Cominimax modules and generalized local cohomology modules

Open Access Full Text Article

Research Article

Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam

Correspondence

Nguyen Minh Tri, Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam

Email: nguyenminhtri@dnpu.edu.vn

History

•Received:2019-07-11

•Accepted:2020-02-11

•Published:2020-03-24 DOI :10.32508/stdj.v23i1.1696

Copyright

© VNU-HCM Press.This is an open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license.

Cominimax modules and generalized local cohomology modules

Bui Thi Hong Cam, Nguyen Minh Tri

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ABSTRACT

The local cohomology theory plays an important role in commutative algebra and algebraic ge- ometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. AnR-moduleMis I-cominimax ifSupp_RM⊆V(I) andExtⁱ_R(R/I,M)is minimax for alli≥0. The aim of this paper is to show some conditions such that the generalized local cohomology moduleH_I^′(M,N)is I-cominimax for alli≥0. We prove that H_Iⁱ(M,K)if is I-cofinite for alli≥0. We prove that ifH_Iⁱ(M,K)is I-cofinite for alli<tand all finitely generatedR-moduleK, thenH_Iⁱ(M,N)isI-cominimax for alli<tand all minimaxR-moduleN. IfM is a finitely generatedR-module,Nis a minimaxR-module andtis a non-negative integer such that dim Supp_RH_Iⁱ(M,N)≤1for alli<t, thenH_Iⁱ(M,N)isI-cominimax for all i < t . WhendimR/I≤1 andH_Iⁱ(N)isI-cominimax for alli≥0, thenH_I^′(M,N)is I-cominimax for alli≥0.

Key words:Generalized local cohomology, I-cominimax

INTRODUCTION

LetRbe a local Noetherian ring,Ian ideal ofRandM a finitely generatedR-module. It is well known that the local cohomology modulesH_Iⁱ(M)are not gen- erally finitely generated for i > 0. In a 1970 paper Hartshorne¹gave the concept of I-cofinite modules.

AnR-moduleKto be I-cofinite ifSupp_RK⊆V(I) andExt_R^j(R/I,K)is finitely generated for all j≥0.

Hartshorne asked which ringsRand idealsIthe mod- ulesH_Iⁱ(M)wereI-cofinite for alliand all finitely gen- erated modulesM.

In 1, if (R,m) is a complete regular local ring and Mis a finitely generatedR-module, thenH_Iⁱ(M)is I -cofinite in two cases:

• I is a nonzero principal ideal, or

• I is a prime ideal withdimR/I=1

In 1991, Huneke and Koh²proved that ifRis a com- plete local Gorenstein domain,Iis a one dimension ideal ofRandMis a finitely generatedR-module, then H_lⁱ(M)isI-cofinite for alli. In 1997, Yoshida in³or Delfino and Marley in⁴extended (b) to all one di- mension idealsIof an arbitrary local ringR. In 1998, Kawasaki⁵ proved (a) in an arbitrary commutative Noetherian ring. The local condition in (b) has been removed by Bahmanpour and Naghipour in⁶. In⁷, Herzog gave a generalizations of the local coho- mology theory. Letjbe a non-negative integer andM a finitely generatedR-module anNanR-module. The

j-th generalized local cohomology module ofMand Nwith respect toIis defined by

H_I^j(M,N)∼=lim

⃗n

(

Ext_R^j(M/IⁿM,N) )

We see that ifM=R, thenH_I^j(M,N) =H_I^j(N)the usual local cohomology module of Grothendieck⁸. Another similar question is: When is the module H_I^j(M,N)I-cofinite for all j≥0?

In 2001, Yassemi [⁹, Theorem 2.8] showed that in a Gorenstein ring,H_I^j(M,N)isI-cofinite for allj≥0 whereIis non-zero principal ideal. In 2004, Divaani- Aazar and Sazeedeh [¹⁰, Theorem 2.8 and Theorem 2.9] have eliminated the Gorenstein hypothesis and showed that if either

1. I is principal, or

2. Ris complete local andIis a prime ideal with dimR/I=1, thenH_I^j(M,N)isI-cofinite for all

j≥0.

When I is a principal ideal, Cuong, Goto and Hoang [¹¹, Theorem 1.1] gave another proof forH_I^j(M,N)is I-cofinite for all j. They also showed that ifdimM≤2 ordimN≤2, thenH_I^j(M,N)isI-cofinite for allj.

An extension ofI-cofinite modules isI-cominimax modules which was introduced in 2009¹². An R- moduleMis calledI-comiminax ifSupp_RM⊆V(I) andExtⁱ_R(R/I,M)is minimax for alli≥0(see [2, 3.1 and 2.2(ii)]). Naturally, we have a question:

Cite this article :Hong Cam B T, Minh Tri N.Cominimax modules and generalized local cohomology modules.Sci. Tech. Dev. J.;23(1):479-483.

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Science & Technology Development Journal, 23(1):479-483

Question: When are the modules

H_Iⁱ(N)orH_Iⁱ(M,N)I-cominimax for alli≥0?

In [2, 3.10], we see that ifNis anI-minimaxR-module andIis a principal ideal, thenH_Iⁱ(N)isI-cominimax for alli≥0. In 2011, Mafi¹³proved that ifN is a minimax R-module, thenH_Iⁱ(N)isI-cominimax for alli≥0when one of the following cases holds:

1. dimR/I≤1;

2. cd(I) =1;

3. dimR≤2.

In¹⁴, the authors showed that, ifMis a minimaxR- module withSupp_R(

H_Iⁱ(M))

≤1for alli≥0andN is a finitely generatedR-module withSupp_RN⊆V(I), thenExt_R^j(

N,H_lⁱ(M))

is minimax for alli≥0.

In [18], the authors proved that in a local ring, if M is a finitely generated R-module and N, L are two minimaxR-modules withSupp_RL⊆V(I), then Ext_R^j(

L,H_lⁱ(M,N))

is minimax for all i and j when one of the following cases holds:

1. dimR/I≤1;

2. cd(I) =1;

3. dimR≤2.

The aim of this paper is to study theI-cominimaxness ofH_Iⁱ(M,N). Theorem 2.2 shows that ifH_Iⁱ(M,K)isI- cofinite for alli<tand all finitely generatedR-module K, thenH_Iⁱ(M,N)isI-cominimax for alli<tand all minimaxR-moduleN. We will see in Theorem 2.4 that ifMis a finitely generatedR-module,Nis a minimax R-module andtis a non-negative integer such that dim Supp_RH_lⁱ(M,N)≤1for alli<t,thenH_Iⁱ(M,N) isI- cominimax for alli<t.WhendimR/I≤1, The- orem 2.9 shows that ifH_Iⁱ(N)isI-cominimax for all i≥0, thenH_lⁱ(M,N)isI-cominimax for alli≥0.

MAIN RESULTS

In¹⁵, Zöschinger introduced the class of minimax modules. AnR-moduleKis said to be a minimax module, if there is a finitely generated submoduleT ofK, such thatK/Tis Artinian.

Remark 2.1There are some elementary properties of minimax modules:

1. The class of minimax modules contains all finitely generated modules and all Artinian modules.

2. Let0→L→M→N→0be an exact sequence ofR-modules. Then,Mis minimax if and only if LandNare both minimax. Thus, any submod- ule and quotient of a minimax module is mini- max. Moreover, ifNis finitely generated andM is minimax, thenExt_R^j(N,M)andTor^R_j(N,M) are minimax for allj≥0.

3. The set of associated primes of any minimaxR- module is finite.

4. IfMis a minimaxR-module and p is a non- maximal prime ideal ofR, thenMpis a finitely generatedRp-module.

Definition 2.1(Azami, Naghipour and Vakili) AnR- module M isI-cominimax ifSupp_RM ⊆V(I)and Extⁱ_R(R/I,M)is minimax for alli≥0

The following result is a generalization of [14, 2.3].

Theorem 2.2Let t be a non-negative integer. Assume thatH_Iⁱ(M,K)is I-cofinite for all i < t and all finitely generated R-module K. ThenH_Iⁱ(M,N)is I-cominimax for all i < t and all minimax R-module N.

Proof. SinceN is a minimax R-module, there is a finitely generatedR-moduleKof such thatN/Kis ar- tinian. From the short exact sequence0→K→N→ N/K→0we get the following exact sequence

···→H_Iⁱ(M,K)→^fi H_Iⁱ(M,N)→^gi H_Iⁱ(M,N/K)→^hi H_Iⁱ⁺¹(M,K)→ ···

Now, the short exact sequence

0→lmfi→H_Iⁱ(M,N)→Imgi→0 induces a long exact sequence

··· →Ext_R^j(R/I,Imfi)→Ext_R^j (

R/I,H_Iⁱ(M,N) )→

Ext_R^j(R/I,Imgi)→Ext_R^j+1(R/I,Imfi)→ ···

gives rise to a long exact sequence I -cofinite

··· →Ext_R^j(R/I,Imh_i−1)→Ext_R^j (

R/I,H_Iⁱ(M,K) )

→Ext_R^j(R/I,Imfi)→Ext_R^j+1(R/I,Imhi−1)→ ···

By the hypothesis,H_Iⁱ(M,K)isI-cofinite for alli≥0.

HenceExt_R^j(

R/I,H_Iⁱ(M,K))

is finitely generated for alli<t,j≥0. SinceN/Kis artinian, it follows from [16, 2.6] thatH_Iⁱ(M,N/K)is artinian for alli≥0. It is easy to see thatExt_R^j(R/I,Imhi−1)is Artinian for alli,j≥0. Consequently,Ext_R^j(R/I,Imfi)is mini- max for alli<t,j≥0. SinceImgiis a submodule ofH_Iⁱ(M,N/K), it follows thatExt_R^j(R/I,Img_i)is ar- tinian for alli,j≥0. ThusExt_R^j(

R/I,H_Iⁱ(M,N)) is minimax for alli<t,j≥0.

Before showing a consequence of Theorem 2.2, we re- call the concept of the local cohomology dimension of an ideal.

Definition 2.3The cohomological dimension ofIin R, denoted bycd(I) is the smallest integernsuch that the local cohomology modulesH_Iⁱ(M) =0for allR- modulesM, and for alli>n.

We show some conditions such that the module H_Iⁱ(M,N)isI-cominimax for alli≥0.

Corollary 2.4 Let M be a finitely generated R-module and N a minimax R-module. If either

1. I is principal, or 2. dimM≤2, or 3. dimN≤2, or 4. cd(I)= 1,

then H_Iⁱ(M, N) is I-cominimax for all i ≥ 0.

Proof.(1),(2)and(3),Combining heorem2.2with [¹¹,1.1]or[¹¹ ,1.3],itfollowsthatH_Iⁱ(M,N)isI- cominimaxforalli ≥ 0.

4. follows from [¹⁶, 2.2] and Theorem 2.2.

Next, we will show some results concerning to small dimensions and R is an arbitrary (not local) commu- tative Noetherian ring.

Theorem 2.4Let M be a finitely generated R-module, N a minimax R-module and t a non-negative inte- ger such thatdim Supp_R(

H_Iⁱ(M,N))

≤1for alli<

t. ThenH_Iⁱ(M,N)is I-cominimax for all i < t and Hom(

R/I,H_I^t(M,N))

is minimax.

Proof. SinceNis minimax, there is a finitely generated R-moduleKof such thatN/Kis artinian. The short exact sequence0→K→N→N/K→0gives rise to a long exact sequence.

··· →H_Iⁱ(M,K)→^fi H_Iⁱ(M,N)→^gi H_Iⁱ(M,N/K)→^hi H_Iⁱ⁺¹(M,K)→ ···

SinceN/K is artinian, it follows from [¹⁷, 2.6] that H_Iⁱ(M,N/K)is artinian for alli≥0. By the assump- tion, we inducedim Supp_R(

H_Iⁱ(M,K))

≤1for alli<

t. It follows from [¹¹, 1.2] thatH_Iⁱ(M,K)isI-cofinite for alli<t. Now, the short exact sequence

0→lmf_i→H_Iⁱ(M,N)→Img_i→0

induces a long exact sequence

··· →Ext_R^j(R/I,Imfi)→Ext_R^j (

R/I,H_Iⁱ(M,N) )

→Ext_R^j(

R/I,lmgj

)→Ext_R^j+1(R/I,Imfi)→ ···

Note thatExt_R^j(R/I,Img_i)is artinian for alli,j≥0.

Leti<t, the short exact sequence

0→lmhi−1→H_Iⁱ(M,K)→Imfi→0

induces a long exact sequence

··· →Ext_R^j(R/I,Imh_i−1)→ Ext_R^j

(

R/I,H_Iⁱ(M,K)

)→Ext_R^j(R/I,Imfi)→

Ext_R^j+1(R/I,Imh_i−1)→ ···. Since H_Iⁱ(M,K) is I-cofinite, it follows that Ext_R^j(

R/I,H_Iⁱ(M,K))

is finitely generated for all j≥0. By the artinianness ofExt_R^j(R/I,Imh_i−1), we can conclude thatExt_R^j(R/I,Imfi)is minimax for all

j≥0. ThereforeExt_R^j(

R/I,H_Iⁱ(M,N))

is minimax for allj≥0. We have two following exact sequences 0→HomR(R/I,Imft)→HomR

(R/I,H_I^t(M,N))

→ HomR(R/I,Imgt)

and HomR

(R/I,H_I^t(M,K))

→HomR(R/I,Imft)→ Ext¹_R(R/I,Imht−1)→ ···

Sincedim Supp_R(

H_Iⁱ(M,K))

≤1for alli<t, it follows from [4, Theorem 1.2] thatHomR

(R/I,H_I^t(M,K)) is finitely generated. On the other hand, Ext¹_R(R/I,Imh_t−1) is an artinian R-module.

Therefore HomR(R/I,Imft) is minimax. We see that HomR(R/I,Img_t) is artinian and then HomR

(R/I,H_I^t(M,N))

is minimax.

In [¹⁸, 3.1 and 3.2], the authors showed thatH_Iⁱ(M,N) isI-cominimax for alli≥0when dimR/I≤1where Ris a local ring. Now we consider thatRis not a local ring.

Corollary 2.5 Let M be a finitely generated R-module, N a minimax R-module and t a non-negative inte- ger. Assume thatdimM/IM ≤1or dimN ≤1or dimR/I≤1. ThenH_Iⁱ(M,N)is I-cominimax for all i≥0.

Corollary 2.6 Let M be a finitely generated R-module, N a minimax R-module and t a non-negative inte- ger. Assume thatSupp_R(

H_Iⁱ(M,N))

is finite for all i < t. Then H_Iⁱ(M,N) is I-cominimax for all i < t andHomR

(R/I,H_I^t(M,N))

is minimax. In particular, Ass(

H_I^t(M,N))

is a finite set.

Proof. Since Supp_R(

H_Iⁱ(M,N))

is a finite set, we can conclude thatdim Supp_R(

H_Iⁱ(M,N))

≤1. It fol- lows from Theorem 2.4 thatH_Iⁱ(M,N)isI-cominimax for alli<tandHomR

(R/I,H_I^t(M,N))

is minimax.

Moreover, we have Ass(

H_I^t(M,N))

=Ass( HomR

(R/I,H_I^t(M,N))) By Remark 2.1.3,Ass(

H_I^t(M,N))

is a finite set.

Corollary 2.7Let N be a non-zero minimax R-module and I an ideal of R. Let t be a non-negative integer such thatdim Supp_RH_Iⁱ(N)≤1for all i < t . Then the fol- lowing statements hold:

1. the R-modulesH_Iⁱ(N)are I-cominimax for all i<

t;

2. the R-moduleHomR

(R/I,H_I^t(N))

is minimax.

Lemma 2.8Let M be a finitely generated R-module such thatSupp_RM⊆V(I)and N an I-cominimax R- module. ThenExtⁱ_R(M,N)is minimax for alli≥0.

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Science & Technology Development Journal, 23(1):479-483

Proof. The proof is by induction oni. SinceNis an I-cominimaxR-module, the moduleExtⁱ_R(R/I,N)is minimax for alli≥0. By Gruson’s theorem, there is a chain of submodules ofM.

0=M₀⊆M₁⊆. . .⊆M_k=M

such that Mj/Mj−1 is a homomorphic image of (R/I)^tfor some positive integert. We consider short exact sequences

0→K→(R/I)^m→M₁→0

and

0→Mj−1→Mi→Mj/Mj−1→0

The first exact sequence induces a long exact sequence 0→HomR(M1,N)→HomR((R/I)^m,N)→

HomR(K,N)→ ···

whereK is a submodule of(R/I)^m for some posi- tive integer numberm. SinceHomR((R/I)^m,N)∼= Hom_R(R/I,N)^m, it follows that Hom_R(M₁,N) is minimax. By similar arguments, we also get that Hom_R(

M_j/M_j−1,N)

is minimax for all1≤i≤k. Now, the exact sequence

0→HomR

(Mj/M_j−1,N)

→HomR

(Mj,N)

→ HomR

(M_j−1,N)

→ ···

deduces thatHomR

(Mj,N)

is minimax for alljand thenHomR(M,N)is minimax. Therefore, we have the conclusion wheni= 0.

Leti> 0. The short exact sequence 0→K→(R/I)^m→M1→0

gives rise to a long exact sequence

··· →Extⁱ_R⁻¹(K,N)→Extⁱ_R(M₁,N)→ Extⁱ_R(

(R/I)^t,N)

···

By the inductive hypothesis, Extⁱ_R⁻¹(K,N) is a minimax R-module. Since Extⁱ_R((R/I)^m,N) ∼= Extⁱ_R(R/I,N)^m, it follows thatExtⁱ_R(M1,N)is mini- max. Analysis similar to the above proof, we have Extⁱ_R(Mk,N) is minimax and which completes the proof.

The following result shows a connection on the I- cominimaxness ofH_Iⁱ(N)andH_Iⁱ(M,N)whenRis not a local ring andNis an arbitraryR-module.

Theorem 2.9Let M be a finitely generated R-module withpd(M)<∞and N an R- module. Let I be an ideal of R withdimR/I=1and t a non-negative integer such thatH_Iⁱ(N)is I-cominimax for all i < t. ThenH_Iⁱ(M,N) is I-cominimax for all i < t.

Proof. We prove by induction onp=pd(M). Ifp= 0, thenMis a projectiveR-module. It follows from [¹⁹, 2.5] thatH_Iⁱ(M,N)∼=HomR

(M,H_Iⁱ(N))

for alli≥0.

By [²⁰, 10.65], we have Ext_R^j

(

R/I,HomR

(

M,H_Iⁱ(N) )∼=Ext_R^j

(

M/IM,H_Iⁱ(N) ) for alli<t,j≥0. ThereforeExt_R^j(

R/I,H_Iⁱ(M,N))∼= Ext_R^j

(

M/IM,H_I^j(N) )

whereExt_R^j(

M/IM,H_Iⁱ(N)) is minimax for allj≥0by Lemma 2.8 and then the as- sertion follows.

Letp> 0 and the statement is true for all finitely gen- eratedR-module with projective dimension less than p. There is a short exact sequence

0→K→P→M→0,

whereKis finitely generated,Pis projective finitely generated. Note thatpd(K) =p−1and then by the inductive hypothesisH_Iⁱ(K,N)isI-cominimax for all i<t. On the other hand, there is a long exact sequence

··· →H_Iⁱ(K,N)→H_Iⁱ⁺¹(M,N)→H_Iⁱ⁺¹(P,N)→ ···

in whichH_Iⁱ(K,N)andH_Iⁱ(P,N)areI-cominimax for alli≥0. It follows from [²¹, 2.6] thatH_Iⁱ(M,N)is also I-comiminax for alli≥0and the proof is complete.

COMPETING INTERESTS

The authors declare that they have no conflicts of in- terest.

AUTHOR CONTRIBUTION

Bui Thi Hong Cam has contributed the Theorem 2.2 and has written the manuscript. Nguyen Minh Tri has contributed the Theorem 2.4, 2.9 and revising the manuscript.

ACKNOWLEDGMENTS

The authors would like to thank the referees for his or her substantial comments. This work was supported by Dong Nai University.

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