A natural map in local cohomology

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DOI: 10.1007/s11512-009-0115-3

A natural map in local cohomology

Moharram Aghapournahr and Leif Melkersson

Abstract. LetRbe a Noetherian ring, a an ideal ofR, M an R-module andna non- negative integer. In this paper we ﬁrst study the ﬁniteness properties of the kernel and the cokernel of the natural mapf: Extⁿ_R(R/a, M)→HomR(R/a,Hⁿ_a(M)), under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.

1. Introduction

ThroughoutRis a commutative Noetherian ring. Our terminology follows the book [5] on local cohomology. Huneke formulated and discussed several problems in [12] about local cohomology modules Hⁿ_a(M) of a finite (by which we mean finitely generated) R-module M. For example when do they have just finitely many associated prime ideals or when are they Artinian? Furthermore the following conjecture was made by Grothendieck in [10].

Conjecture1.1. For every ideala⊂Rand each ﬁniteR-moduleM, the module Hom_R(R/a,Hⁿ_a(M)) is ﬁnite for alln≥0.

This was thought of as a substitute for the Artinianness of Hⁿ_a(M), in the case when a is the maximal ideal of a local ring. Moreover the finiteness of HomR(R/a,Hⁿ_a(M)) implies the finiteness of the set of associated prime ideals of Hⁿ_a(M). Although the conjecture is not true in general as was shown by Hart- shorne in [11], there are some attempts to show that under some conditions, for some number n, the module Hom_R(R/a,Hⁿ_a(M)) is finite, see [1, Theorem 3], [8, Theorem 6.3.9] and [7, Theorem 2.1].

In Section 3, in view of Grothendieck’s conjecture and [8, Theorem 6.3.9], we are led to study ﬁniteness properties of the kernel and the cokernel of the natural

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homomorphism

f: Extⁿ_R(R/a, M)−→HomR(R/a,Hⁿ_a(M)).

We give conditions under which Kerf (resp. Cokerf) belongs to a Serre sub- categorySof the category ofR-modules. A class ofR-modules closed under taking submodules, quotients and extensions is called a Serre subcategory. Examples are given by the class of ﬁnitely generated modules, the class of Artinian modules and the class consisting of the zero modules.

To obtain the results about the natural map above we prove a quite general result, Proposition3.1, in the setting of abelian categories. We were also able to do this using spectral sequence techniques, but we preferred to give a more elementary treatment.

We will apply this to ﬁnd some relations between the ﬁniteness of the sets of associated primes of Extⁿ_R(R/a, M) and Hⁿ_a(M), generalizing [13, Proposition 1.1(b)], [2, Lemma 2.5(c)] and [4, Proposition 2.2]. Also an application is given to show that under certain conditions Extⁿ_R(R/a, M) is Artinian if and only if Hⁿ_a(M) is.

One of our main results of this paper is Corollary4.2.

(a) If Extⁿ_R⁻^j(R/a,H^j_a(M))=0 for allj <n, thenf is injective.

(b) If Extⁿ⁺¹_R ⁻^j(R/a,H^j_a(M))=0 for allj <n, thenf is surjective.

(c) If Ext^t_R⁻^j(R/a,H^j_a(M))=0 for t=n, n+1 and for all j <n, then f is an isomorphism.

In the last section we will study the asymptotic behavior of the kernel and the cokernel of the above mentioned natural map in the graded case.

2. A homological lemma

Let F: A→B be an additive covariant functor between abelian categories.

Given a morphismf:M→N inA we get inBinduced morphisms λf: CokerF f−→FCokerf and ˇf:FKerf−→KerF f .

Consider the exact sequence 0→Kerf→^uM→^f N→^v Cokerf→0 inA. The equal- itiesF f¨F u=0 andF v¨F f=0, will allow the construction ofˇf andλf.

Next we generalize the useful result [14, Lemma 3.1].

Lemma 2.1. Let S, T:A→B be additive covariant functors between abelian categories, such that whenever we have in A a short exact sequence 0→X→^uX→^v X→0 we will have inB an exact sequence

SX−−^Su→SX−−^Sv→SX−−→T X−−^{T u}→T X−−^{T v}→T X.

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Let S be a Serre subcategory of B and let f: M→N be a morphism in A. Consider the induced morphisms

λf: CokerSf−→SCokerf and ˇf:TKerf−→KerT f . (a) If TKerf is in S,then Kerλ_f is also inS.

(b) IfSCokerf is in S, then Cokerˇf is also inS. (c) If TImf is inS,then Cokerλf is also inS. (d) IfSImf is inS,then Kerˇf is also in S. Proof. Consider the exact sequences

0−−→K−−→M−−^g→I−−→0 and 0−−→I−−^h→N−−→C−−→0, whereK=Kerf,I=Imf, C=Cokerf andf=h¨g.

There is a commutative diagram with the ﬁrst row exact:

CokerSg // CokerSf

λ_f

// CokerSh

λh

// 0

SCokerf

∼=

// SCokerh.

From the exactness of SI^Sh→SN^Sv→SC, we get Kerλh=0. Hence there is an epimorphism CokerSg→Kerλf→0. Furthermore the exactness ofSM^Sg→SI→T K yields a monomorphism 0→CokerSg→T K. Hence if T K is in S then Kerλ_f is inS, too.

The assertion in (b) follows by dualizing the proof of (a). There is the following commutative diagram with exact bottom row:

TKerg

ˇg

∼=

// TKerf

ˇf

0 // KerT g // KerT f // KerT h.

Since T K→T M^{T g}→T I is exact, ˇg is an epimorphism and therefore there is a monomorphism 0→Cokerˇf→KerT h.

The exactness of SC→T I^{T h}→T N yields an epimorphismSC→KerT h→0. It follows that ifSC is inS, then so is Cokerˇf. This proves (b).

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From the exactness of SI^Sh→SN→SC→T I, we get an exact sequence 0→ CokerSh→^λ^hSC→T I and ﬁnally we get a monomorphism 0→Cokerλ_h→T I. Hence ifT I is in S, then Cokerλf=Cokerλh is inS. This proves (c).

The exactness ofSI→T K^{T g}→T I yields the exactness ofSI→TKerg→^ˇ^gKerT g.

We get an epimorphismSI→Kerˇg→0. But Kerˇg=Kerˇf. Hence ifSIbelongs toS then so does Kerˇf, and (d) is proved.

3. A morphism between connected sequences of functors

Let A, B and C be abelian categories and suppose that both A and B have enough injectives. Let G:A→BandF:B→Cbe two left exact covariant functors such that for each injective objectE inA,Fⁿ(GE)=0 for alln≥1. We denote the nth right derived functors ofF,Gand their compositeF GbyFⁿ,Gⁿ and (F G)ⁿ, respectively. Then there is a natural morphism

{θn}n:{(F G)ⁿ}n−→{F Gⁿ}n

between connected sequences of functors. Next we will give conditions for the kernel or the cokernel ofθn(M) to belong to a given Serre subcategoryS ofC. HereM is a ﬁxed object inC andn is a ﬁxed natural number.

Proposition 3.1.

(a) If Fⁿ⁻ⁱGⁱM belong toS for all i<n, then Kerθn(M)belongs to S. (b) If Fⁿ⁺¹⁻ⁱGⁱM belong toS for all i<n, then Cokerθ_n(M) belongs toS. Proof. Let ﬁrstn=1 and let 0→M→E→^η N→0 be exact, whereEis injective.

We get the exact sequences

0−−→GM−−→GE−−^Gη→GN−−→G¹M−−→0 and

0−−→F GM−−→F GE−−−^{F Gη}→F GN−−→(F G)¹M−−→0.

Hence θ1(M) : (F G)¹M→F G¹M coincides withλGη: CokerF Gη→FCokerGη.

SinceF¹KerGη∼=F¹GM, which belongs toS, we can apply Lemma 2.1with S=F andT=F¹. This settles the casen=1 for the statement (a).

The statement (b) forn=1 follows from case (c) in Lemma2.1. PutI=ImG_η and applyF to the exact sequence 0→GM^G→^ηGE→I→0. SinceF¹GE=0 we get F¹I∼=F²GM.

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For the induction step we use the commutative diagram (n≥2):

0 // (F G)ⁿ⁻¹N

θ_n−1(N)

// (F G)ⁿM

θ_n(M)

// 0

0 // F Gⁿ⁻¹(N) // F Gⁿ(M) // 0.

Consequently, Kerθn(M)∼=Kerθn−1(N). When 1≤k≤n−1 we have isomor- phismsF⁽ⁿ⁻¹⁾⁻^kG^kN∼=Fⁿ⁻^(k+1)G^k+1M. Moreover since Fⁿ⁻¹GE=0 and FⁿGE

=0, the exactness of the sequences 0→GM→GE→I→0 and 0→I→GN→ G¹M→0 implies that Fⁿ⁻¹I∼=FⁿGM (remember that n>1). The second exact sequence, i.e. 0→I→GN→G¹M→0, also implies the exactness of Fⁿ⁻¹I→ Fⁿ⁻¹GN→Fⁿ⁻¹G¹M. Since the endterms are inS, it follows that Fⁿ⁻¹G^kN is inS also fork=0.

Consequently we are able to apply the induction hypothesis for (a) toN. The case (b) is treated in the same way (just changenton+1).

4. Applications to local cohomology

Theorem 4.1. Let R be a Noetherian ring, a be an ideal of R and M be an R-module. Let n be a non-negative integer and S be a Serre subcategory of the category of R-modules. Consider the natural homomorphism

f: Extⁿ_R(R/a, M)→HomR(R/a,Hⁿ_a(M)).

(a) If Extⁿ_R⁻^j(R/a,H^j_a(M))belongs toS for allj <n,then Kerf belongs toS. (b) If Extⁿ⁺¹_R ⁻^j(R/a,H^j_a(M))belongs to S for all j <n, then Cokerf belongs toS.

(c) If Ext^t_R⁻^j(R/a,H^j_a(M))belongs to S fort=n, n+1 and for all j <n, then Kerf and Cokerf both belong toS. Thus Extⁿ_R(R/a, M)belongs to S if and only if HomR(R/a,Hⁿ_a(M))belongs toS. (See also[8, Theorem6.3.9].)

Proof. We apply Proposition 3.1withF X=HomR(R/a, X) andGX=Γa(X).

Observe thatF G=F.

The following corollary is one of our main results of this paper. It is well known, by using a spectral sequence argument as in [13, Proposition 1.1(b)] or another method as in [2, Lemma 2.5(b)], (whenM is a ﬁnite module andn=depth_aM, or

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more generally when the local cohomology vanishes below n, for example use [15, Theorem 11.3]) that the natural map

f: Extⁿ_R(R/a, M)→HomR(R/a,Hⁿ_a(M))

is an isomorphism. In the following corollary we show that without any condition on M it is not necessary that these local cohomology modules are zero for the natural mapf to be an isomorphism.

Corollary 4.2.

(a) If Extⁿ_R⁻^j(R/a,H^j_a(M))=0 for allj <n,then f is injective.

(b) If Extⁿ⁺¹_R ⁻^j(R/a,H^j_a(M))=0 for allj <n,then f is surjective.

(c) If Ext^t_R⁻^j(R/a,H^j_a(M))=0 for t=n, n+1 and for all j <n, then f is an isomorphism.

Proof. In Proposition 4.1setS={0}.

Corollary 4.3. If Hⁱ_a(M)=0for all i<n(in particular,whenM is ﬁnite and n=depth_aM) then

Extⁿ_R(R/a, M)∼= HomR(R/a,Hⁿ_a(M)).

Corollary 4.4. If Extⁿ_R⁻^j(R/a,H^j_a(M)) and Extⁿ⁺¹_R ⁻^j(R/a,H^j_a(M)) are fi- nite for all j <n,then the module Extⁿ_R(R/a, M) is finite if and only if the module HomR(R/a,Hⁿ_a(M))is finite.

Corollary 4.5. If Ext^t_R⁻^j(R/a,H^j_a(M))is Artinian for t=n, n+1and for all j <n, then Extⁿ_R(R/a, M)is Artinian if and only if Hⁿ_a(M) is Artinian.

Proof. This is immediate by using Theorem4.1(c) and [5, Theorem 7.1.2].

Corollary 4.6.

(a) If Extⁿ_R⁻^j(R/a,H^j_a(M))=0 for allj <n,then

AssR(Hⁿ_a(M))⊂AssR(Extⁿ_R(R/a, M))∪AssR(Cokerf).

(b) If Extⁿ⁺¹_R ⁻^j(R/a,H^j_a(M))=0 for allj <n,then

AssR(Hⁿ_a(M))⊂AssR(Extⁿ_R(R/a, M))∪Supp_R(Kerf).

(c) If Ext^t_R⁻^j(R/a,H^j_a(M))=0 fort=n, n+1 and for allj <n,then AssR(Extⁿ_R(R/a, M)) = AssR(Hⁿ_a(M)).

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For the proof of part (b) in Corollary4.6we need the following lemma.

Lemma 4.7. For any exact sequence N→L→T→0 of R-modules and R-homomorphisms we have

Ass_R(T)⊂Ass_R(L)∪Supp_R(N).

Proof. Let p∈AssR(T)\Supp_R(N). Then Np=0 and therefore Lp∼=Tp. Now pRp∈AssR_p(Tp). HencepRp∈AssR_p(Lp) and consequently p∈AssR(L).

Divaani-Aazar and Mafi introduced in [9] weakly Laskerian modules. An R-moduleM isweakly Laskerian if for any submoduleN ofM the quotientM/N has finitely many associated primes. The weakly Laskerian modules form a Serre subcategory of the category of R-modules. In fact it is the largest Serre subcat- egory such that each module in it has just finitely many associated prime ideals.

In the following corollary we give some conditions for the ﬁniteness of the sets of associated primes of Extⁿ_R(R/a, M) and Hⁿ_a(M).

Corollary 4.8.

(a) If AssR(Hⁿ_a(M)) is a ﬁnite set and Extⁿ_R⁻^j(R/a,H^j_a(M)) is weakly Laskerian for all j <n (e.g. if H^j_a(M) is weakly Laskerian for all j <n), then Ass_R(Extⁿ_R(R/a, M))is a ﬁnite set. (See also [8, Corollary6.3.11]).

(b) If Extⁿ_R(R/a, M) and Extⁿ⁺¹_R ⁻^j(R/a,H^j_a(M)) for all j <n are weakly Laskerian(e.g.ifM and H^j_a(M)forj <nare weakly Laskerian),thenAssR(Hⁿ_a(M)) is a ﬁnite set. (Compare with [9, Corollary2.7].)

Proof. Note that AssR(Hⁿ_a(M))=AssR(HomR(R/a,Hⁿ_a(M))). We use the exact sequence

0−−→Kerf−−→Extⁿ_R(R/a, M)−−^f→HomR(R/a,Hⁿ_a(M))−−→Cokerf−−→0.

(a) By Theorem4.1(a), Kerf is weakly Laskerian.

(b) By Theorem4.1(b), Cokerf is weakly Laskerian.

5. Asymptotic behaviors We now turn to the graded case. Let R=

i≥0Ri be a homogeneous graded Noetherian ring andM=

i∈ZM_i be a gradedR-module. The module M is said to have

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(a) the property ofasymptotic stability of associated primes if there exists an integer n0 such that AssR₀(Mn)=AssR₀(Mn₀) for alln≤n0;

(b) the property ofasymptotic stability of supports if there exists an integern0

such that Supp_R

0(M_n)=Supp_R

0(M_n₀) for alln≤n₀.

(c) the property oftameness ifMi=0 for alli0 or else Mi=0 for alli0.

Note that ifM has the property of (a) thenM has the property of (b) and if M has the property of (b) then M is tame i.e. has the property of (c).

Let s be a non-negative integer. In order to reﬁne and complete [6, The- orem 4.4] we study the asymptotic behavior of the kernel and cokernel of the natural graded homomorphism between the graded modules Ext^s_R(R/R+, M) and Hom_R(R/R+,H^s_R

+(M)). For more information on the notions in (a), (b) and (c), see [3].

Deﬁnition5.1. A graded moduleM over a homogeneous graded ringRis called asymptotically zero ifMi=0 for alli0.

All ﬁnite gradedR-modules are asymptotically zero.

Theorem 5.2. Assume that M is a graded R-module and let n be a ﬁxed non-negative integer such that the modules

Extⁿ_R⁻^j(R/R+,H^j_R

+(M)) and Extⁿ_R⁻^j+1(R/R+,H^j_R

+(M)), wherej < s, are asymptotically zero (e.g.they might be ﬁnite). Then the kernel and the cokernel of the natural homomorphism

f: Ext^s_R(R/R+, M)→HomR(R/R+,H^s_R₊(M)) are asymptotically zero.

Therefore Ext^s_R(R/R+, M)has one of the properties of (a), (b)and (c)if and only if HomR(R/R+,H^s_R₊(M))has.

Proof. Note that the class of graded modules which are asymptotically zero is a Serre subcategory of the category of graded R-modules and we can use Proposi- tion3.1.

References

1. Asadollahi, J., Khashyarmanesh, K. andSalarian, S., A generalization of the coﬁniteness problem in local cohomology modules, J. Aust. Math. Soc. 75 (2003), 313–324.

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2. Asadollahi, J. andSchenzel, P., Some results on associated primes of local cohomology modules,Japan J.Math.29(2003), 285–296.

3. Brodmann, M. P., Asymptotic behavior of cohomology: tameness, supports and associated primes, inCommutative Algebra and Algebraic Geometry(Bangalore, 2003), Contemp. Math.390, pp. 31–61, Amer. Math. Soc., Providence, RI, 2005.

4. Brodmann, M. P.,Rotthaus, C. andSharp, R. Y., On annihilators and associated primes of local cohomology modules,J.Pure Appl.Algebra153(2000), 197–

227.

5. Brodmann, M. P. and Sharp, R. Y.,Local Cohomology:An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.

6. Dibaei, M. T. and Nazari, A., Graded local cohomology: attached and associated primes, asymptotic behaviors,Comm.Algebra35(2007), 1567–1576.

7. Dibaei, M. T. andYassemi, S., Associated primes and coﬁniteness of local cohomology modules,Manuscripta Math.117(2005), 199–205.

8. Dibaei, M. T. andYassemi, S., Associated primes of the local cohomology modules, in Abelian Groups,Rings,Modules,and Homological Algebra, pp. 49–56, Chapman and Hall/CRC, London, 2006.

9. Divaani-Aazar, K. andMafi, A., Associated primes of local cohomology modules, Proc.Amer.Math.Soc.133(2005), 655–660.

10. Grothendieck, A.,Cohomologie locale des faisceaux cohérents et théorème de Lef- schetz locaux et globaux(SGA2), North-Holland, Amsterdam, 1968.

11. Hartshorne, R., Aﬃne duality and coﬁniteness,Invent.Math.9(1970), 145–164.

12. Huneke, C., Problems on local cohomology, in Free Resolutions in Commutative Al- gebra and Algebraic Geometry(Sundance, UT,1990), pp. 93–108, Jones and Bartlett, Boston, MA, 1992.

13. Marley, M., The associated primes of local cohomology modules over rings of small dimension,Manuscripta Math.104(2001), 519–525.

14. Melkersson, L., Modules coﬁnite with respect to an ideal, J.Algebra 285(2005), 649–668.

15. Rotman, J., An Introduction to Homological Algebra, Academic Press, New York, 1979.

Moharram Aghapournahr Arak University

Beheshti St P.O. Box 879 Arak Iran

m-aghapour@araku.ac.ir m.aghapour@gmail.com

Leif Melkersson

Department of Mathematics Link¨oping University SE-581 83 Link¨oping Sweden

lemel@mai.liu.se

Received November 13, 2008 published online December 12, 2009