SOME RESULTS ON LOCAL COHOMOLOGY MODULES WITH RESPECT TO A PAIR OF IDEALS

Abstract. We show that if M and Hi I,J (M) are weakly Laskerian for all i < d, then AssR(Hom(RI, Hd I,J (M))) is finite. If Hi I,J (M) is (I, J)weakly cofinite for all i 6= d, then Hd I,J (M)) is also (I, J)weakly cofinite. We also study some properties of Hi I,J (M) concerning Serre subcategory.

MODULES WITH RESPECT TO A PAIR OF IDEALS

TRAN TUAN NAM AND NGUYEN MINH TRI

Abstract We show that if M and H I,Ji (M ) are weakly

Laske-rian for all i < d, then Ass R (Hom(R/I, HI,Jd (M ))) is finite If

H i

I,J (M ) is (I, J )-weakly cofinite for all i 6= d, then H d

I,J (M ))

is also (I, J )-weakly cofinite We also study some properties of

HI,Ji (M ) concerning Serre subcategory.

Key words: local cohomology, weakly Laskerian, weakly cofinite

2000 Mathematics subject classification: 13D45 Local cohomology

1 Introduction Throughout this paper, R is a commutative Noetherian ring and

I, J are two ideals of R In [7] Takahashi, Yoshino and Yoshizawa in-troduced the definition of local cohomology modules with respect to a pair of ideals (I, J ) which is a generalization of the definition of local cohomology modules with respect to an ideal I of Grothendieck Let

M be an R-module, the (I, J )-torsion submodule ΓI,J(M ) of M is the subset consisting of all elements x of M such that Inx ⊆ J x for some

n ∈ N For an integer i, the local cohomology functor Hi

I,J with respect

to a pair of ideals (I, J ) is the i-th right derived functor of ΓI,J Note that if J = 0, then HI,Ji coincides with the ordinary local cohomology functor Hi

I of Grothendieck

In [5] Grothendieck gave a conjecture that: For any ideal I of R and any finitely generated R-module M, the module HomR(R/I, Hi

I(M ))

is finitely generated, for all i One year later, Hartshorne provided a counterexample to Grothendieck’s conjecture He defined an R-module

M to be I-cofinite if Supp(M ) ⊆ V (I) and ExtiR(R/I, M ) is finitely generated for all i and asked: For which rings R and ideals I are the modules HIi(M ) I-cofinite for all i and all finitely generated modules

M ?

The organization of the paper is as follows In next section, we will be concerned with Grothendieck’s conjecture Theorem 2.2 shows that if M and Hi

I,J(M ) are weakly Laskerian R−modules for all i < d,

1

then the set AssR(Hom(R/I, HI,Jd (M ))) is finite Let denote W (I, J ) = {p ∈ Spec(R) | In ⊆ p + J for some positive integer n} An R-module

M is said to be (I, J )-weakly cofinite if Supp(M ) ⊆ W (I, J ) and ExtiR(R/I, M ) is weakly Laskerian for all i ≥ 0 We prove in Theo-rem 2.6 that if HI,Ji (M ) is (I, J )-weakly cofinite for all i ≤ d, then ExtiR(R/I, M ) is weakly Laskerian for all i ≤ d On the other hand, if

Hi

I,J(M ) is (I, J )-weakly cofinite for all i 6= d, then Hd

I,J(M )) is also (I, J )-weakly cofinite (Theorem 2.7)

The last section is devoted to study some properties of Hi

I,J(M ) concerning Serre subcategory Theorem 3.1 says that if HI,Ji (M ) ∈ S for all i < d, then ExtiR(R/I, M )) ∈ S for all i < d In case (R, m)

is a local ring and M is a finitely generated R−module we see that if

Hi

I,J(M ) ∈ S for all i < d, then HomR(R/m, Hd

I,J(M )) ∈ S (Theorem 3.2)

2 Weakly Laskerian modules and cofinite modules

We begin by recalling the definition of weakly Laskerian modules ([3, 2.1]) An R-module M is said to be weakly Laskerian if the set of associated primes of any quotient module of M is finite

Lemma 2.1 ([3, 2.3])

i) Let 0 −→ L −→ M −→ N −→ 0 be an exact sequence of R−modules Then M is weakly Laskerian if and only if L and N are both weakly Laskerian Thus any subquotient of a weakly Laskerian module as well as any finite direct sum of weakly Laskerian mod-ules is weakly Laskerian

ii) Let M and N be two R−modules If M is weakly Laskerian and

N is finitely generated, then ExtiR(N, M ) and TorRi (N, M ) are weakly Laskerian for all i ≥ 0

The following theorem answers the question concerning Grothendieck’s conjecture: When is the set AssR(Hom(R/I; HI,Jd (M ))) finite?

Theorem 2.2 Let M be a weakly Laskerian R−module and d a non-negative integer If Hi

I,J(M ) is a weakly Laskerian R−module for all

i < d, then AssR(Hom(R/I; HI,Jd (M ))) is a finite set

Proof Let us consider functors F = HomR(R/I, −) and G = ΓI,J(−) Then F G = Hom(R/I, −) We have a Grothendieck spectral sequence

by [6, 11.38]

E2p,q = ExtpR(R/I, HI,Jq (M )) ⇒

p Extp+qR (R/I, M )

We consider homomorphisms of the spectral

Ek−k,d+k−1d−k,d+k−1−→ Ek0,d−→Ed0,d kk,d+1−k Since Ek−k,d+k−1 = 0 for all k ≥ 2, Ker d0,dk = Ek+10,d It follows an exact sequence

0 −→ Ek+10,d −→ Ek0,d−→Ed0,d kk,d+1−k Hence

Ass(Ek0,d) ⊆ Ass(Ek+10,d ) ∪ Ass(Ekk,d+1−k)

By iterating this for all k = 2, , d + 1, we get

Ass(E20,d) ⊆ (

d+1

[

k=2

Ass(Ekk,d+1−k))[Ass(Ed+20,d )

It is clear that

Ed+20,d = Ed+30,d = = E∞0,d Therefore

Ass(E20,d) ⊆ (

d+1

[

k=2

Ass(Ekk,d+1−k))[Ass(E∞0,d)

For all k = 2, , d + 1 as HI,Jd+1−k(M ) a weakly Laskerian R−module,

so is E2k,d+1−k = ExtkR(R/I, HI,Jd+1−k(M )) by 2.1 (ii) As Ekk,d+1−k is a subquotient of E2k,d+1−k, it follows from 2.1 (i) that Ekk,d+1−kis a weakly Laskerian R−module Thus

d+1

S

k=2

Ass(Ekk,d+1−k) is a finite set

The proof is completed by showing that the set Ass(E∞0,d) is finite Indeed, there is a filtration Φ of Hp+q = Extp+qR (R/I, M ) with

0 = Φp+q+1Hp+q ⊆ Φp+qHp+q ⊆ ⊆ Φ1Hp+q⊆ Φ0Hp+q = Extp+qR (R/I, M )

and

E∞k,p+q−k = ΦkHp+q/Φk+1Hp+q, 0 ≤ k ≤ p + q

It follows that Ep,q

∞ is a weakly Laskerian R−module, so Ass(Ep,q

∞) is finite for all p, q In particular, Ass(E0,d

Note that finitely generated modules or modules that have finite support are weakly Laskerian modules So we have an immediate con-sequence

Corollary 2.3 Let M be a finitely generated R−module and d a non-negative integer If Hi

I,J(M ) is finitely generated or Supp(Hi

I,J(M )) is

a finite set for all i < d, then AssR(HomR(R/I, HI,Jd (M ))) is finite

An R-module M is said (I, J )-cofinite if Supp(M ) ⊆ W (I, J ) and ExtiR(R/I, M ) is finitely generated for all i ≥ 0 ([8]) The following definition is an extension of the definitions of (I, J )-cofinite modules and I−weakly cofinite modules ([4])

Definition 2.4 An R-module M is said (I, J )-weakly cofinite if Supp(M ) ⊆ W (I, J ) and ExtiR(R/I, M ) is weakly Laskerian for all

i ≥ 0

From the definition of (I, J )-weakly cofinite modules we have the following immediate consequence

Corollary 2.5 i) Every (I, J )-cofinite module is a (I, J )-weakly

cofinite module

ii) If Supp(M ) ⊆ W (I, J ) and M is weakly Laskerian, then M is (I, J )-weakly cofinite

Theorem 2.6 Let M be an R-module and d a non-negative inte-ger such that Hi

I,J(M ) is (I, J )-weakly cofinite for all i ≤ d Then ExtiR(R/I, M ) is weakly Laskerian for all i ≤ d

Proof We now proceed by induction on d When d = 0, the short exact sequence

0 → ΓI,J(M ) → M → M/ΓI,J(M ) → 0 induces an exact sequence

0 → HomR(R/I, ΓI,J(M )) → HomR(R/I, M ) → HomR(R/I, M/ΓI,J(M ))

Since M/ΓI,J(M ) is (I, J )-torsion free, it is also I-torsion free and then HomR(R/I, M/ΓI,J(M )) = 0 It follows

HomR(R/I, M ) ∼= HomR(R/I, ΓI,J(M ))

Hence HomR(R/I, M ) is a weakly Laskerian R-module

Let d > 0 Note that HI,Ji (M ) ∼= HI,Ji (M/ΓI,J(M )) for all i > 0 Let

M = M/ΓI,J(M ) and E(M ) denote the injective hull of M From the short exact sequence

0 → M → E(M ) → E(M )/M → 0

we get

ExtiR(R/I, E(M )/M ) ∼= Exti+1R (R/I, M ) and

HI,Ji (E(M )/M ) ∼= HI,Ji+1(M )

for all i ≥ 0 It follows from the hypothesis that Hi

I,J(E(M )/M ) is (I, J )-weakly cofinite for all i ≤ d − 1 By the inductive hypothesis ExtiR(R/I, E(M )/M ) is weakly Laskerian for all i ≤ d − 1 and then ExtiR(R/I, M ) is also weakly Laskerian for all i ≤ d Now the short exact sequence

0 → ΓI,J(M ) → M → M → 0 gives rise to a long exact sequence

· · · → Exti

R(R/I, ΓI,J(M )) → ExtiR(R/I, M ) → ExtiR(R/I, M ) → · · · Since ΓI,J(M ) is (I, J )-weakly cofinite, ExtiR(R/I, ΓI,J(M )) is weakly Laskerian for all i ≥ 0 Finally, it follows from the long exact sequence that ExtiR(R/I, M ) is also weakly Laskerian for all i ≤ d  Theorem 2.7 Let M be R-module such that ExtiR(R/I, M ) is weakly Laskerian for all i and d a non-negative integer If HI,Ji (M ) is (I, J )-weakly cofinite for all i 6= d, then Hd

I,J(M )) is also (I, J )-weakly cofi-nite

Proof We use induction on d When d = 0, set M = M/ΓI,J(M ), then the short exact sequence

0 → ΓI,J(M ) → M → M → 0 gives rise a long exact sequence

· · · → ExtiR(R/I, ΓI,J(M )) → ExtiR(R/I, M ) → ExtiR(R/I, M ) → · · ·

We have Hi

I,J(M ) ∼= HI,Ji (M ) for all i > 0 and H0

I,J(M ) = 0 From the hypothesis, Hi

I,J(M ) is (I, J )-weakly cofinite for all i ≥ 0 It follows from 2.6 that ExtiR(R/I, M ) is weakly Laskerian for all i ≥ 0 There-fore, combining the long exact sequence and the hypothesis gives that ExtiR(R/I, ΓI,J(M )) is weakly Laskerian This implies that H0

I,J(M ) is (I, J )-weakly cofinite

Let d > 0 The short exact sequence

0 → M → E(M ) → E(M )/M → 0 yields

ExtiR(R/I, E(M )/M ) ∼= Exti+1R (R/I, M ) and

HI,Ji (E(M )/M ) ∼= HI,Ji+1(M ) for all i ≥ 0 Then HI,Ji (E(M )/M ) is (I, J )-weakly cofinite for all

i 6= d − 1 Note that ExtiR(R/I, ΓI,J(M )) is weakly Laskerian and then

ExtiR(R/I, E(M )/M ) is weakly Laskerian for all i ≥ 0 By the induc-tive hypothesis HI,Jd−1(E(M )/M ) is (I, J )-weakly cofinite Therefore

Combining 2.1(ii) with 2.7 we obtain the following consequence Corollary 2.8 Let M be a weakly Laskerian R-module and d a non-negative integer If HI,Ji (M ) is (I, J )-weakly cofinite for all i 6= d, then

HI,Jd (M )) is also (I, J )-weakly cofinite

Corollary 2.9 Let I be a principal ideal of R and M a weakly Laske-rian module Then Hi

I,J(M ) is (I, J )-weakly cofinite for all i ≥ 0 Proof It follows from [7, 4.11] that HI,Ji (M ) = 0 for all i > 1 More-over, H0

I,J(M ) is a weakly Laskerian R−module, since H0

I,J(M ) is a submodule of M That means Hi

I,J(M ) is (I, J )-weakly cofinite for all

i 6= 1 Now the conclusion follows from 2.7 

3 On Serre subcategory Recall that the class S of R-modules is a Serre subcategory of the category of R-modules if it is closed under taking submodules, quotients and extensions

Theorem 3.1 Let M be an R−module and d a non-negative integer

If HI,Ji (M ) ∈ S for all i < d, then ExtiR(R/I, M )) ∈ S for all i < d Proof We begin by considering functors F = HomR(R/I, −) and G =

ΓI,J(−) It is clear that F G = Hom(R/I, −) By [6, 11.38] we have a Grothendieck spectral sequence

E2p,q = ExtpR(R/I, HI,Jq (M )) ⇒

p Extp+qR (R/I, M )

Since Hi

I,J(M ) ∈ S for all i < d, E2p,q ∈ S for all p ≥ 0, 0 ≤ q < d

We consider homomorphisms of the spectral for all p ≥ 0, 0 ≤ t < d and i ≥ 2

Eip−i,t+i−1 d

p−i,t+i−1 i

−→ Eip,t d

p,t i

−→ Eip+i,t−i+1 Note that Eip,t = Ker dp,ti−1/ Im dp−i+1,t+i−2i−1 and Eip,j = 0 for all j < 0 This implies Ker dp,t−pt+2 ∼= Ep,t−p

t+2 ∼= ∼= Ep,t−p

∞ for all 0 ≤ p ≤ t We now have a filtration Φ of Ht = ExttR(R/I, M ) such that

0 = Φt+1Ht⊆ ΦtHt⊆ ⊂ Φ1Ht⊆ Φ0Ht = ExttR(R/I, M ) and

ΦiHt/Φi+1Ht= E∞i,t−i

for all 0 ≤ i ≤ t Then there is a short exact sequence

0 → Φi+1Ht→ ΦiHt→ Ei,t−i

From the proof above we have Ei,t−i

∞ ∼= Ei,t−i

t+2 ∼= Ker di,t−i

t+2 a subquotient

of E2i,t−i and E2i,t−i ∈ S for all 0 ≤ i ≤ t It follows that Ei,t−i

∞ ∈ S for all 0 ≤ i ≤ t By induction on i we get ΦiHt ∈ S for all 0 ≤ i ≤ t Finally ExttR(R/I, M ) ∈ S for all t < d  Theorem 3.2 Let M be a finitely generated module over a local ring (R, m) and d a non-negative integer If Hi

I,J(M ) ∈ S for all i < d, then HomR(R/m, HI,Jd (M )) ∈ S

Proof The proof is by induction on d When d = 0, since M is finitely generated, so is HI,J0 (M ) Hence HomR(R/m, HI,J0 (M ))) has finite length and then HomR(R/m, H0

I,J(M ))) ∈ S by [1, 2.11]

Let d > 0 It follows from [7, 1.13 (4)] that Hi

I,J(M ) ∼= HI,Ji (M/ΓI,J(M )) for all i > 0 Thus we can assume, by replacing M with M/ΓI,J(M ), that M is (I, J )-torsion-free Since ΓI(M ) ⊆ ΓI,J(M ) = 0, it follows that M is also I-torsion-free Hence, there exists an element x ∈ I which is non-zerodivisor on M Set M = M/xM, the short exact se-quence

0 → M → M → M → 0.x gives rise to a long exact sequence

· · · → Hd−1

I,J (M )→ H.x d−1

I,J (M )→ Hf d−1

I,J (M )→ Hg d

I,J(M )→ H.x d

I,J(M ) → · · ·

As Hi

I,J(M ) ∈ S for all i < d, Hi

I,J(M ) ∈ S for all i < d − 1 Then HomR(R/m, HI,Jd−1(M )) ∈ S by the inductive hypothesis Applying the functor HomR(R/m, −) to the short exact sequence

0 → Im f → HI,Jd−1(M ) → Im g → 0

we get a long exact sequence

0 → HomR(R/m, Im f ) → HomR(R/m, HI,Jd−1(M )) →

→ HomR(R/m, Im g) → Ext1R(R/m, Im f ) → · · · Note that Ext1R(R/m, Im f ) ∈ S, so HomR(R/m, Im g)) ∈ S Now from the exact sequence

0 → Im g → HI,Jd (M )→ H.x d

I,J(M )

we obtain an exact sequence

0 → HomR(R/m, Im g) → HomR(R/m, HI,Jd (M ))→ Hom.x R(R/m, HI,Jd (M ))

It is clear that Im(HomR(R/m, HI,Jd (M ))→ Hom.x R(R/m, HI,Jd (M ))) =

0 Hence HomR(R/m, Im g) ∼= HomR(R/m, Hd

I,J(M )) and then HomR(R/m, Hd

I,J(M )) ∈ S The proof is complete 

It should be mentioned that if M is a finitely generated module over

a local ring (R, m) with Supp(M ) ⊆ {m}, then M is artinian From 3.2 we obtain the following consequence

Corollary 3.3 Let M be a finitely generated module over a local ring (R, m) and d a non-negative integer If HI,Ji (M ) is finitely generated for all i < d, then HomR(R/m, Hd

I,J(M )) has finite length

Proof It follows from 3.2 that HomR(R/m, HI,Jd (M )) is finitely gener-ated Moreover Supp(HomR(R/m, Hd

I,J(M ))) ⊆ {m} Therefore HomR(R/m, Hd

I,J(M )) is an artinian R-module and then it has finite

Acknowledgments We would like to express our gratitude to Pro-fessor Nguyen Tu Cuong for his support and advice The first author

is partially supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam

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Department of Mathematics-Informatics, Ho Chi Minh University

of Pedagogy, Ho Chi Minh city, Viet Nam.

E-mail address: namtuantran@gmail.com

Department of Natural Science Education, Dong Nai University, Dong Nai, Viet Nam.

E-mail address: triminhng@gmail.com