Graded local cohomology modules with respect to the linked ideals
aa r X i v : . [ m a t h . A C ] F e b GRADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THELINKED IDEALS
MARYAM JAHANGIRI ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI Abstract.
Let R = ⊕ n ∈ N R n be a standard graded ring, M be a finitely generated graded R -module and R + := ⊕ n ∈ N R n denotes the irrelevant ideal of R . In this paper, consideringthe new concept of linkage of ideals over a module, we study the graded components H i a ( M ) n when a is an h-linked ideal over M . More precisely, we show that H i a ( M ) is tame in each ofthe following cases:(i) i = f R + a ( M ), the first integer i for which R + * p H i a ( M );(ii) i = cd( R + , M ), the last integer i for which H iR + ( M ) = 0, and a = b + R + where b isan h-linked ideal with R + over M .Also, among other things, we describe the components H i a ( M ) n where a is radically h- M -licci with respect to R + of length 2. introduction Throughout the paper, R = L n ∈ N R n is a standard graded Noetherian ring, i.e. R is acommutative Noetehrian ring and R is generated, as an R -algebra, by finitely many elementsof degree one, R + = L n ∈ N R n is the irrelevant ideal of R and a and b are homogeneous idealsof R . Also, M denotes a finitely generated graded R -module.For i ∈ N , the set of non-negative integers, and n ∈ Z , the set of integers, let H i a ( M ) n denotes the n -th component of graded local cohomology module H i a ( M ) of M with respectto a (our terminology on local cohomology comes from [3]). It is well-known that H iR + ( M ) n is a finitely generated R -module for all n ∈ Z and H iR + ( M ) n = 0 for all n ≫ H iR + ( M ) n when n → −∞ has been studied bymany authors, too. See for example [1], [2], [5] and [12]. But, we know not much aboutthe graded components H i a ( M ) n where a is an arbitrary homogeneous ideal of R . Although,there are some studies in this topic, see for example [4], [10] and [16].In a recent paper ([9]), the authors introduced the concept of linkage of ideals over amodule, which is a generalization of its classical concept introduced by Peskine and Szpiro([15]). Mathematics Subject Classification.
Key words and phrases.
Graded local cohomology modules, linkage of ideals, finiteness dimension, coho-mological dimension, relative Cohen-Macaulay. ∗ Corresponding author. ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI Let c and d be ideals of the commutative Noetherian ring A with 1 = 0 and N be a finitelygenerated A -module. Assume that c N = N = d N and I ⊆ c ∩ d is an ideal generated byan N -regular sequence. Then the ideals c and d are said to be linked by I over N , denotedby c ∼ ( I ; N ) d , if c N = IN : N d and d N = IN : N c . In [7] and [8], the authors studied somecohomological properties of linked ideals.In this paper, we consider the above concept in the graded case. More precisely, thehomogeneous ideals a and b are said to be homogeneously linked (or h-linked) by I over M ,denoted by a h ∼ ( I ; M ) b , if I is generated by a homogeneous M -regular sequence and a and b are linked by I over M . We consider homogeneously linked ideals and study some of theircohomological properties.This paper is divided into three sections. In Section 2, we study some basic properties ofhomogeneously linked ideals. We Show by examples that if a h ∼ ( I ; M ) b , this doesn’t implythat a ∩ R ∼ ( I ∩ R ; M n ) b ∩ R for all n ∈ Z and vice versa. Although, in some cases it does(see 2.3 and 2.4). Due to the importance of irrelevant ideal in a standard graded ring, itis natural to ask whether a homogeneous ideal could be h-linked with R + . We answer thisquestion in some cases, see 2.5 and 2.8.Section 3, which is the main part of the paper, is devoted to study the graded componentsof local cohomology modules H i a ( M ) n where a is an h-linked ideal over M . Let N = L n ∈ Z N n be a graded R -module, end ( N ) is defined to be the last integer n for which N n = 0. Aswe mentioned above, end ( H iR + ( M )) < ∞ for all i ∈ N . Also, in [10], it is shown that end ( H i a ( M )) < ∞ for all i ∈ N and all homogeneous ideal a ⊇ R + . In Theorem 3.9,we show that if a is an h-linked ideal with R + over M , then end ( H i a ( M )) < ∞ for any i = grade( a , M ) and that end ( H grade( a ,M ) a ( M )) < ∞ or H grade( a ,M ) a ( M ) n = 0 for all n ≫ a , M ) denotes the length of a maximal M -regular sequence in a . f R + a ( M ) is defined to be the first integer i for which R + * p H i a ( M ). This invariant wasstudied in [12] as well in [3, § R -module N = L n ∈ Z N n is said to be tameif { n ∈ Z | N n = 0 and N n +1 = 0 } is a finite set. Tameness of local cohomology modules isone of the most fundamental concepts concerning this modules and attracts lots of interests,see for example [1], [2], [4], [5] and [18]. In [5, 2.2], the authors show that H f R + a ( M ) a ( M ) istame whenever a ⊇ R + . In theorem 3.14, we prove it without any restriction on a .In theorem 3.15, it is shown that if ( R , m ) is local, then f R + a + R + ( M ) is finite and that H cd( R + ,M ) a + R + ( M ) is tame in the case where a is an h-linked ideal with respect to R + over M .Here, cd( R + , M ) is the cohomological dimension of M with respect to R + , that is the lastinteger i for which H iR + ( M ) = 0.We keep the notations introduced in the introduction, throughout the paper.2. Homogenously linked ideals over a module
We start by the basic concept of the paper.
RADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THE LINKED IDEALS 3
Definition 2.1.
Assume that a M = M = b M and I ⊆ a ∩ b be an ideal generated by ahomogeneous M -regular sequence. Then we say that the ideals a and b are homogeneouslylinked (or h-linked) by I over M , denoted a h ∼ ( I ; M ) b , if b M = IM : M a and a M = IM : M b . The ideals a and b are said to be geometrically h-linked by I over M if, inaddition, a M ∩ b M = IM . Also, we say that the ideal a is h-linked over M if there existhomogeneous ideals b and I of R such that a h ∼ ( I ; M ) b . a is h- M -selflinked by I if a h ∼ ( I ; M ) a . Remark 2.2.
Note that, this definition is a special case of linkage of ideals over a module,studied in [9] . Moreover, if a and b are h-linked by I over M and grade( a , M ) = t then thefollowing statements hold. (i) If a M ∩ b M = IM , then IM : M ( a + b ) = IM . So, ( a + b ) ⊆ Z ( M/IM ) , the set ofzero divisors of M/IM , that results grade( a + b , M ) = t . (ii) If a M ∩ b M = IM (i.e. a and b are geometrically h-linked), then, by [7, 2.9] , grade( a + b , M ) = t + 1 . In the next example, we show that there is no bilateral relation between h-linkedness ofideals a and b by I over M with linkage of a ∩ R and b ∩ R by I ∩ R over homogeneous com-ponents of M . Example 2.3.
Let ( R , m ) be local with depth R > , m = m + R + be the homoge-neous maximal ideal of R and x , x , . . . , x s ( s ≥ ) be a homogeneous R -regular sequence in m . Assume that ≤ l < s such that deg x i = 0 for all ≤ i ≤ l and deg x i ≥ for all l < i ≤ s . • Set a := ( x , x , . . . , x s ) , I := ( x , . . . , x l , x l +1 , . . . , x s ) , a := a ∩ R = ( x , . . . , x l ) R and I := I ∩ R = ( x , . . . , x l ) R . So, by [9, 2.2] , a is h- R -selflinked by I . But, since a = I , a is not R n -selflinked by I , for all n. • Again, set a := ( x , x , . . . , x s ) , I := ( x , x , . . . , x s − ) , a := a ∩ R = ( x , . . . , x l ) R and I := I ∩ R = ( x , x , . . . , x l ) R . Then, grade( a , R ) = s = grade( I, R ) , so, by [9, 2.6(i)] , a is not h- R -selflinked by I . But, a is R n -selflinked by I for all n , using [9, 2.2] and the fact that x , x , . . . , x l is an R n -regular sequence for all n . Remark 2.4.
Assume that ( R , m ) is local, a and b are generated by elements of degreezero. Then, despite the above example, a h ∼ (0; M ) b if and only if ( a ∩ R ) ∼ (0; M n ) ( b ∩ R ) forall n ∈ Z . In view of the importance of the irrelevant ideal in a standard graded ring, it is naturalto study homogeneous ideals which are h-linked with R + .If R = R [ x , . . . , x n ] is a polynomial ring graded in the usual way, then R + = ( x , . . . , x n )is h- R -selflinked by ( x , x , . . . , x n ), using [9, 2.2]. In the next example, we find some homo-geneous ideals that are h-linked with R + . It will be used in the next section, too. Example 2.5.
MARYAM JAHANGIRI ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI (1) Let R = R [ x ] and x / ∈ Z ( M ) , then R + h ∼ (( r x t ); M ) ( r x t − ) for all r ∈ R \ Z ( M ) which is not unit and all t ≥ .(2) Let R = R [ x, y ] and r x, r y be an M -regular sequence where r , r ∈ R . Then R + h ∼ (( r x t ,r y t ′ ); M ) ( r x t , r y t ′ , r r x t − y t ′ − ) for all t, t ′ ≥ .Proof. The first case is obvious. To prove (2), we have to show that(i) ( x, y ) M = ( r x t , r y t ′ ) M : M ( r x t , r y t ′ , r r x t − y t ′ − ),(ii) ( r x t , r y t ′ , r r x t − y t ′ − ) M = ( r x t , r y t ′ ) M : M ( x, y ).Since r x, r y is an M -regular sequence, x, y and r x i , r y j are M -regular sequence for all i, j ≥
0. On the other hand, if x , x , . . . , x n is an M -regular sequence and m , m , . . . , m n ∈ M such that m x + . . . + m n x n = 0, then m ∈ ( x , . . . x n ) M .(i) Assume that m ∈ M such that m ( r x t , r y t ′ , r r x t − y t ′ − ) ⊆ ( r x t , r y t ′ ) M . So, mr r x t − y t ′ − ∈ ( r x t , r y t ′ ) M , then there exist m , m ∈ M such that ( mr y t ′ − − xm ) r x t − − r y t ′ m = 0. Thus mr y t ′ − − xm ∈ r y t ′ M and so there exists m ′ ∈ M such that ( m − m ′ y ) r y t ′ − ∈ M x . Hence, m − m ′ y ∈ M x and as a result m ∈ ( x, y ) M .(ii) Now, let m ∈ M such that m ( x, y ) ⊆ ( r x t , r y t ′ ) M . So, mx ∈ ( r x t , r y t ′ ) M and my ∈ ( r x t , r y t ′ ) M . Then, there exist m , m ∈ M such that ( m − r x t − m ) x − r y t ′ m = 0. Thus m − r x t − m ∈ r y t ′ M . So, we have m = r x t − m + r y t ′ m ′′ where m ′′ ∈ M . Same as above, m = r x t m ′′ + r y t ′ − m ′ where m ′ , m ′′ ∈ M . Hence,(*) m = r x t − m + r y t ′ m ′′ = r x t m ′′ + r y t ′ − m ′ . So, m − xm ′′ ∈ M ( r y t ′ − ) which implies that m = xm ′′ + nr y t ′ − for some n ∈ M .As a result, by (*), m ∈ ( r x t , r y t ′ , r r x t − y t ′ − ) M . (cid:3) It’s natural to ask whether a homogeneous ideal which is linked could be an h-linked ideal?In the following, we answer it in a special case.
Remark 2.6.
Let R be reduced and R + be a linked ideal by I over R . As R is reduced, R + is radical. Also, in view of [3, 16.1.2] , there exists an ideal, say I ′ , generated by ahomogeneous R -regular sequence of length t in R + , where t := grade R + and I ′ = R + . So,using [9, 2.8] , [6, Theorem 1] and [19, 1.4] , Ass
R/R + ⊆ Ass
R/I ∩ V ( R + ) = Ass Hom R ( R/R + , R/I )= Ass Ext tR ( R/R + , R ) = Ass R/I ′ ∩ V ( R + ) , where V ( R + ) denotes the set of prime ideals of R containing R + . This implies that R + isan h-linked ideal by I ′ , by [8, 2.5] . RADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THE LINKED IDEALS 5
Definition 2.7.
Following [14, 2.1] , a sequence x , x , . . . , x t of homogeneous elements of b issaid to be a homogeneous b -filter regular sequence on M if x i / ∈ p for all p ∈ Ass( M ( x ,...,x i − ) M ) \ V ( b ) and all i = 1 , . . . , t . Assume that a is generated by elements of positive degrees and a ⊆ b . By [10, 1.5],if Supp( M a M ) * V ( b ), then all maximal homogeneous b -filter regular sequences in a on M have the same finite length, that is denoted by f-grade( b , a , M ). Also f-grade( b , a , M ) := ∞ whenever Supp( M a M ) ⊆ V ( b ). Note that grade( a , M ) ≤ f-grade( b , a , M ).Moreover, Chu and Gu in [4, 2.4] in the case where b = R + , show that if Supp( M a M ) * V ( R + )then f-grade( R + , a , M ) = max { i | H j a ( M ) n = 0 , for all n ≫ j < i } . In the following proposition, we consider a polynomial ring and see whether a homoge-neous ideal could be h-linked with R + . Proposition 2.8.
Let ( R , m ) be a regular local ring containing a field of characteristic zeroand R = R [ x , . . . , x t ] be the polynomial ring graded in the usual way, that is deg( x i ) = 1 forall i = 1 , . . . , t . Then R + can’t be h-linked with any ideal a ) R + . Moreover, if a h ∼ ( I ; R ) R + and a * R + , then a and R + are geometrically h-linked by I over R .Proof. Let a h ∼ ( I ; R ) R + and suppose to the contrary that R + ( a . Since t := grade R + f − grade( a , R + , R ), so H t a ( R ) n is a finitely generated R -module for all n ∈ Z , by [10, 1.7].Thus H t a ( R ) = 0, using [16, 8.1], that is a contradiction, in view of [9, 2.6(i)].Now, assume that a h ∼ ( I ; R ) R + and a * R + . By [10, 1.7] and [16, 8.1], H i a + R + ( R ) = 0 forall i ≤ f − grade( a + R + , R + , R ). So, f − grade( a + R + , R + , R ) (cid:12) grade( a + R + ). Thusgrade R + (cid:12) grade( a + R + ) and, by 2.2(i), a ∩ R + = I . (cid:3) In the following proposition, we study the set Ass R ( M/ a M ) where a is an h-linked idealover M . It will be used later in the paper,too. Proposition 2.9.
Assume that a and b are geometrically h-linked by I over M and b ⊇ R + .Then (i) Ass R ( M/ a M ) = Ass R ( M/IM ) T V ( a ) ; (ii) Ass R ( M/ a M ) T Ass R ( M/ b M ) = ∅ ; (iii) Ass R ( M/ b M ) T V ( a ) = ∅ .The first case also holds if a and b are just h-linked and Ass R ( M/IM ) = minAss R ( M/IM ) (e.g. M is a Cohen-Macaulay module).Proof. (i) By [9, 2.9] and [13, Exercise 6.7], Ass R ( M/ a M ) = { p ∩ R | p ∈ Ass R ( M/IM ) T V ( a ) } and that Ass R ( M/IM ) = { p ∩ R | p ∈ Ass R ( M/IM ) } . This implies thatAss R ( M/ a M ) ⊆ Ass R ( M/IM ) T V ( a ).Now, let p ∈ Ass R ( M/IM ) T V ( a ). Then, there exists p ∈ Ass R ( M/IM ) such that
MARYAM JAHANGIRI ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI p ∩ R = p . √ M + I = p M/IM ⊆ p . Thus, by [7, 2.2] and the assumption, √ M + a ∩ R + ⊆ p . So, p ⊇ a and, again by [9, 2.9], p ∈ Ass R ( M/ a M ).(ii) Let p ∈ Ass R ( M/ b M ) then, by [9, 2.9], there exists p ∈ Ass R ( M/IM ) T V ( b )such that p ∩ R = p and p / ∈ Ass R ( M/ a M ). So, p / ∈ V ( a ). On the other hand, p ⊇ b ⊇ R + thus p + a and by (i), p / ∈ Ass R ( M/ a M ).(iii) Follows from (i) and (ii). (cid:3) If we remove the condition b ⊇ R + , then the above proposition does not hold any more,as the following example shows. Example 2.10.
Let a and R + be geometrically h-linked by I over R , then Ass R ( R/R + ) =Ass R ( R/I ) .Proof. Since I = a ∩ R + , so a + R + . Assume that Ass R ( R/R + ) = Ass R ( R/I ), then by2.9(i), Ass R ( R/ a ) ⊆ Ass R ( R/R + ) that is a contradiction, by 2.9(ii). (cid:3) Graded components of H i a ( M ) where a is an h-linked ideal In this section, which is the main part of the paper, we study the graded components of H i a ( M ) where a is h-linked with the irrelevant ideal R + over M .For a graded R -module N = L n ∈ Z N n , set end ( N ) := sup { n ∈ Z | N n = 0 } . Note that end ( N ) could be ∞ and that the supremum of the empty set is to be taken as −∞ .The following lemma, which consider a case where end ( H i a ( M )) < ∞ , will be used laterin the paper, too. Lemma 3.1.
Let t ∈ N and assume that end ( H i a ( M )) < ∞ for all i = t . Then for all n ≫ and all i ∈ N , H i a + b ( M ) n ∼ = (cid:26) H i − t b ( H t a ( M )) n , i ≥ t i (cid:12) t. Proof.
We have the following convergence of spectral sequences, by [17, 11.38],( E i,j ) n = H i b ( H j a ( M )) n i ⇒ H i + j a + b ( M ) n . Since end ( H j a ( M )) < ∞ for all j = t , H j a ( M ) is R + -torsion for all j = t . So, by [3, 2.1.9], H i b ( H j a ( M )) ∼ = H i b + R + ( H j a ( M )) ∼ = H i b R ( H j a ( M )) for all j = t and all i ≥ , where b := b ∩ R . Hence, by [3, 14.1.12] and the assumption, ( E i,j ) n = H i b ( H j a ( M ) n ) = 0for all j = t and all n ≫
0. As a result, H i a + b ( M ) n ∼ = ( E i − t,t ) n for all i ≥ t and that H i a + b ( M ) n = 0 for all i (cid:12) t , when n ≫ (cid:3) RADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THE LINKED IDEALS 7
The following corollary, which is immediate by the above lemma, generalizes [10, 1.1].
Corollary 3.2.
Let end ( H i a ( M )) < ∞ for all i ∈ N . Then for any homogeneous ideal b ⊇ a , end ( H i b ( M )) < ∞ for all i ∈ N . The following lemma will be used several times in the paper.
Lemma 3.3.
Let a be linked by I over M . Then Supp H t a ( M ) = Supp M/ a M , where t :=grade( a , M ) .Proof. By [9, 2.8], [6, Theorem 1] and [19, 1.4],Ass M/ a M ⊆ Ass
M/IM ∩ V ( a ) = Ass Hom R ( R/ a , M/IM ) = Ass Ext tR ( R/ a , M ) = Ass H t a ( M ) . On the other hand, Supp H t a ( M ) ⊆ Supp M/ a M , which proves the claim. (cid:3) In the following, we show some equivalent conditions for end ( H i a ( M )) < ∞ , where a is anh-linked ideal over M . Proposition 3.4.
Let a be an h-linked ideal by I over M with grade( a , M ) = t . Then thefollowing statements are equivalent. (i) end ( H i a ( M )) < ∞ for all i ∈ N , (ii) end ( H t a ( M )) < ∞ , (iii) Supp M/ a M ⊆ V ( R + ) .Also, if a h ∼ ( I ; M ) b and one of the above conditions holds, then H i b ( M ) n ∼ = (cid:26) i = tH tI ( M ) n i = t, for all i and all n ≫ .Proof. ”( ii ) ⇒ ( iii )” Since end ( H t a ( M )) < ∞ , H t a ( M ) is R + -torsion and Ass H t a ( M ) ⊆ V ( R + ). So, the result follows from 3.3.”( iii ) ⇒ ( i )” Since √ a + 0 : M ⊇ √ R + , using [3, 2.1.9], [3, 16.1.5(ii)] and 3.2, the statementholds.The last statement follows from [7, 2.2(i)], 3.2 and the following homogeneous Mayer-Vietoris sequence . . . −→ H i a + b ( M ) −→ H i a ( M ) ⊕ H i b ( M ) −→ H iI ( M ) −→ H i +1 a + b ( M ) −→ . . . . (cid:3) Note that if b ⊆ R + and one of the above conditions holds, then a can’t be geometricallyh-linked with b . Otherwise, by 3.4(iii), H i a ( M ) ∼ = H i a + R + ( M ) for all i , so grade( a , M ) =grade( a + b , M ), that is a contradiction by 2.2. Definition 3.5.
MARYAM JAHANGIRI ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI • We say that the ideal I is generated by an M -regular sequence under radical if thereexists an M -regular sequence x = x , . . . , x t such that √ I + 0 : M = √ x + 0 : M . • a and b are said to be in an h- M -linkage class of length n if there exist n ∈ N andhomogeneous ideals a , . . . , a n and I , . . . , I n such that a = a h ∼ ( I ; M ) a h ∼ . . . h ∼ ( I n ; M ) a n = b . If, in addition, b is generated by a homogeneous M -regular sequence underradical, then a is called radically h- M -licci with b of length n . Corollary 3.6. If a h ∼ ( I ; M ) b and t := grade( a , M ) , then the following statements areequivalent. (i) max { end ( H t a ( M )) , end ( H t b ( M )) } < ∞ , (ii) Supp M/IM ⊆ V ( R + ) .In particular, if b = R + and end ( H t a ( M )) < ∞ , then a is radically h- M -licci with R + oflength 1.Proof. The results follow from [9, 2.6(iii)], 3.4 and the fact that end ( H iR + ( M )) < ∞ for all i ∈ N . (cid:3) The above corollary shows, if a h ∼ ( I ; M ) b , then end ( H i a ( M )) or end ( H i b ( M )) is infinite forsome i ∈ N if and only if Supp M/IM * V ( R + ). Proposition 3.7.
Assume that a h ∼ ( I ; M ) b h ∼ ( J ; M ) c and end ( H tI ( M )) < ∞ , where t :=grade( I, M ) . Then H i c ( M ) n = (cid:26) i = tH tJ ( M ) n i = t, for all i and all n ≫ . Proof.
Since end ( H tI ( M )) < ∞ , H tI ( M ) is R + -torsion. So, by [6, Theorem 1], [19, 1.4]and [9, 2.6], Supp( M/ b M ) ⊆ V ( R + ), in other words R + ⊆ √ b + 0 : M . Thus H i b ( M ) n ∼ = H i b + R + ( M ) n = 0 for all i and all n ≫
0, by 3.2. Also, in view of [7, 2.2], we have the followinghomogeneous Mayer-Vietoris sequence . . . −→ H i b + c ( M ) −→ H i b ( M ) ⊕ H i c ( M ) −→ H iJ ( M ) −→ H i +1 b + c ( M ) −→ . . . . This, in conjunction with 3.2 and [9, 2.6], follows the result. (cid:3)
Using 3.7, we can describe the components H i a ( M ) n where a is radically h- M -licci with R + of length 2, as follows: Corollary 3.8. If a is radically h- M -licci with R + of length 2, i.e. a h ∼ ( I ; M ) b h ∼ ( J ; M ) R + and √ R + + 0 : M = √ J + 0 : M , then H i a ( M ) n = (cid:26) i = tH tI ( M ) n i = t, RADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THE LINKED IDEALS 9 for all i and all n ≫ where t := grade( I, M ) .Proof. By hypothesis H tJ ( M ) ∼ = H tR + ( M ). So, using [3, 16.1.5] and 3.7, the statementholds. (cid:3) M is called relative Cohen-Macaulay with respect to a of degree n if H i a ( M ) = 0 for all i = n . Theorem 3.9. If a h ∼ ( I ; M ) R + and t := grade( R + , M ) , then end ( H i a ( M )) < ∞ for all i = t and end ( H t a ( M )) < ∞ or H t a ( M ) n = 0 for all n ≫ .In a special case, H t a ( M ) n is a finitely generated R -module for all n ∈ Z .Proof. The case i = t follows from 3.4 and [3, 16.1.5(ii)]. Also, we have H t a ( M ) n ∼ = H tI ( M ) n for all n ≫
0. We consider two cases:case 1: Let Supp(
M/IM ) * V ( R + ). By 3.6 and [3, 16.1.5(ii)], end ( H t a ( M )) = ∞ . Now, weprove, by induction on t , that H t a ( M ) n = 0 for all n ≫ t = 0, then Γ a ( M ) n = Γ ( M ) n = M n for all n ≫
0. On the other hand, by [11,Theorem 1], R M n = M n +1 for all n ≫
0, thus Γ a ( M ) n = 0 for all n ≫ t > t −
1. Let I = ( x , x , . . . , x t )and deg ( x ) = l . Now, the homogeneous exact sequence0 −→ M .x −→ M ( l ) −→ ( M/x M )( l ) −→ R -modules for all n ∈ Z ,0 −→ H t − a ( M/x M ) n + l −→ H t a ( M ) n .x −→ H t a ( M ) n + l −→ . . . . Since x ∈ I , a / ( x ) h ∼ ( I/ ( x ); M/x M ) R + / ( x ) and, by the inductive hypothesis, H t − a ( x ( M/x M ) n = 0 for all n ≫
0. Hence, H t a ( M ) n = 0 for all n ≫ M/IM ) ⊆ V ( R + ). Then √ M + I = √ M + R + and H iI ( M ) ∼ = H iR + ( M ) for all i . So, by [3, 16.1.5], we have(1) end ( H tI ( M )) < ∞ ,(2) H tI ( M ) n is finitely generated R -module for all n ,(3) M is relative Cohen-Macaulay with respect to R + of degree t .Therefore, in view of 3.4, end ( H t a ( M )) < ∞ . Also, using [14, 3.4], there are homoge-neous isomorphisms(3.1) H i a ( M ) ∼ = H i − t a ( H tI ( M )) ∼ = H i − t a + R + ( H tI ( M )) ∼ = H i a + R + ( M ) for all i ≥ t. Using [9, 2.6], t ≤ f-grade( a + R + , R + , M ). Therefore, H t a + R + ( M ) n is a finitelygenerated R -module for all n ∈ Z , by [10, 1.7]. As a result, by (3.1), H t a ( M ) n is afinitely generated R -module for all n ∈ Z . (cid:3) ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI As wee have seen in the proof of 3.9, if R + is h-linked by I over M and Supp( M/IM ) ⊆ V ( R + ) then M is relative Cohen-Macaulay with respect to R + . However, the converse doesnot hold any more, as the following example shows. Although, it does in some special cases,see Proposition 3.11 and 3.12. Example 3.10.
Assume that ( R , m ) is a domain and dim R = 2 . Set R = R [ x ] . So,there exists a non zero prime ideal p of R such that p ( m . By 2.5, for any = r ∈ p , ( r ) h ∼ (( r x ); R ) ( x ) and Supp( R/ ( r x )) = V ( r ) S V ( x ) * V ( x ) , while R is relative Cohen-Macaulay with respect to ( x ) of degree 1. The following proposition considers a case where the irrelevant ideal can be generated byan M -regular sequence under radical. Proposition 3.11.
Let ( R , m ) be local and M be relative Cohen-Macaulay with respect to R + of degree t . Then there exists a maximal homogeneous M -regular sequence I in R + suchthat Supp(
M/IM ) ⊆ V ( R + ) . In other words, R + can be generated by a homogeneous M -regular sequence under radical.Proof. Assume that t = 0. By [1, 2.3], dim M/ m M = 0. Therefore, M/ m M is Artinian and end ( M/ m M ) < ∞ , using [11, Theorem 1]. Hence, by Nakayama Lemma, end ( M ) < ∞ .This implies that M is R + -torsion and that Supp M ⊆ V ( R + ).Now, let t > t −
1. As t > R + * S p ∈ ( MinAss ( M/ m M ) ∪ Z ( M )) p , where Z ( M ) denotes the set of zero divisors on M . So, by[3, 16.1.2], there exists a homogeneous element x ∈ R + \ [ MinAss ( M/ m M ) ∪ Z ( M ) p . Therefore, dim
M/xM m ( M/xM ) = t − R + , M/xM ). In other words, using [1, 2.3], M/xM is relative Cohen-Macaulay with respect to R + of degree t − M/xM -regular sequence I ′ in R + such thatSupp M ( I ′ +
Let I be an ideal generated by a maximal M -regular sequence in R + .Then the following statements hold. (i) If R is a field then Supp
M/IM ⊆ V ( R + ) if and only if M is a Cohen-Macaulymodule. (ii) Supp M/IM * V ( R + ) if and only if grade( I, M ) = f-grade( R + , I, M ) .Proof. (i) If Supp M/IM ⊆ V ( R + ) then √ M + I = √ M + R + and H iI ( M ) ∼ = H iR + ( M ) for all i ∈ N . It follows from [3, 6.2.9] and the fact that I is generatedby M -regular sequence, that M is Cohen-Macaulay. Now, assume that M is Cohen-Macaulay. Then dim M = grade( I, M ). Hence, dim
M/IM = 0 and Supp
M/IM ⊆{ R + } . RADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THE LINKED IDEALS 11 (ii) The result follows from [4, 2.4] and [3, 3.3.1]. (cid:3)
Definition and Remark 3.13. (i)
Let N = L n ∈ Z N n be a graded R -module. Then following [12] , • N is called finitely graded if N n = 0 for all but finitely many n ∈ Z ; • g a ( N ) := sup { k ∈ N | H i a ( N ) is finitely graded for all i < k } ; • f R + a ( N ) := sup { k ∈ N | R + ⊆ p H i a ( N ) for all i < k } ; • N is called tame, if the set { n ∈ Z | N n = 0 , N n +1 = 0 } is finite.Note that, by [12, 2.3] , if N is finitely generated, then g a ( N ) = f R + a ( N ) . (ii) Let a h ∼ ( I ; M ) R + and Supp
M/IM * V ( R + ) , then using [9, 2.6(i)] , 3.9 and (i), wehave f R + a ( M ) = grade( R + , M ) . In [5, 2.2], the authors studied tameness of H f R + a ( M ) a ( M ) under the assumption that a ⊇ R + . In the following Theorem, we consider this problem without any restriction on a .Although, the proof is a modification of [5, 2.2], we bring it here for the reader’s convenience.It will be used later in the paper, too. Theorem 3.14.
Let ( R , m ) be local and f R + a ( M ) < ∞ . Then H f R + a ( M ) a ( M ) is tame.Proof. Let y be an indeterminate. Set R ′ = R [ y ] m [ y ] , R ′ = R N R R ′ and M ′ = M N R R ′ . R ′ is a faithfully flat R -algebra so, by [3, 16.2.2(iv)], H i a ( M ) n N R R ′ ∼ = H i a R ′ ( M ′ ) n for all i ∈ N and all n ∈ Z . This results f R + a ( M ) = f R ′ + a R ′ ( M ). Thus, replacing R by R ′ , wecan assume that R / m is an infinite filed. Now, we prove the assertion by induction on f := f R + a ( M ).Let f = 0. By [11, Theorem 1], Γ a ( M ) n = 0 for all n ≪ R Γ a ( M ) n = Γ a ( M ) n +1 forall n ≫ a ( M ) n = 0 for all n ≫
0. Now, assume that f ≥ f − R + ( M ) is a finitely graded R -module so, using [12, 2.2], H i a (Γ R + ( M )) is finitely graded forall i ∈ N . Hence, the exact sequence . . . −→ H i a (Γ R + ( M )) n −→ H i a ( M ) n −→ H i a ( M/ Γ R + ( M )) n −→ H i +1 a (Γ R + ( M )) n −→ . . . implies that H i a ( M ) n ∼ = H i a ( M/ Γ R + ( M )) n for all i ∈ N and all n ∈ Z \ X , where X is afinite set. So, replacing M with M/ Γ R + ( M ), we can assume that M is R + -torsion free andthat there exists a homogeneous element x ∈ R which is a non-zero deviser on M , by [3,16.1.4(ii)]. Now, consider the homogeneous exact sequence0 −→ M x −→ M (1) −→ ( M/xM )(1) −→ . It yields the following exact sequence of R -modules . . . −→ H i − a ( M ) n +1 −→ H i − a ( M/xM ) n +1 −→ H i a ( M ) n x −→ H i a ( M ) n +1 −→ . . . . ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI Therefore, by 3.13, f R + a ( M/xM ) ≥ f R + a ( M ) − f R + a ( M/xM ) (cid:13) f −
1, by 3.13, 0 −→ H f a ( M ) n x −→ H f a ( M ) n +1 is exact for all n ≫ n ≪ H f a ( M ) is tame.Also, if f R + a ( M/xM ) = f −
1, again by 3.13, we have the following exact sequence0 −→ H f − a ( M/xM ) n +1 −→ H f a ( M ) n x −→ H f a ( M ) n +1 for all n ≫ n ≪
0. But, by induction, H f − a ( M/xM ) is tame, which requires H f a ( M ) is tame. (cid:3) Regards t := grade( a , M ) ≤ f R + a ( M ) and 3.13, H t a ( M ) is tame.Let N be an R -module. The cohomological dimension of N with respect to a is definedto be cd( a , N ) := sup { i ∈ Z | H i a ( N ) = 0 } . The following Theorem considers tameness of H cd( R + ,M ) a + R + ( M ) with some restrictions on M or linkedness of a with R + . Theorem 3.15.
Let ( R , m ) be local and cd( R + , M ) = 0 . Then f R + a + R + ( M ) < ∞ and H cd( R + ,M ) a + R + ( M ) is tame in each of the following cases: (i) M is relative Cohen-Macaulay with respect to R + ; (ii) a h ∼ ( I ; M ) R + .Proof. (i) Let t := cd( R + , M ) = grade( R + , M ). Since H iR + ( M ) n is a finitely gener-ated R -module for all i and all n ([3, 16.1.5]), by [18, 2.6], a H tR + ( M ) n = H tR + ( M ) n for all n ≪
0, where a := a ∩ R . Therefore, using [3, 6.2.7],(3.2) for all n ≪ k n ∈ N such that H k n a ( H tR + ( M ) n ) = 0 . Now, considering the following Grothendieck’s spectral sequence ([17, 11.38]) E i,j = H i a ( H jR + ( M )) i ⇒ H i + j a + R + ( M ) , we have E i,j = 0 for all i and all j = t . This implies that(3.3) H i a ( H tR + ( M )) ∼ = H i + t a + R + ( M ) for all i ∈ N . On the other hand, by [3, 14.1.12] and the fact that H tR + ( M ) is R + -torsion, wehave(3.4) H i a ( H tR + ( M )) n ∼ = H i a + R + ( H tR + ( M )) n ∼ = H i a R ( H tR + ( M )) n ∼ = H i a ( H tR + ( M ) n )for all i and all n . So, if f R + a + R + ( M ) = ∞ , then by (3.4), (3.3) and 3.13, H i a ( H tR + ( M ) n ) ∼ = H i + t a + R + ( M ) n = 0 RADED LOCAL COHOMOLOGY MODULES WITH RESPECT TO THE LINKED IDEALS 13 for all i and all n ≪ f R + a + R + ( M ) < ∞ .In addition, t = grade( R + , M ) ≤ grade( a + R + , M ) ≤ f R + a + R + ( M ). So, by 3.13 and3.14, H t a + R + ( M ) is tame.(ii) Set t := grade( R + , M ). By the homogeneous Mayer-Vietoris sequence and [7, 2.2],we have the homogeneous exact sequence . . . −→ H i − I ( M ) −→ H i a + R + ( M ) −→ H i a ( M ) ⊕ H iR + ( M ) −→ H iI ( M ) −→ H i +1 a + R + ( M ) −→ . . . . It yields(3.5) H i a + R + ( M ) ∼ = H i a ( M ) ⊕ H iR + ( M ) for all i (cid:13) t + 1and, by [3, 6.2.7], the exact sequence(3.6)0 −→ H t a + R + ( M ) −→ H t a ( M ) ⊕ H tR + ( M ) −→ H tI ( M ) −→ H t +1 a + R + ( M ) −→ H t +1 a ( M ) ⊕ H t +1 R + ( M ) −→ . Therefore, in view of (3.5), (3.6) and [18, 2.6], H cd( R + ,M ) a + R + ( M ) n = 0 for all n ≪
0. So,by [10, 1.1], H cd( R + ,M ) a + R + ( M ) is tame and, using 3.13, f R + a + R + ( M ) ≤ cd( R + , M ) < ∞ . (cid:3) Remark 3.16. (i)
Here is another situation for the finiteness of f R + a ( M ) . Assume that a and R + aregeometrically h-linked over M . Then, by [9, 2.9(iii)] , Supp( M/ a M ) * V ( R + ) andthere exists a homogeneous prime ideal p ∈ Supp M ∩ V ( a ) \ V ( R + ) . Hence, p + a + R + = p + R + = p ∩ R + R + = R . Therefore, by [3, 9.3.7] , f R + a + R + ( M ) ≤ depth M p + ht ( a + R + + p ) / p < ∞ . (ii) Assume that b ⊇ a . Then b can be represented as b = a + ( b , . . . , b s ) for somehomogeneous elements b , . . . , b s ∈ R . Using [3, 14.1.11] and induction on s , one cansee that f R + a ( M ) ≤ f R + b ( M ) . The following proposition presents possibilities for f R + a ( M ) and f R + ( M ) in the case where a is h-linked with R + . Proposition 3.17.
Let a h ∼ ( I ; M ) R + , then f R + a ( M ) , f R + ( M ) ∈ { grade( R + , M ) , f R + a + R + ( M ) } .Proof. Set t := grade( R + , M ). By [9, 2.6(i)] and [3, 6.2.7], t ≤ f R + a ( M ) , f R + ( M ). Also, by3.16(ii), f R + a ( M ) , f R + ( M ) ≤ f R + a + R + ( M ).If f R + a + R + ( M ) ≤ t + 1, the result follows. So, let f R + a + R + ( M ) (cid:13) t + 1. By (3.5) and (3.6), forall i ≥ t + 1, H i a ( M ) and H iR + ( M ) are finitely graded if and only if H i a + R + ( M ) is finitelygraded and this proves the claim. (cid:3) ∗ , AZADEH NADALI, AND KHADIJEH SAYYARI References [1] Brodmann M. Asymptotic behaviour of cohomology: tameness, supports and associated primes. In:Ghorpade S, Srinivasan H, Verma J (editors). Commutative Algebra and Algebraic Geometry. AmericanMathematical Society, 2005, pp. 31-61.[2] Brodmann M, Hellus M. Cohomological patterns of coherent sheaves over projective schemes. Journalof Pure and Applied Algebra 2002; 172: 165-182.[3] Brodman M, Sharp RY. Local cohomology: An algebraic introduction with geometric applications. NY,USA: Cambridge University Press, 2012.[4] Chu L, Gu Y. A problem of local cohomology modules. Communications in Algebra 2008; 36 (4):1603-1607.[5] Hassanzadeh SH, Jahangiri M, Zakeri H. Asymptotic behaviour and Artinian property of graded localcohomology modules. Communications in Algebra 2009; 37 (11): 4095-4102.[6] Hellus M. On the set of associated primes of a local cohomology module. Journal of Algebra 2001; 237(1): 406-419.[7] Jahangiri M, Sayyari K. Cohomological dimension with respect to the linked ideals. Journal of Algebraand Its Applications. doi: 10.1142/s0219498821501048.[8] Jahangiri M, Sayyari K. Characterization of some special rings via linkage. Journal of Algebra andRelated Topics 2020; 8 (1): 67-81.[9] Jahangiri M, Sayyari K. Linkage of ideals over a module. Journal of Algebraic Systems 2021; 8 (2):267-279.[10] Jahangiri M, Zakeri H. Local cohomology modules with respect to an ideal containing the irrelevantideal. Journal of Pure and Applied Algebra 2009; 213 (4): 573-581.[11] Kirby D. Artinian modules and Hilbert polynomials. The Quarterly Journal of Mathematics 1973; 24(1): 47-57.[12] Marley T. Finitely graded local cohomology and the depths of graded algebras. Proceedings of theAmerican Mathematical Society 1995; 123 (12): 3601-3607.[13] Matsumura H. Commutative ring theory. New Rochelle, NY, USA: Cambridge University Press, 1989.[14] Nagel U, Schenzel P. Cohomological annihilators and Castelnuovo-Mumford regularity. CommutativeAlgebra 1994; 159: 307-328.[15] Peskine C, Szpiro L. Liaison des vari´et´es alg´ebriques. I. Inventiones mathematicae 1974; 26 (4): 271-302.[16] Puthenpurakal T. Graded components of local cohomology modules. accepted for publication in Col-lectena Math 2020.[17] Rotman JJ. An introduction to homological algebra. London, UK: Academic Press Limited, 1979.[18] Rotthaus C, S¸ega LM. Some properties of graded local cohomology modules. Journal of Algebra 2005;283 (1): 232-247.[19] Singh AK, Walther U. Local cohomology and pure morphisms. Illinois Journal of Mathematics 2007;51 (1): 287-298.
Department of Mathematics, Faculty of Mathematical Sciences and Computer, KharazmiUniversity, Tehran, Iran.
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