Communications in Algebra®, 37: 1748–1757, 2009Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802216396

A FINITENESS RESULT FOR CO-ASSOCIATEDAND ASSOCIATED PRIMES OF GENERALIZEDLOCAL HOMOLOGY AND COHOMOLOGY MODULES

Tran Tuan NamDepartment of Mathematics – Informatics, Ho Chi Minh Universityof Pedagogy, Ho Chi Minh City, Vietnam

We show that if M is a finitely generated R-module, N is an I-stable semi-discrete linearly compact R-module and G is a closed R-submodule of the generalizedlocal homology module HI

i �M�N� such that HIj �M�N� is I-stable for all j < i and

HIi �M�N�/G is I-stable, then the set Coass G is finite. As an immediate consequence,

the first non-I-stable generalized local homology module of N�M with respect to I

has only finitely many co-associated primes. By duality, we get a new result for thefiniteness of associated primes of (generalized) local cohomology modules.

Key Words: Associated prime; Co-associated prime; (Generalized) Local homology; (Generalized)Local cohomology; Linearly compact module.

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

1. INTRODUCTION

The local cohomology theory of Grothendieck has proved to be an importanttool in algebraic geometry, commutative algebra, and algebraic topology. Its dualtheory of local homology is also studied by many mathematicians: Greenlees andMay (1992), Tarrio et al. (1997), and Cuong and Nam (2001), etc. Recently, in2006 we have studied the generalized local homology for linearly compact moduleswhich is in some sense dual to Herzog’s (1970) generalized local cohomology andin fact a generalized one of the usual local homology. It should be mentionedthat the class of linearly compact modules is great, it contains important classes ofmodules in algebra. For example, artinian modules are linearly compact and discrete(Macdonald, 1962, 3.10). Moreover, if R is a complete local noetherian ring and M isa finitely generated R-module, then M is semi-discrete (that means every submodulesof M is closed) and linearly compact (Macdonald, 1962, 7.3). Note that (generalized)local cohomology was essentially studied on finitely generated modules. Fromproperties of (generalized) local homology, by duality, we get new properties andextend some well-known properties of (generalized) local cohomology modules ofGrothendieck on linearly compact modules.

Received October 19, 2007; Revised April 1, 2008. Communicated by I. Swanson.Address correspondence to Nam Tran, Department of Mathematics – Informatics, 280 An

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

1748

A FINITENESS RESULT 1749

Huneke (1992) asked the interesting question: If M is finitely generated, is thenumber of associated primes of local cohomology modules Hi

I�M� always finite?Some authors studied about that such as: Brodmann and Faghani (2000), Cuongand Nam (2001), Divaani-Aazar and Mafi (2004) etc. It is clear that the finitenessof associated primes of (generalized) local cohomology modules is closely related tothe finiteness of co-associated primes of (generalized) local homology modules. Theaim of this article is to study the finiteness of co-associated and associated primesof (generalized) local homology and cohomology modules. The organization of thearticle is as follows.

Section 2 is devoted to recall some basic properties of linearly compactmodules and generalized local homology modules that we shall use.

In the last section, we define an I-stable module as a module M suchthat for each element x ∈ I� there is a positive integer n and xtN = xnN for allt≥ n. Theorem 3.3 shows that if M is a finitely generated R-module, N is anI-stable semidiscrete linearly compact R-module, and G is a closed R-submodule ofHI

i �M�N� such that HIj �M�N� is I-stable for all j < i and HI

i �M�N�/G is I-stable,then the set CoassG is finite. As an immediate consequence, we deduce that thefirst non-I-stable generalized local homology module of N�M with respect to I

has only finitely many co-associated primes (Corollary 3.4). By duality, we get anew result for the finiteness of associated primes of (generalized) local cohomologymodules (Corollaries 3.5, 3.6). Note that the finiteness of associated primes of localcohomology modules is closely related to the local-global-principle for finitenessdimensions of Faltings (1978).

Throughout this article, �R��� is a local noetherian commutative (nonzeroidentity) ring with the �-adic topology.

2. PREMILINARIES

To establish the context, we recall briefly definitions and basic properties oflinearly compact modules and generalized local homology modules that we shalluse.

We first recall the concept of linearly compact modules by terminology ofMacdonald (1962). Let M be a topological R-module. M is said to be linearlytopologized if M has a base of neighborhoods of the zero element � consisting ofsubmodules. M is called Hausdorff if the intersection of all the neighborhoods ofthe zero element is 0. A Hausdorff linearly topologized R-module M is said to belinearly compact if � is a family of closed cosets (i.e., cosets of closed submodules) inM which has the finite intersection property, then the cosets in � have a nonemptyintersection.

It is clear that artinian R-modules are linearly compact and discrete, moreoverthe only linear topology that an artinian module can carry is the discrete topology(Macdonald, 1962, 3.10).

If N is a closed submodule of a Hausdorff linearly topologized R-moduleM� then M is linearly compact if and only if N and M/N are linearly compact(Macdonald, 1962, 3.5). If �Mi�i∈I is a family of linearly compact R-modules, then∏

i∈I Mi is linearly compact with the product topology (Macdonald, 1962, 3.6).

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Lemma 2.1 (Macdonald, 1962, 3.13). Let M be a linearly compact R-module, N aclosed submodule of M� �Pi� a family of closed submodules of M , such that for eachpair Pi� Pj , there is a Pk ⊆ Pi ∩ Pj . Then

⋂i�N + Pi� = N +⋂

i Pi.

Denote by lim←−i

t

then ith right derived functor of the inverse limit lim←−t

� If �Mt�

is an inverse system of linearly compact modules with continuous homomorphisms,the lim←−1

t

Mt = 0 by Jensen (1972, 7.1). Therefore, we have the following immediate

result.

Lemma 2.2. Let 0 −→ �Mt� −→ �Nt� −→ �Pt� −→ 0 be a short exact sequence ofinverse systems of R-modules. If �Mt� is an inverse system of linearly compact moduleswith continuous homomorphisms, then the sequence of inverse limits

0 −→ lim←−t

Mt −→ lim←−t

Nt −→ lim←−t

Pt −→ 0

is exact.

Let I be an ideal of the ring R and M�N R-modules. The ith generalized localhomology module HI

i �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 Herzog’s (1970) definition of generalizedlocal cohomology nodules and in fact a generalized one of the usual local homology.Denote by �I�N� = lim←−

t

N/ItN the I-adic completion of N . When i = 0,

HI0�M�N� � lim←−

t

�R/It ⊗R M ⊗R N�

� lim←−t

�M ⊗R N�/It�M ⊗R N� � �I�M ⊗R N��

Following is some basic properties of generalized local homology modules.

Lemma 2.3 (Nam, 2006, 3.3(i)). If M is a finitely generated R-module and N isa linearly compact R-module, then for all i ≥ 0� HI

i �M�N� is a linearly compactR-module.

We have the positive strongly connected sequence of functors �HIi �M�−�� on

the category of linearly compact R-module.

Lemma 2.4 (Nam, 2006, 3.5, 3.7). Let M be a finitely generated R-module. If

0 −→ N ′ −→ N −→ N ′′ −→ 0

A FINITENESS RESULT 1751

is a short exact sequence of linearly compact modules with the continuoushomomorphisms, then we have a long exact sequence of generalized local homologymodules with the continuous homomorphisms

· · · → HII+1�M�N ′′� → HI

i �M�N ′� → HIi �M�N� → HI

i �M�N ′′� → · · · �By the same argument as in the proof of Cuong and Nam (2001, 3.3), but

replacing the module R/It with the module M/ItM� we have the following lemma.

Lemma 2.5. 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 M is a finitely generated 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/�� theinjective envelope of the residue ring R/�.

Lemma 2.6 (Nam, 2006, 3.12). Let M be a finitely generated module and N alinearly compact R-module. If N is complete in I-adic topology (i.e., �I�N� � N), thenthere is an isomorphism for all i ≥ 0�

TorRi �M�N� � HIi �M�N��

A Hausdorff linearly topologized R-module M is called semidiscrete if everysubmodule of M is closed (Macdonald, 1962, 5.1). Thus a discrete R-module issemi-discrete. The class of semidiscrete linearly compact R-modules contains allartinian R-modules. Moreover, in case R is a complete local noetherian ring, R issemidiscrete and linearly compact as R-module, since R = lim←−

t

R/�t and each R/�t

is artinian (Macdonald, 1962, 3.7, 6.2, 6.4). Let M be a finitely generated moduleover the complete local noetherian ring R, we have a continuous epimorphismRn −→ M which is an open map, hence M is semidiscrete and linearly compact(Macdonald, 1962, 7.3). If the finitely generated R-module M is discrete, then M isartinian and then has finite length (see Macdonald, 1962, 5.3, 5.4). Therefore, theclass of semidiscrete linearly compact modules contains also all finitely generatedmodules over a complete ring.

If an R-module N can be written as a finite sum of sum-irreducible R-modules,then IN = N if and only if there is an x ∈ I such that xN = N (Chambless,1981, 2.9). It follows from Zöschinger (1983, 1(L3)) that every semidiscrete linearlycompact R-modules satisfies the condition of Chambless (1981, 2.9), hence we havethe following lemma.

Lemma 2.7 (Nam, 2006, 3.9). Let N be a semidiscrete linearly compact R-module.Then IN = N if and only if there is an x ∈ I such that xN = N�

1752 NAM

Lemma 2.8. Let M be a finitely generated R-module and N a semidiscrete linearlycompact R-module. Then for all i ≥ 0�TorRi �M�N� is a semidiscrete linearly compactR-module.

Proof. We first note that if F is a finitely generated free R-module, then F ⊗R

N � Nn for some positive integer n. By Macdonald (1962, 3.6), F ⊗R N is linearlycompact and semidiscrete, as N is a semidiscrete linearly compact R-module. LetF• be a free resolution of M with finitely generated free R-modules Fi� we have acomplex F• ⊗R N of semidiscrete linearly compact R-modules with the continuousdifferentiations. Therefore, TorRi �M�N� = Hi�F• ⊗R N� a quotient of submodulesof the semidiscrete linearly compact module Fi ⊗R N� Finally, TorRi �M�N� is asemidiscrete linearly compact R-module for all i ≥ 0� �

3. THE FINITENESS OF CO-ASSOCIATEDAND ASSOCIATED PRIMES

Let us first recall the concept of co-associated primes (Chambless, 1981;Yassemi, 1995; Zöschinger, 1983, etc.). A prime ideal � is called co-associatedto a nonzero R-module M if there is an artinian homomorphic image T of Mwith � = AnnRT� The set of co-associated primes of M is denoted by CoassR�M�.It should be noted that, if M is a semidiscrete linearly compact R-module, then theset CoassR�M� is finite (Zöschinger, 1983, 1(L4)). If 0 −→ N −→ M −→ K −→ 0is an exact sequence of R-modules, then CoassR�K� ⊆ CoassR�M� ⊆ CoassR�N� ∪CoassR�K� (Yassemi, 1995, 1.10).

To state the finiteness of co-associated primes of generalized local homologymodules, we need the definition of I-stable modules.

Definition 3.1. An R-module N is called I-stable if for each element x ∈ I� thereis a positive integer n such that xtN = xnN for all t ≥ n�

There are many I-stable modules. For example, for an R-module M thequotient module N = M/IM is I-stable. Especially, every artinian module is discretelinearly compact and I-stable.

Lemma 3.2. Let 0 −→ M −→ N −→ P −→ 0 be a short exact sequence in whichthe modules M�N� P are I-separated. Then the module N is I-stable if and only if themodules M�P are I-stable.

Proof. �⇒� Assume that N is I-stable. For each element x ∈ I� there is a positiveinteger n such that xnN = ⋂

t>0 xtN = 0� It follows xnM = xnP = 0� thus M�P are

I-stable.

�⇐� As M�P are I-stable, for each element x ∈ I� there is a positive integer msuch that xmM = xmP = 0� By Brodmann and Sharp (1998, 9.1.1), there is a positiveinteger r such that xrN = 0� Hence N is I-stable. �

Theorem 3.3. Let M be a finitely generated R-module and N an I-stable semidiscretelinearly compact R-module. Let i be a non-negative integer. If HI

j �M�N� is I-stable for

A FINITENESS RESULT 1753

all j < i and G is a closed R-submodule of HIi �M�N� such that HI

i �M�N�/G is I-stable,then the set CoassG is finite.

Proof. We prove by induction on i.When i = 0� as M is a finitely generated, M ⊗R N is a semidiscrete linearly

compact module by 2.8. Now the canonical epimorphism of inverse systems oflinearly compact modules �M ⊗R N�t −→ ��M ⊗R N�/It�M ⊗R N��t induces by 2.2and Macdonald (1962, 5.5) a continuous (and open) surjective homomorphismM ⊗R N −→ HI

0�M�N�� Thus HI0�M�N� can be regarded as a quotient module of the

semidiscrete linearly compact module M ⊗R N� so is semidiscrete linearly compact.If G is a submodule of HI

0�M�N�� G is also semidiscrete linearly compact. HenceCoassG is finite.

Let i > 0� Note that R is the noetherian ring, I is finitely generated. From 3.1there is a positive integer n such that ItN = InN for all t ≥ n� Set K = InN� fromMacdonald (1962, 3.14) K is linearly compact. Now the short exact sequence oflinearly compact R-modules with continuous homomorphisms

0 −→ K −→ N −→ N/K −→ 0

induces by 2.4 an exact sequence of generalized local homology modules withcontinuous homomorphisms

HIi+1�M�N/K� −→ HI

i �M�K�f−→ HI

i �M�N�g−→ HI

i �M�N/K�� (∗)It is clear that N/K is complete in I-adic topology. From 2.6, there is anisomorphism TorRi �M�N/K� � HI

i �M�N/K� for all i ≥ 0� As M is finitely generatedand N/K is I-stable, TorRi �M�N/K� is I-stable and so is HI

i �M�N/K� for all i ≥ 0�By the hypothesis, HI

j �M�N� is I-stable for all j < i� In virtue of 3.2, HIj �M�K� is

also I-stable for all j < i�We now prove that if S is a closed submodule of HI

i �M�K� such thatHI

i �M�K�/S is I-stable, then Coass S is finite. Indeed, as IK = K� there is an elementy ∈ I such that yK = K by 2.7. Since HI

i �M�K�/S is I-stable, there is a positiveinteger r such that

yr(HI

i �M�K�/S) = ⋂

t>0

yt�HIi �M�K�/S� �

(⋂t>0

�ytHIi �M�K�+ S�

)/S�

From 2.3 and Macdonald (1962, 3.14), HIi �M�K� is a linearly compact module

and �ytHIi �M�K� is a family of closed submodules of HI

i �M�K�� Now, combining2.1 with 2.5(i) yields

⋂t>0�y

tHIi �M�K�+ S� � ⋂

t>0 ytHI

i �M�K�+ S = S� It followsyr�HI

i �M�K�/S� = 0 and then yrHIi �M�K� ⊆ S. Set yr = x� the short exact sequence

of linearly compact modules

0 −→ 0 �K x −→ K�x−→ K −→ 0

induces an exact sequence of generalized homology modules

· · · −→ HIi �M�K�

�x−→ HIi �M�K�

−→ HIi−1�M� 0 �K x�

−→ HIi−1�M�K� −→ · · · �

1754 NAM

It follows from Zöschinger (1983, Co. 1(b0)) that 0 �K x is artinian, so isI-stable and semidiscrete linearly compact. As HI

j �M�K� is I-stable for all j < i�HI

j �M� 0 � Kx� is also I-stable, for all j < i− 1� On the other hand, we have an exactsequence

HIi �M�K�/S

−→ HIi−1�M� 0 � Kx�/�S�

−→ HIi−1�M�K��

in which the homomorphisms � is induced from the homomorphisms and �Since HI

i �M�K�/S and HIi−1�M�K� are I-stable, HI

i−1�M� 0 � Kx�/�S� is also I-stable.From Macdonald (1962, 3.4), �S� is closed. Then the set Coass �S� is finiteby the inductive hypothesis. Let � ∈ Coass S − Coass �S�, we will show that � ∈Coass�HI

i �M�K��� Indeed, we can write � = Ann�S/L� for some artinian quotientmodule S/L� As xHI

i �M�K� ⊆ S� there is a short exact sequence

0 −→ �xHIi �M�K�+ L�/L −→ S/L

�−→ �S�/�L� −→ 0�

in which � is defined by ��u+ L� = �u�+ �L� for all u+ L ∈ S/L� It follows fromYassemi (1995, 1.10) that

Coass�S/L� ⊆ Coass��xHIi �M�K�+ L�/L� ∪ Coass��S�/�L��

⊆ Coass��xHIi �M�K�+ L�/L� ∪ Coass �S��

It is clear that � ∈ Coass S/L and � � Coass �S�, so

� ∈ Coass��xHIi �M�K�+ L�/L��

Moreover, we have an epimorphism

HIi �M�K�

�x−→ �xHIi �M�K�+ L�/L�

It follows that � ∈ Coass�HIi �M�K�� and then

Coass S − Coass �S� ⊆ Coass�HIi �M�K���

Thus Coass S will be finite if we show that Coass�HIi �M�K�� is finite. Indeed,

we have HIi−1�M� 0 �K x�/Im � Im ⊆ HI

i−1�M�K� an I-stable module. ThenHI

i−1�M� 0 �K x�/Im is I-stable. By the hypothesis, Coass Im is finite, that meansCoass HI

i �M�K�/xHIi �M�K� is finite. Set H = HI

i �M�K� and let � ∈ CoassRH� thereis an artinian quotient R-module H/W of H with � = Ann�H/W�� Remind that,by Macdonald (1962, 3.10), the topology on the artinian module H/W is onlythe discrete topology. Thus W is open, so is closed. Hence there is a positiveinteger s such that xs�H/W� = ⋂

t>0 xt�H/W� � �

⋂xtH +W�/W = 0 by 2.1 and 2.5.

Thus xsH ⊆ W� so H/W may be considered as an artinian quotient R-module ofH/xsH� Then � ∈ CoassR�H/xsH�� that means CoassR�H� ⊆ CoassR�H/xsH�� Fromthe epimorphim H/xH

�x−→ xH/x2H , we have CoassR�xH/x2H� ⊆ CoassR�H/xH��Now, the short exact sequence

0 −→ xH/x2H −→ H/x2H −→ H/xH −→ 0

A FINITENESS RESULT 1755

gives CoassR�H/x2H� ⊆ CoassR�xH/x2H� ∪ CoassR�H/xH� ⊆ CoassR�H/xH�� soCoassR�H/x

2H� is finite. We continue in this fashion obtaining CoassR�H/xsH� isfinite, and then CoassR�H� is finite. It follows Coass S is finite.

We now have exact sequences induced from the exact sequence

0 −→ HIi �M�K�/f−1�G�

f−→ HIi �M�N�/G

0 −→ ff−1�G� −→ G −→ g�G� −→ 0� (∗)

By the hypothesis, HIi �M�N�/G is I-stable, so is HI

i �M�K�/f−1�G�� ThenCoass f−1�G� is finite by the argument above, so is Coass ff−1�G�� Moreover,g�G� is an submodule of the module HI

i �M�N/K�� Recall that HIi �M�N/K� �

TorRi �M�N/K� and TorRi �M�N/K� is semidiscrete linearly compact by 2.8, thusCoass g�G� is finite. Finally, the finiteness of CoassG follows from the last shortexact sequence. �

Corollary 3.4. Let M be a finitely generated R-module and N an I-stable semidiscretelinearly compact R-module. Let i be a non-negative integer. If HI

j �M�N� is I-stable forall j < i� then the set CoassHI

i �M�N� is finite.

Proof. It follows from 3.3 by replacing G with HIi �M�N�� �

We now use Matlis duality to get a finiteness result for associated primes of(generalized) local cohomology modules.

Corollary 3.5. Let M be a finitely generated R-module and N an I-stable semidiscretelinearly compact R-module. Let i be a non-negative integer. If Hj

I �M�N� is I-stable forall j < i� then the set AssHi

I�M�N� is finite.

Proof. At first we assume in addition that �R��� is a complete ring. From 2.5(ii),we have the following isomorphism for all i ≥ 0�

HIi �M�D�N�� � D�Hi

I�M�N���

In virtue of Macdonald (1962, 5.8, 9.13), D�N� is semidiscrete linearly compact,and it is clear that D�M� is I-stable. By 3.3, CoassHI

i �M�D�N�� is finite, so isCoassD�Hi

I�M�N��� But AssHiI�M�N� ⊆ CoassD�Hi

I�M�N�� by Yassemi (1995, 1.8),so AssHi

I�M�N� is finite.Let �R��� be a local ring. From Suzuki (1978, 1.3, 1.5, 1.6) we have

an isomorphim DR�HiI�M�N�� � DR�H

i

IR�M� N ��, where M and N are the �-adic

completions of M and N� By our claim above CoassDR�Hi

IR�M� N �� is finite,

so is CoassDR�HiI�M�N��� Then AssR�H

iI �M�N�� is finite. On the other hand,

the natural homomorphism of rings f � R −→ R induces a map f ∗ � Spec�R� −→Spec�R�� Moreover, it follows from Matsumura (1980, 9.B) that AssR�H

iI�M�N�� =

f ∗�AssR�HiI �M�N��� Therefore, AssR�H

iI�M�N�� is finite. �

In the special case M = R� it should be noted that HIi �R�N� = HI

i �N� andHi

I�R�N� = HiI�N�� Thus from Corollaries 3.4 and 3.5, replacing the module M with

1756 NAM

the ring R� we have the immediate consequence which are finiteness results forco-associated and associated primes of local homology and cohomology modules ofGrothendieck.

Corollary 3.6. Let N be an I-stable semidiscrete linearly compact R-module and i anon-negative integer.

(i) If HIj �N� is I-stable for all j < i� then the set CoassHI

i �N� is finite;(ii) If Hj

I �N� is I-stable for all j < i� then the set AssHiI�N� is finite.

Remark 3.7. (i) There are many I-stable semidiscrete linearly compactR-modules which are neither artinian nor finitely generated. For example, in case�R��� is complete, let X1 be an artinian R-module but not finitely generatedand X2 a finitely generated R-module but not artinian. Then both X1 and X2 aresemi-discrete linearly compact, so is the module X = X1 ⊕ X2� But X is neitherartinian nor finitely generated. Finally, let N = X/IX� thus N is an I-stablesemi-discrete linearly compact R-module.

(ii) Brodmann and Faghani (2000, 2.2) stated that “if N is a finitely generatedR-module and i a non-negative integer such that the local cohomology moduleH

jI �N� is finitely generated for all j < i� then the set AssR�H

iI�N�� is finite.” When

N is an I-stable semidiscrete linearly compact R-module, the condition HjI �N� is

I-stable for all j < i may not follow HjI �N� are finitely generated for all j < i� For

example, let A be an artinian R-module such that the module N = 0 � AI is notfinitely generated. Thus N is I-stable semidiscrete linearly compact. It is clear thatH0

I �N� � N� Then H0I �N� is I-stable, but H0

I �N� is not finitely generated. However,the converse is true, moreover, when N is a finitely generated R-module, theseconditions are equivalent (Brodmann and Sharp, 1998, 9.1.2).

Similarly, the condition HjI �N� is I-stable for all j < i may not follow H

jI �N�

are artinian for all j < i� For example, let �R��� be a complete local ring and L afinitely generated R-module such that the module N = 0 �L I is not artinian. ThenN is an I-stable semidiscrete linearly compact R-module. As H0

I �N� � N� H0I �N� is

I-stable, but H0I �N� is not artinian.

ACKNOWLEDGMENTS

The author acknowledges support by the Abdus Salam International Centrefor Theoretical Physics, Trieste, Italy and the Basis Research Program in NaturalScience of Vietnam. I would like to express my gratitude to Prof. Nguyen Tu Cuongand Prof. M. P. Brodmann for their support and advice. The author is also deeplygrateful to the refree for careful reading of the manuscript and for the helpfulsuggestions.

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A FINITENESS RESULT 1757

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