亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Co fi niteness of Local Cohomology Modules with Respect to a Pair of Ideals

        2014-03-03 00:49:14

        (Department of Mathematics,Soochow University,Suzhou,Jiangsu,215006)

        Co fi niteness of Local Cohomology Modules with Respect to a Pair of Ideals

        GU YAN

        (Department of Mathematics,Soochow University,Suzhou,Jiangsu,215006)

        Communicated by Du Xian-kun

        Let R be a commutative Noetherian ring,I and J be two ideals of R,and M be an R-module.We study the co fi niteness and fi niteness of the local cohomology module(M)and give some conditions for the fi niteness of HomR(R/I,and(R/I,(M)).Also,we get some results on the attached primes of HdimMI,J(M).

        local cohomology,co fi nite module,attached prime

        1 Introduction

        Throughout this paper,we always assume that R is a commutative Noetherian ring,I and J are two ideals of R,and M is an R-module.Takahashiet al.[1]introduced the concept of local cohomology module(M)with respect to a pair of ideals(I,J).The set of elements x of M such that In?Ann(x)+J for some integer n?1 is said to be(I,J)-torsion submodule of M and is denoted byΓI,J(M).For an integer i≥0,the local cohomology functorwith respect to(I,J)is de fi ned to be the i-th right derived functor ofΓI,J.Note that,if J=0,thencoincides withWhen M is fi nitely generated,we know that(M)=0 for i>dimM from Theorem 4.7 in[1].

        Hartshorne[2]de fi ned an R-module M to be I-co fi nite if

        is fi nitely generated for all i≥0.Also,he asked the following question:

        QuestionIf M is fi nitely generated,when is ExtjR(R/I,HiI(M)) fi nitely generated forall i≥0 and j≥0(considering Supp((M))?V(I),so(R/I,(M))is fi nitely generated if and only ifis I-co fi nite).

        Hartshorne[2]showed that if(R,m)is a complete regular local ring and M is fi nitely generated,thenis I-co fi nite in two cases:

        (a)I is a non-zero principal ideal;

        (b)I is a prime ideal with dimR/I=1.

        Yoshida[3],Del fi no and Marley[4]extended(b)to all dimension one ideals I of any local ring R,and Kawasaki[5]proved(a)for any ring R.

        Let

        As a generalization of I-co fi nite module,we give the following de fi nition:

        De fi nition 1.1AnR-moduleMis said to be(I,J)-co fi nite ifSuppM ?W(I,J)and(R/I,M)is fi nitely generated for alli≥0.

        For an R-module M,the cohomological dimension of M with respect to I and J is de fi ned as

        When J=0,then cd(I,J,M)coincides with cd(I,M).

        In this paper,we mainly consider the(I,J)-co fi niteness ofSince

        we focus on the fi niteness of(R/I,(M)).

        In Section 2,we discuss the fi niteness of HomR(R/I,(M))(see Theorem 2.1),which generalizes Theorem 2.1 in[6]and Theorem B(β)in[7].In addition,when M is fi nitely generated and I is a principal ideal or cd(I,J,M)=1,we get the(I,J)-co fi niteness offor all i≥0,which generalizes the corresponding results in[5]and[9],respectively. In Proposition 2.3(iii)of[10],it is proved that if

        then

        In this paper,we get the corresponding result for the local cohomology module with respect to(I,J).In Section 3,we prove the(I,J)-co fi niteness ofwhich is a generalization of Theorem 3 in[4].

        2 The Co fi niteness of(M)

        First,we give a theorem which is a generalization of Theorem 2.1 in[6]and Theorem B(β) in[7].It is also a main result of this paper.

        Theorem 2.1Assume thatMis anR-module,andsis a non-negative integer,such that(R/I,M)is fi nitely generated.If(M)is(I,J)-co fi nite for alli<s,thenHomR(R/I,(M))is fi nitely generated.In particular,Ass((M))∩V(I)is a fi nite set.

        Proof.We use induction on s.Let s=0.Then

        This result is clear.Now we assume that s>0,and the result has been proved for smaller values of s.Since(M)is(I,J)-co fi nite,(R/I,(M))is fi nitely generated for all i≥0.The short exact sequence

        induces the exact sequence

        It is easy to see that

        So,we may assume that

        Then

        Let E be an injective hull of M and put N=E/M.Then

        From the short exact sequence

        we have that

        and

        Applying the inductive hypothesis to N,it yields the fi niteness of HomR(R/I,(N)). Hence HomR(R/I,(M))is fi nitely generated.It follows that Ass((M))∩V(I)is a fi nite set.

        Proposition 2.1Assume thatMis anR-module,andsis a non-negative integer,such that(R/I,M)is fi nitely generated.If(R/I,(M))is fi nitely generated for alli<s,then(R/I,(M))is fi nitely generated.

        Proof.We prove the result by induction on s.When s=0,from the short exact sequence

        we get the exact sequence

        Now we suppose that s>0,and the claim has been proved for smaller values of s.From the exact sequence

        Proposition 2.2Assume thatMis fi nitely generated,andsis a non-negative integer, such that(M)is Artinian for alli<s.Then(M)is(I,J)-co fi nite for alli<s.

        Proof.We use induction on i.When i=0,since M is fi nitely generated,the result is clear. Now we suppose that i>0,and the result has been proved for smaller values of i.By the inductive hypothesis,(M)is(I,J)-co fi nite for all j<i.Hence,HomR(R/I,(M)) is fi nitely generated by Theorem 2.1,that is,0I is fi nitely generated,and thus

        where I=(a1,a2,···,at)for all k≥0 by Theorem 5.1 in[8].By the same proof with Theorem 5.5 in[9],we deduce that

        By Theorem 2.1 and Proposition 2.2,we have the following results.

        Proposition 2.3Assume thatMis fi nitely generated,andsis a non-negative integer, such that(M)is Artinian for alli<s.ThenHomR(R/I,(M))is fi nitely generated.In particular,Ass((M))∩V(I)is a fi nite set.

        Proposition 2.4Assume thatMis anR-module,andsis a non-negative integer,such that(R/I,(M))is fi nitely generated for alliandj(respectively fori≤sand allj).Then(R/I,M)is fi nitely generated for alli(respectively fori≤s).

        Proof.The case s=0 is clear.Now we suppose that s>0 and the case s?1 is settled. Set N=M/(M).Then we have the long exact sequence

        and

        Furthermore,we get the fi niteness of(R/I,L)for all i(respectively for i≤s?1)by the inductive hypothesis.So(R/I,M)is fi nitely generated for all i(respectively for i≤s).The proof is completed.

        Corollary 2.1Assume thatMis anR-module,andsis a non-negative integer,such that(M)is(I,J)-co fi nite for alli(respectively for alli≤s).Then(R/I,M)is fi nitely generated for alli(respectively for alli≤s).

        Proposition 2.5Assume thatMis anR-module,sis a non-negative integer,such that(R/I,M)is fi nitely generated for alli≥0,and(M)is(I,J)-co fi nite for alli=s. Then(M)is(I,J)-co fi nite.

        Proof.We prove the result by induction on s.Let N=M/(M).Then we have that

        we get the following long exact sequence:

        Now we assume that s>0 and the claim holds for s?1.We see that(R/I,N)is fi nitely generated for all i≥0.By using the above long exact sequence and similar argument to that of Theorem 2.1 and letting L=E(M)/M,we can get

        Corollary 2.2Assume thatMis a fi nitely generatedR-module,andcd(I,J,M)=1. Then(M)is(I,J)-co fi nite for alli≥0.

        Proposition 2.6Assume thatMis a fi nitely generatedR-module,andIis a principal ideal.Then(M)is(I,J)-co fi nite for alli≥0.

        Proof.Note that(M)=0 for all i>ara(IRˉ)by Proposition 4.11 in[1],where

        So when I is a principal ideal,(M)=0 for all i>1.Since(M)is fi nitely generated,(M)is(I,J)-co fi nite for all i=1.The result is clear.

        Proposition 2.7Assume thatMis anR-module.Then we have

        Proof.Let E be the injective hull of M/(M).Put N=E/(M/(M)).Since

        we have

        From the exact sequence

        we have

        and

        Hence

        Next we give a proposition,which generalizes Proposition 2.3(iii)of[10].

        Proposition 2.8Assume thatMis anR-module,andsis a positive integer such that

        Then

        Proof.When s=1,(M)=0.The result is clear,since

        Suppose that s>1 and the claim holds for s?1.Let E be an injective hull of M.Put N=E/M.Since(M)=0,we have(E)=0.By the exact sequence

        we have

        So

        Thus

        by the induction hypothesis.Hence

        3 The Co fi niteness of(M)

        In this section,we always assume that(R,m)is a commutative Noetherian local ring,I and J are two ideals of R,and M is an R-module.

        Assume that M is fi nitely generated.We know that(M)is Artinian by Theorem 2.1 in[11].Next,we discuss Att(M))and the co fi niteness of(M).

        Lemma 3.1[12]Suppose thatMis a non-zero fi nitely generatedR-module of dimensionn.Then

        Next,we show the co fi niteness of(M).The following proposition is the key for this fact.

        Proposition 3.1Assume thatRis a complete ring andMis a fi nitely generatedR-module of dimensionn.Then

        Proof.Since(M)is Artinian,we have

        Hence

        by Theorem A in[13]and Lemma 3 in[4],therefore

        Now the result is clear by Lemma 3.1.

        Next,we give the main result of this section.

        Theorem 3.1Assume thatMis a fi nitely generatedR-module of dimensionn.ThenHIn,J(M)is(I,J)-co fi nite.In fact,(R/I,(M))has fi nite length for alli≥0. Proof.Considering that(M)is Artinian,we know that(R/I,(M))is Artinian and

        So we assume that R is complete.We know that D((M))is fi nitely generated and Att((M))is a fi nite set.Suppose that

        Then

        From Matlis Duality Theorem,we get

        Since

        by Theorem 11.57 in[14],it is enough for us to show that

        In fact

        by Proposition 3.1.

        [1]Takahashi R,Yoshino Y,Yoshizawa T.Local cohomology based on a nonclosed support de fi ned by a pair of ideals.J.Pure Appl.Algebra,2009,213:582–600.

        [2]Hartshorne R.Affine duality and co fi niteness.Invent.Math.,1970,9:145–164.

        [3]Yoshida K I.Co fi niteness of local cohomology modules for ideals of dimension one.Nagoya Math.J.,1997,147:179–191.

        [4]Del fi no D,Marley T.Co fi nite modules and local cohomology.J.Pure Appl.Algebra,1997, 121:45–52.

        [5]Kawasaki K I.Co fi niteness of local cohomology modules for principal ideals.Bull.London Math.Soc.,1998,30:241–246.

        [6]Dibaei M,Yassemi S.Associated primes and co fi niteness of local cohomology modules. Manuscripta Math.,2005,117:199–205.

        [7]Khashyarmanesh K,Salarian S.On the associated primes of local cohomology modules.Comm. Algebra,1999,27:6191–6198.

        [8]Melkersson L.Some applications of a criterion for artinianness of a module.J.Pure Appl. Algebra,1995,101:291–303.

        [9]Melkersson L.Modules co fi nite with respect to an ideal.J.Algebra,2005,285:649–668.

        [10]Hassanzadeh S H,Vahidi A.On vanishing and co fi niteness of generalized local cohomology modules.Comm.Algebra,2009,37:2290–2299.

        [11]Chu L Z,Wang Q.Some results on local cohomology modules de fi ned by a pair of ideals.J. Math.Kyoto.Univ.,2009,49:193–200.

        [12]Chu L Z.Top local cohomology modules with respect to a pair of ideals.Proc.Amer.Math. Soc.,2011,139:777–782.

        [13]Dibaei M T,Yassemi S.Attached primes of the top local cohomology modules with respect to an ideal.Arch.Math.,2005,84:292–297.

        [14]Rotman J.An Introduction to Homological Algebra.Orlando,FL:Academic Press,1979.

        tion:13D45,13E15

        A

        1674-5647(2014)01-0033-08

        Received date:April 15,2011.

        Foundation item:The NSF(BK2011276)of Jiangsu Province,the NSF(10KJB110007,11KJB110011)for Colleges and Universities in Jiangsu Province and the Research Foundation(Q3107803)of Pre-research Project of Soochow University.

        E-mail address:guyan@suda.edu.cn(Gu Y).

        中文字幕一区二区三区久久网| 午夜免费福利一区二区无码AV | 又爽又猛又大又湿的视频| 视频一区二区三区黄色| 亚洲av日韩aⅴ无码色老头| 久久久久久av无码免费看大片 | av高潮一区二区三区| 97人伦影院a级毛片| 久久精品亚洲中文字幕无码网站| 亚洲激情人体艺术视频| 不卡av一区二区在线| 真实的国产乱xxxx在线| 最近中文字幕在线mv视频在线| 人妻无码ΑV中文字幕久久琪琪布| av网站免费观看入口| 亚洲日韩国产av无码无码精品| 国产亚洲精品久久久久秋霞| 久久婷婷国产综合精品| 亚洲国产综合人成综合网站| 国产精品永久免费| 免费a级毛片在线观看| 女同久久精品国产99国产精| 天堂av在线美女免费| 国产农村乱子伦精品视频| 久久久久久一级毛片免费无遮挡 | 亚洲AV秘 片一区二区三| 热综合一本伊人久久精品| 日本污ww视频网站| 精品久久久无码中文字幕| 久久久久久人妻一区二区无码Av| 一区二区三区午夜视频在线| 东北女人毛多水多牲交视频| 人妻无码一区二区在线影院| 国产av精品一区二区三区视频| 国产97色在线 | 国产| 好大好硬好爽免费视频 | 99久久婷婷国产精品综合网站| 亚洲熟妇丰满多毛xxxx| 久久麻豆精品国产99国产精| 日本高清免费播放一区二区| 国语自产精品视频在线看|