H

i�̂�M̂� N̂ �� = 0 and so

DR̂�Hi��M�N�� = 0 for all i > dimR. Therefore, Hi

��M�N� = 0 for all i > dimR. �

ACKNOWLEDGMENTS

The author acknowledges support by the Abdus Salam International Centrefor Theoretical Physics, Trieste, Italy, and the Basis Research Program in NaturalScience of Vietnam.

REFERENCES

[1] Alonso, L. T., Jeremias, A. L., Lipman, J. (1997). Local homology and cohomologyon schemes. Ann. Scient. Ec. Norm. Sup. 4c Serie 1(30):1–39.

[2] Atiyah, M. F., Macdonald, I. G. (1969). Introduction to Commutative Algebra. London:Addison Wesley.

[3] Brodmann, M. P., Sharp, R. Y. (1998). Local Cohomology: An Algebraic Introductionwith Geometric Applications. London: Cambridge University Press.

[4] Cuong, N. T., Hoang, N. V. (2008). On the vanishing and the finiteness of supportsof generalized local cohomological modules. Manuscripta Math. 126:59–72.

[5] Cuong, N. T., Nam, T. T. (2001). The I-adic completion and local homology forArtinian modules. Math. Proc. Camb. Phil. Soc. 131:61–72.

[6] Cuong, N. T., Nam, T. T. (2001). On the co-localization, co-support and co-associatedprimes of local homology modules. Vietnam J. Math. 29(4):359–368.

[7] Cuong, N. T., Nam, T. T. (2008). A local homology theory for linearly compactmodules. J. Algebra 319:4712–4737.

Dow

nloa

ded

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The

Aga

Kha

n U

nive

rsity

] at

07:

53 1

0 O

ctob

er 2

014

LEFT-DERIVED FUNCTORS 453

[8] Greenlees, J. P. C., May, J. P. (1992). Derived functors of I-adic completion and localHomology. J. Algebra 149:438–453.

[9] Grothendieck, A. (1966). Local Cohomology. Lect. Notes in Math. (20). Berlin–Heidelberg–New York: Springer-Verlag.

[10] Herzog, J. (1970). Komplexe, Auflösungen und dualität in der localen algebra.Habilitationschrift Univ. Regensburg. (Doctoral thesis, 165 pgs.)

[11] Herzog, J., Zamani, N. (2003). Duality and vanishing of generalized local cohomology.Arch. Math. 81:512–519.

[12] Jensen, C. U. (1972). Les Foncteurs Dérivés de lim←− et leurs Applications en Théorie desModules. Berlin–Heidelberg–New York: Springer-Verlag, 1972.

[13] Macdonald, I. G. (1962). Duality over complete local rings. Topology 1:213–235.[14] Matsumura, H. (1986). Commutative Ring Theory. Cambridge University Press.[15] Melkersson, L., Schenzel, P. (1995). The co-localization of an Artinian module.

proceed. Ed. Math. Soc. 38:121–131.[16] Nam, T. T. (2008). Co-support and coartinian modules. Algebra Colloquium

15(1):83–96.[17] Nam, T. T. Generalized local homology for artinian modules. Preprint.[18] Nam, T. T. Generalized local homology and cohomology for linearly compact

modules. Preprint.[19] Sharp, R. Y. (1989). A method for the study of Artinian modules with an application

to asymptotic behavior. In: Commutative Algebra (Math. Siences Research Inst. Publ.New York: Springer-Verlag), 15:443–465.

[20] Simon, A. M. (1990). Some homological properties of complete modules. Math. Proc.Cambr. Phil. Soc. 108:231–246.

[21] Suzuki, N. (1978). On the generalized local cohomology and its duality. J. Math. KyotoUniv. (JAKYAZ) 18-1:71–85.

[22] Rotman, J. J. (1979). An Introduction to Homological Algebra. London: Academic Press.[23] Weibel, C. A. (1994). An Introduction to Homological Algebra. Cambridge University

Press.[24] Yassemi, S. (1994). Generalized section functors. J. Pure and Applied Algebra

95:103–119.

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