Artinianness of local cohomology modules

DOI: 10.1007/s11512-013-0187-y

c 2013 by Institut Mittag-Leffler. All rights reserved

Artinianness of local cohomology modules

Moharram Aghapournahr and Leif Melkersson

Abstract. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

1. Introduction

Throughout, A is a commutative noetherian ring. As a general reference to homological and commutative algebra we use [7] and [11]. The main problems in the study of local cohomology modules are to determine when they are artinian, finite, zero and non-zero and when their sets of associated primes are finite. Recently some results have been proved about the artinianness of graded local cohomology modules in [4], [5], [14] and [15].

In those papers the local cohomology modules Hⁱ_R

+(M) of a finitely generated graded moduleM=

n∈ZM_nover a noetherian homogeneous ringR=_∞

n=0R_nare studied. HereR+is the irrelevant homogeneous ideal and the base ring (R0,m0) is assumed to be local.

We prove, with uniform proofs, some general results about artinianness of local cohomology modules in the context of an arbitrary noetherian ring A. They have the special cases in the above references as immediate consequences.

In [15, Theorem 2.4] Sazeedeh showed that ifn is a non-negative integer such that Hⁱ_R

+(M) is artinian for all i>n, then Hⁿ_R

+(M)/m0Hⁿ_R

+(M) is an artinian R-module.

We generalize this in two ways. First we replace the graded ring R by an arbitrary noetherian ring A and consider ideals a and b such that A/(a+b) is artinian, in Corollary 2.3. We get this result as a corollary of a general theorem, Theorem2.1, in which we consider arbitrary Serre subcategories of the category of A-modules.

A Serre subcategory of the category ofA-modules is a full subcategory closed under taking submodules, quotient modules and extensions. An example is given by the class of artinianA-modules. A useful method to prove that a certain module belongs to such a Serre subcategory is to apply the homological principle of [13, Corollary 3.2]:

• Suppose we are given a (strongly) connected sequence {Tⁱ}^∞i=0 of functors between the abelian categoriesAandBand a Serre subcategoryS ofB(i.e., a full subcategory such that whenever 0→X→Y→Z→0 is an exact sequence inB, then X and Z both belong to S if and only if Y belongs to S), then for f: X→Y, a morphism inAandi≥0, ifTⁱCokerf andTⁱ⁺¹Kerf are inS, then also CokerTⁱf and KerTⁱ⁺¹f are inS.

LetAbe a noetherian ring anda be an ideal ofA. AnA-moduleM is called a-cofiniteif Supp_A(M)⊂V(a) and Extⁱ_A(A/a, M) are finiteA-modules (i.e., finitely generatedA-modules) for alli. This notion was introduced by Hartshorne in [9]. For more information about cofiniteness with respect to an ideal, see [10], [8] and [13].

2. Main results

Theorem 2.1. Let S be a Serre subcategory of the category ofA-modules. Let M be a finite A module, a be an ideal of A and n be a non-negative integer such that Hⁱ_a(M) belong to S for all i>n. If bis an ideal of A such that Hⁿ_a(M/bM) belongs toS,then the module Hⁿ_a(M)/bHⁿ_a(M)belongs to S.

Proof. Let b=(b₁, ..., b_r) and consider the map f:M^r→M that is defined by f(x1, ..., xr)=r

i=1bixi. Then Imf=bM and Cokerf=M/bM. Since Hⁱ_a(M) is in S for alli>n and Supp_AKerf⊂Supp_AM it follows from [1, Theorem 3.1] that Hⁿ⁺¹_a (Kerf) is also inS. By hypothesis Hⁿ_a(Cokerf) belongs to S. Hence by [13, Corollary 3.2] Coker Hⁿ_a(f), which equals Hⁿ_a(M)/bHⁿ_a(M), is inS.

Corollary 2.2. Let a and b be two ideals of A. Let M be a finite A-module andnbe a non-negative integer. If Hⁱ_a(M)is artinian fori>n and Hⁿ_a(M/bM)is artinian, then Hⁿ_a(M)/bHⁿ_a(M) is artinian.

Proof. In Theorem2.1take S as the category of artinianA-modules.

The next corollary generalizes [15, Theorem 2.4], as said in the introduction.

Corollary 2.3. Let a and bbe two ideals ofAsuch thatA/(a+b)is artinian and let M be a finite A-module and n be a non-negative integer. If Hⁱ_a(M) is artinian for i>n, then Hⁿ_a(M)/bHⁿ_a(M) is artinian.

Proof. Note that Hⁿ_a(M/bM)∼=Hⁿ_a+b(M/bM), which is artinian.

Corollary 2.4. Let M be a finite A-module such that Hⁱ_a(M) is artinian for i>n, wheren is a positive integer,then Hⁿ_a(M)/aHⁿ_a(M)is artinian.

Proof. Note that Hⁿ_a(M/aM)=0 whenn≥1.

Remark 2.5. In Corollary 2.4we must assume thatn≥1. Take any ideala in a ringA such that A/a is not artinian and letM=A/a. Then Hⁱ_a(M)=0 fori≥1, and Γ_a(M)=M. On the other hand M/aM∼=M. Thus H⁰_a(M)/aH⁰_a(M) is not artinian.

Corollary 2.6. Let aand bbe two ideals ofAand letM be a finiteA-module and n be a non-negative integer. If Hⁱ_a(M) is a-cofinite artinian (resp. has finite support)fori>n,and Hⁿ_a(M/bM)is a-cofinite artinian (resp.has finite support) thenHⁿ_a(M)/bHⁿ_a(M)isa-cofinite artinian(resp.has finite support). In particular, ifn≥1 then Hⁿ_a(M)/aHⁿ_a(M)has finite length (resp.has finite support).

Proof. The category ofa-cofinite artinian modules and the category of modules with finite support are Serre subcategories of the category ofA-modules.

Corollary 2.7. If c=cd(a, M)>0,then H^c_a(M)=aH^c_a(M).

Proof. TakeS to consist of the zero module.

As a corollary we recover Yoshida’s theorem [16, Proposition 3.1].

Corollary 2.8. If c=cd(a, M)>0,then H^c_a(M)is not finite.

Proof. From Corollary2.7, we get that H^c_a(M)=aⁿH^c_a(M) for alln. But if the module H^c_a(M) is finite, then it is annihilated by aⁿ for some n and we get the contradiction H^c_a(M)=0.

Proposition 2.9. Let a and b be ideals of A with A/(a+b) artinian. If the module M isa-cofinite, thenHⁱ_b(M)is artinian for alli.

Proof. Supp_A(Hⁱ_b(M)) is contained in V(a+b), which consists of finitely many maximal idealsm. Since M isa-cofinite, Extⁱ_A(A/m, M) is finite for each such m.

From [13, Theorem 5.5, (i)⇔(iv)] we get that the module Hⁱ_b(M) is artinian for alli.

With the notation as in the introduction, the modules Hⁱ_m0R(H^j_R

+(M)) were shown to be artinian for alliandj when dimR0=1 in [4, Theorem 2.5(b)], or when the idealR+ is principal in [14, Proposition 2.6]. This is obtained as a special case of the following general result.

Corollary 2.10. Let a and b be ideals of A such that A/(a+b) is artinian.

Let M be a finite module such that dimM/aM≤1 orcda=1. Then Hⁱ_b(H^j_a(M))is an artinian module for alli andj.

Proof. The modules H^j_a(M) area-cofinite by [3, Corollary 2.7] resp. [13, Corol- lary 3.14].

In [5, Proposition 5.10] Brodmann, Rohrer and Sazeedeh (with the nota- tion as in the introduction) showed that H¹_m

0R(Hⁱ_R

+(M)) is artinian for all i if dimR0≤2. They also give an example [5, Example 5.11] where this does not hold if dimR0>2 and in [4, Example 4.2] there is an example with dimR0=2 but where Γ_m0R(H²_R

+(M)) not artinian.

In the next theorem we generalize this result.

Theorem 2.11. Let a and b be ideals of A such that A/(a+b) is artinian.

LetM be a finite module such thatdimM/aM≤2. ThenH¹_b(Hⁱ_a(M))is an artinian module for alli.

Proof. Takex∈boutside all prime idealsp⊃a+Ann(M), such that dimA/p=2.

Then dimM/(a+xA)M≤1.

Consider for a fixedithe exact sequence [6, Proposition 8.1.2]

Hⁱ_a+xA(M)−−^g→Hⁱ_a(M)−−^f→Hⁱ_a(M)x−−→Hⁱ⁺¹_a+xA(M).

PutK=Kerf andL=Kerg. From the short exact sequence 0−→L−→Hⁱ_a+xA(M)−→K−→0 we get the exact sequence

H¹_b(Hⁱ_a+xA(M))−→H¹_b(K)−→H²_b(L).

Since

Supp_AL⊂V(a+xA+AnnM)

and dim (A/(a+xA+AnnM))≤1, we get H²_b(L)=0. Hence H¹_b(K) is a homo- morphic image of H¹_b(Hⁱ_a+xA(M)), which is artinian by Corollary2.10. Moreover,

Γ_b(Cokerf) is isomorphic to a submodule of Γ_b(H_a+xAⁱ⁺¹ (M)), which is artinian by Corollary2.10.

Since we now have shown that H¹_b(Kerf) and Γ_b(Cokerf) are artinian, we can apply [13, Lemma 3.1] in order to deduce that the kernel of H¹_b(f) is artinian.

However, x∈b, so the codomain of H¹_b(f) : H¹_b(Hⁱ_a(M))→H¹_b(Hⁱ_a(M)x) is 0. Hence Ker H¹_b(f)=H¹_b(Hⁱ_a(M)), which therefore is artinian.

The following lemma is a generalization of [12, Proposition 1.8].

Lemma 2.12. Let a and b be ideals of A such that A/(a+b) is artinian.

If M is a module such that Supp_A(M)⊂V(a) and 0:_Ma is finite, then Γ_b(M) is a-cofinite artinian.

Proof. The module 0:_Γb(M)a is contained in 0:Ma and is therefore finite. But the support of 0:_Γb(M)a is contained in V(a+b), and therefore 0:_Γb(M)a has finite length. Moreover, Supp_A(Γ_b(M))⊂Supp_A(M)⊂V(a). We conclude that Γ_b(M) is a-cofinite artinian, by [13, Proposition 4.1].

Corollary 2.13. Let a and b be ideals of A such that A/(a+b) is artinian.

If M is a module such that

Extⁿ_A(A/a, M) and Extⁿ⁺¹_A ^−j(A/a,H^j_a(M)) are finite for allj <n, then Γ_b(Hⁿ_a(M))is a-cofinite artinian.

Proof. Note that by [2, Theorem 4.1(b)], Hom_A(A/a,Hⁿ_a(M)) is finite. Hence by Lemma2.12, Γ_b(Hⁿ_a(M)) isa-cofinite artinian.

Next we generalize [15, Theorem 2.9], where it was shown that (with the nota- tion as in the introduction) if Hⁱ_R

+(M) isR+-cofinite fori<s, then Γ_m0R(Hⁱ_R

+(M)) is artinian fori<s.

Corollary 2.14. Let aand bbe two ideals ofAsuch thatA/(a+b)is artinian.

IfM is a finite module such that Hⁱ_a(M) isa-cofinite for i<n,then Γ_b(Hⁱ_a(M))is a-cofinite artinian for all i≤n.

Proof. The hypothesis in Corollary2.13is satisfied.

The next theorem is a generalization of [14, Theorem 2.2], where it is shown (with the notation as in the introduction) that H¹_m

0R(H¹_R+(M)) is artinian.

Theorem 2.15. Let a and b be ideals of A such that A/(a+b) is artinian.

For each finite moduleM, the modules Γ_b(H¹_a(M))and H¹_b(H¹_a(M)) are artinian.

Proof. Corollary 2.14withn=1 implies that Γ_b(H¹_a(M)) is artinian. We may assume that Γ_a(M)=0, so there is anM-regular elementx in a. From the exact sequence 0→M→^xM→M/xM→0, we get the exact sequence

0−−→Γ_a(M/xM)−−→H¹_a(M)−−^f→H¹_a(M)−−→H¹_a(M/xM), where the mapf is defined as multiplication withxon H¹_a(M).

We get

H¹_b(Kerf)∼= H¹_b(Γ_a(M/xM))∼= H¹_a+b(Γ_a(M/xM)), which is artinian. We also get the exact sequence

0−→Γ_b(Cokerf)−→Γ_b(H¹_a(M/xM)).

Since Γ_b(H¹_a(M/xM)) is artinian by Corollary 2.14, Γ_b(Cokerf) is artinian. We use [13, Lemma 3.1] with S=Γ_b(−) andT=H¹_b(−) to conclude that Ker H¹_b(f) is artinian. But H¹_b(f) is multiplication by the element x∈a on H¹_b(H¹_a(M)). Again using [13, Lemma 3.1], we conclude that H¹_b(H¹_a(M)) is artinian.

Recall that thearithmetic rank, ara(a), of an ideal a in a noetherian ringAis defined as the least number of elements ofArequired to generate an ideal with the same radical asa.

In [14, Theorem 2.3] it was shown that (with the notation given in the intro- duction) if ara(R+)=2, then fori≥0, the module Hⁱ_m0R(H²_R+(M)) is artinian if and only if Hⁱ⁺²_m

0R(H¹_R

+(M)) is artinian. Before we state and prove a result of this kind in general form, we need a homological lemma, which might also have independent interest.

Lemma 2.16. Let {Tⁱ}^∞i=0 be a strongly connected sequence of functors be- tween the abelian categoriesAandBand letS be a Serre subcategory ofB. Suppose f:L→M is a morphism inA andi is a positive integer such that

KerTⁱ⁺¹f , KerTⁱf , CokerTⁱf and CokerTⁱ⁻¹f are all inS. ThenTⁱ⁺¹(Kerf)is in S if and only if Tⁱ⁻¹(Cokerf)is inS.

Proof. Factorize f asf=h◦g, where 0→K→L→^gI→0 and 0→I→^hM→C→0 are exact. Thus there are exact sequences

Tⁱ⁻¹I ^T

i−1h

−−−−→Tⁱ⁻¹M−−→Tⁱ⁻¹C−−→TⁱI ^T

ih

−−→TⁱM and

TⁱL ^T

ig

−−→TⁱI−−→Tⁱ⁺¹K−−→Tⁱ⁺¹L ^T

i+1g

−−−−→Tⁱ⁺¹I.

Hence we get the exact sequences

0−−→CokerTⁱ⁻¹h−−→Tⁱ⁻¹C−−→KerTⁱh−−→0 and

0−−→CokerTⁱg−−→Tⁱ⁺¹K−−→KerTⁱ⁺¹g−−→0.

From the compositionsT^jf=T^jh◦T^jg, where j isi−1,iori+1, we get the exact sequences

CokerTⁱ⁻¹f−−→CokerTⁱ⁻¹h−−→0, KerTⁱf−−→KerTⁱh−−→CokerTⁱg−−→CokerTⁱf and

0−−→KerTⁱ⁺¹g−−→KerTⁱ⁺¹f .

Applying our hypothesis we get that CokerTⁱ⁻¹h and KerTⁱ⁺¹g are in S. Moreover we get that KerTⁱhis inS if and only if CokerTⁱg is inS. Comparing this with the two exact sequences we have previously obtained, we conclude that Tⁱ⁻¹C is in S if and only if KerTⁱh is in S, and Tⁱ⁺¹K is in S if and only if CokerTⁱg is in S. Summing up,Tⁱ⁻¹C is in S precisely whenTⁱ⁺¹K is.

Theorem 2.17. Let aandbbe two ideals ofAsuch thatA/(a+b)is artinian.

Assume thatara(a)=2and thatM is a finiteA-module. Then for a given i>0,the moduleHⁱ_b⁻¹(H²_a(M))is artinian if and only if the moduleHⁱ⁺¹_b (H¹_a(M))is artinian.

Proof. We may assume that Γ_a(M)=0. Thena can be generated by twoM- regular elementsxandy, [11, Exercise 16.8]. Therefore there is an exact sequence [6, Proposition 8.1.2]

0−−→H¹_a(M)−−→H¹_xA(M)−−^f→H¹_xA(My)−−→H²_a(M)−−→0.

In order to apply Lemma2.16, we prove that Ker H^j_b(f) and Coker H^j_b(f) are ar- tinian for allj.

By [6, Proposition 8.1.2], there is an exact sequence H^j_b+yA(H¹_xA(M))−−→H^j_b(H¹_xA(M)) ^H

j b(f)

−−−−→H^j_b(H¹_xA(M)_y)−−→H^j+1_b+yA(H¹_xA(M)).

The outer modules are artinian, by Corollary 2.10. Hence Ker H^j_b(f) and Coker H^j_b(f) are artinian for allj.

We now extend [14, Proposition 2.8], where it was shown (for notation see the introduction) that if dimR₀=dand Hⁱ_R

+(M)=0 for alli>c, then H^d_m

0R(H^c_R

+(M)) is artinian. We will now again use Lemma2.16in our proof.

Theorem 2.18. Let aandbbe two ideals ofAsuch thatA/(a+b)is artinian.

LetM be a finiteA-module andnbe a non-negative integer such thatH^j_a(M)=0for all j >n. If dimA/a=d,then the modules H^d_b(Hⁿ_a(M))and H^d_b⁻¹(Hⁿ_a(M))are ar- tinian. Moreover H^d_b⁻²(Hⁿ_a(M))is an artinian module if and only if H^d_b(Hⁿ_a⁻¹(M)) is artinian.

Proof. Since dimA/a=d, H^j_b(X)=0 for all j >d and each A-module X with Supp_A(X)⊂V(a). Hence ifX→X→X→0 is an exact sequence, where the mod- ules have support in V(a), then the sequence H^d_b(X)→H^d_b(X)→H^d_b(X)→0 is ex- act.

Let 0→M→E^0∂

→0E^1∂

→1E²→... be a minimal injective resolution of M. The modules Γa+b(Eⁱ) are artinian, since they are finite direct sums of modules of the form E(A/m), wherem is a maximal ideal containinga+b.

LetL=Ker∂ⁿ andN=Ker∂ⁿ⁻¹. Since H^j_a(M)=0 forj >n, 0−−→Γ_a(L)−−→Γ_a(Eⁿ)−−→Γ_a(Eⁿ⁺¹)−−→...

is an injective resolution of Γ_a(L). For each j, the module H^j_b(Γ_a(L)) is a sub- quotient of Γ_a+b(E^n+j) and is therefore an artinian module. Also the modules Hⁱ_b(Γ_a(Eⁿ⁻¹) are artinian for alli, actually zero for i>0.

Thus we can apply Lemma2.16to the exact sequence

0−−→Γ_a(N)−−→Γ_a(Eⁿ⁻¹)−−^f→Γ_a(L)−−→Hⁿ_a(M)−−→0.

Consequently, for eachithe module Hⁱ_b(Hⁿ_a(M)) is artinian if and only if the module Hⁱ⁺²_b (Γ_a(N)) is artinian. Since the latter module even vanishes if i=d or i=d−1, we conclude that the former is artinian ifi=dori=d−1.

The exact sequence Γ_a(Eⁿ⁻²)→Γ_a(N)→Hⁿ_a⁻¹(M)→0 gives the exact sequence H^d_b(Γ_a(Eⁿ⁻²))→H^d_b(Γ_a(N))→H^d_b(Hⁿ_a⁻¹(M))→0, by the right exactness of H^d_b(−)

on modules with support at V(a). Since the first module is artinian the next two modules are artinian simultaneously. However we showed above that H^d_b(Γa(N)) is artinian if and only if the module H^d_b⁻²(Hⁿ_a(M)) is artinian.

Acknowledgement. We thank the referee for the careful reading of our paper.

References

1. Aghapournahr, M. and Melkersson, L., Local cohomology and Serre subcate- gories,J.Algebra320(2008), 1275–1287.

2. Aghapournahr, M.andMelkersson, L., A natural map in local cohomology,Ark.

Mat.48(2010), 243–251.

3. Bahmanpour, K.andNaghipour, R., Cofiniteness of local cohomology for ideals of small dimension,J.Algebra321(2009), 1997–2011.

4. Brodmann, M.,Fomasoli, S.andTajarod, R., Local cohomology over homogeneous rings with one-dimensional local base ring,Proc.Amer.Math.Soc.131(2003), 2977–2985.

5. Brodmann, M.,Rohrer, F.andSazeedeh, R., Multiplicities of graded components of local cohomology modules,J.Pure Appl.Algebra197(2005), 249–278.

6. Brodmann, M. P.andSharp, R. Y.,Local Cohomology:An Algebraic Introduction with Geometric Applications, 2nd ed., Cambridge University Press, Cambridge.

2013.

7. Bruns, W.andHerzog, J.,Cohen–Macaulay Rings, Revised ed., Cambridge Univer- sity Press, Cambridge, 1998.

8. Delfino, D.andMarley, T., Cofinite modules and local cohomology,J.Pure Appl.

Algebra121(1997), 45–52.

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

10. Huneke, C.and Koh, J., Cofiniteness and vanishing of local cohomology modules, Math.Proc.Cambridge Philos.Soc.110(1991), 421–429.

11. Matsumura, H.,Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

12. Melkersson, L., Properties of cofinite modules and applications to local cohomology, Math.Proc.Cambridge Philos.Soc.125(1999), 417–423.

13. Melkersson, L., Modules cofinite with respect to an ideal,J. Algebra285(2005), 649–668.

14. Sazeedeh, R., Finiteness of graded local cohomology modules,J.Pure Appl.Algebra 212(2008), 275–280.

15. Sazeedeh, R., Artinianness of graded local cohomology modules,Proc.Amer.Math.

Soc.135(2007), 2339–2345.

16. Yoshida, K. I., Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math.J.147(1997), 179–191.

Artinianness of local cohomology modules

Moharram Aghapournahr Department of Mathematics Faculty of Science

Arak University 8156-8-8349 Arak Iran

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

Leif Melkersson

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

leif.melkersson@liu.se

Received May 6, 2011

in revised form August 29, 2013 published online December 5, 2013