A theory of relative generalized Cohen-Macaulay modules
Kamran Divaani-Aazar, Akram Ghanbari Doust, Massoud Tousi, Hossein Zakeri
aa r X i v : . [ m a t h . A C ] J a n A THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES
KAMRAN DIVAANI-AAZAR, AKRAM GHANBARI DOUST, MASSOUD TOUSIANDHOSSEIN ZAKERI
Abstract.
Let a be a proper ideal of a commutative Noetherian ring R with identity. Using the notionof a -relative system of parameters, we introduce a relative variant of the notion of generalized Cohen-Macaulay modules. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum andsurjective Buchsbaum modules. We reveal several interactions between these types of modules thatextend some of the existing results in the classical theory to the relative one. Contents
1. Introduction 12. Koszul complexes and filter regular sequences 33. Relative generalized Cohen-Macaulay modules 124. Relative Buchsbaumness 14References 18
1. Introduction
Throughout this paper, R is a commutative Noetherian ring with identity, a is an ideal of R and M isa finitely generated R -module. A relative theory of system of parameters is introduced in [DGTZ]. Recallthat if R is local with the maximal ideal m and dim M = d , then a sequence x , . . . , x d of elements of m forms a system of parameters if and only ifRad (cid:0) h x , . . . , x d i + Ann R M (cid:1) = Rad ( m + Ann R M ) . Although the Krull dimension seems suitable to define system of parameters, but in the relative case, weappeal to the cohomological dimension c := cd ( a , M ). Indeed, a sequence x , . . . , x c of elements of a issaid to be an a -relative system of parameters, a -Rs.o.p, of M ifRad (cid:0) h x , . . . , x c i + Ann R M (cid:1) = Rad ( a + Ann R M ) . Contrary to systems of parameters which always exist, relative systems of parameters may don’t exist. Itis easy to see that a contains an a -Rs.o.p of M if and only if ara ( a , M ) = cd ( a , M ); see [DGTZ, Lemma2.2]. (Here, ara ( a , M ) is the arithmetic rank of a with respect to M which is defined as the infimum Mathematics Subject Classification.
Key words and phrases.
Arithmetic rank; cohomological dimension; Koszul complex; local cohomology; relative Buchs-baum module; relative generalized Cohen-Macaulay module; relative quasi-Buchsbaum module; relative surjective Buchs-baum module; relative system of parameters. of the integers n ∈ N for which there exist x , . . . , x n ∈ R such that Rad (cid:0) h x , . . . , x n i + Ann R M (cid:1) =Rad ( a + Ann R M ) . )The notion of relative system of parameters enhanced the theory of relative Cohen-Macaulay modules.For instance, when a is contained in the Jacobson radical of R and ara ( a , M ) = cd ( a , M ), it is shownthat M is a -relative Cohen-Macaulay if and only if every a -Rs.o.p of M is an M -regular sequence if andonly if there exists an a -Rs.o.p of M which is an M -regular sequence; see [DGTZ, Theorem 3.3].There are several extensions of the notion of Cohen-Macaulay modules. One of them is the notionof generalized Cohen-Macaulay modules which is very well studied; see e.g. [T2]. Quasi Buchsbaum,Buchsbaum and surjective Buchsbaum modules are special types of generalized Cohen-Macaulay modulesthat have been studied extensively in late 80’s; see e.g. [SV]. Among many characterizations of thesetypes of generalized Cohen-Macaulay modules, there are some well-known one, using system of parameters.This provides hope for having a relative theory of these types of modules. This is what we are going to doin this paper. The notions of filter regular sequences, weak sequences and d -sequences play an importantrole in commutative algebra. Indeed, we impose some suitable properties of those sequences on relativesystem of parameters to develop a theory for modules which are introduced below. Definition 1.1.
Let a be an ideal of R and M a finitely generated R -module. (i) We know that for every i ≥ , H i a ( M ) ∼ = lim −→ n Ext iR (cid:0) R/ a n , M (cid:1) . Thus, there is a natural map ϕ iM : Ext iR (cid:0) R/ a , M (cid:1) −→ H i a ( M ) for all i ≥ . We say that M is a -relative surjective Buchsbaum if ϕ iM is surjective for all i < cd ( a , M ) . (ii) We say that M is a -relative Buchsbaum if cd ( a , M ) = ara ( a , M ) and every a -Rs.o.p of M is an a -weak sequence on M . (iii) We say that M is a -relative quasi Buchsbaum if cd ( a , M ) = ara ( a , M ) and every a -Rs.o.p of M contained in a is an a -weak sequence on M . (iv) We say that M is a -relative generalized Cohen-Macaulay if cd ( a , M ) ≤ ; or cd ( a , M ) = f a ( M ) . Next, we will describe the organization of the paper together with its main results.Section 2 is devoted to some preliminaries on properties of filter regular sequences, weak sequences and d -sequences. In Remark 2.3, we reveal some of the existing relationships in the literature between thesethree types of sequences. Then, we present some basic characterizations of these sequences. Theorems2.6 and 2.7 are the main results of this section. The Koszul complexes, which assist us through the paper,are the main ingredient of this section.In Section 3, we establish some of the main properties of relative generalized Cohen-Macaulay modules.Cuong, Schenzel and Trung in [CST, Satz 3.3] showed that every system of parameters of a generalizedCohen-Macaulay R -module M is a filter regular sequence on M . Now, the question arises whether every a -Rs.o.p of an a -relative generalized Cohen-Macaulay R -module M is an a -filter regular sequence on M ?Among other things, Theorem 3.3 gives a positive answer to this question.In Section 4, we investigate relative Buchsbaumness. Assume that the ideal a is contained in theJacobson radical of R . In Theorem 4.1, we provide a characterization of a -relative quasi Buchsbaum THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 3 modules via local cohomology modules. Theorem 4.2, among other things, indicates that a given finitelygenerated R -module is a -relative Buchsbaum if there exists a generating set of a satisfying some certainproperties. Also in Corollary 4.3, we show that the classes of modules introduced in Definition 1.1 aresubset of each other from top to bottom; respectively. Theorem 4.8 reveals a close connection between a -relative Buchsbaum R -modules and a -relative quasi Buchsbaum R -modules.
2. Koszul complexes and filter regular sequences
The Koszul complexes play an important role in this paper. In Definition and Remark 2.1, for thereaders convenience, we briefly state the construction of the Koszul complexes and recall some resultsabout these complexes. We obey the notation [BS, 5.2], which shall be used throughout of the paperwithout further comments.
Definition and Remark 2.1.
Let x := x , . . . , x n be a sequence of elements of R , a := h x i and M an R -module. (i) For k ∈ N with ≤ k ≤ n , we shall set I ( k, n ) := { ( i , . . . , i k ) ∈ N k | ≤ i < i < . . . < i k ≤ n } . Let i ∈ I ( k, n ) . For every ≤ j ≤ k , we denote the j -th component of i by i j , so that i =( i , i , . . . , i k ) . If k < n , then by the n -complement of i , we mean the sequence j ∈ I ( n − k, n ) such that { , , . . . , n } = { i , . . . , i k , j , . . . , j n − k } . Let e , . . . , e n be n new symbols. We define a complex K • ( x, M ) as follows: Set K ( x, M ) := M and K p ( x, M ) := 0 if p is not in the range ≤ p ≤ n . For ≤ p ≤ n , let K p ( x, M ) := L i ∈I ( p,n ) M e i e i . . . e i p . Thus, when ≤ p ≤ n , K p ( x, M ) is effectively a direct sum of (cid:0) np (cid:1) copiesof M and a typical element of it has a unique representation of the form X i ∈I ( p,n ) m i i ...i p e i e i . . . e i p , where m i i ...i p belongs to M for all i ∈ I ( p, n ) .The differential map d p : K p ( x, M ) −→ K p − ( x, M ) , where < p ≤ n is defined by setting d p (cid:16) me i . . . e i p (cid:17) = p X r =1 ( − r − x i r me i . . . ˆ e i r , . . . e i p . This complex is called Koszul complex of M with respect to x and is denoted by K • ( x, M ) . Its j th homology module is denoted by H j ( x, M ) . It is known that a H j ( x, M ) = 0 for all j . (ii) An exact sequence −→ L −→ M −→ N −→ of R -homomorphisms gives rise to a long exact sequence of Koszul homology modules: · · · −→ H i ( x, L ) −→ H i ( x, M ) −→ H i ( x, N ) −→ H i − ( x, L ) −→ · · · . (iii) Let i ∈ N . We set K i ( x, M ) := K n − i ( x, M ) , and we denote the induced co-complex with K • ( x, M ) . Then one has: H i ( x, M ) = H i (cid:0) K • ( x, M ) (cid:1) = H n − i ( x, M ) . K. DIVAANI-AAZAR, A. GHANBARI DOUST, M. TOUSI AND HOSSEIN ZAKERI (iv)
Let u, v ∈ N with u ≤ v . There is a morphism (cid:0) ψ vu (cid:1) = (cid:16)(cid:0) ψ vu (cid:1) k (cid:17) k ∈ Z : K • (cid:0) x u , . . . , x un , M (cid:1) −→ K • (cid:0) x v , . . . , x vn , M (cid:1) of complexes such that (cid:0) ψ vu (cid:1) n is the identity mapping of M , (cid:0) ψ vu (cid:1) is the endomorphism of M given by multiplication by ( x . . . x n ) v − u and for every k = 1 , . . . , n − and i ∈ I ( k, n ) , (cid:0) ψ vu (cid:1) k (cid:0) me i . . . e i k (cid:1) = (cid:16) x j . . . x j n − k (cid:17) v − u me i . . . e i k , where j ∈ I ( n − k, n ) is the n -complement of i . Then { K • (cid:0) x u , . . . , x un , M (cid:1) , ψ vu } u ≤ v ∈ N is a directsystem in the category of complexes. For any i ∈ N and any R -module M , there is a natural R -isomorphism H i a ( M ) ∼ = lim −−→ u ∈ N H n − i (cid:0) x u , . . . , x un , M (cid:1) , and so there are the canonical maps λ iM : H i ( x, M ) −→ H i a ( M ) . (v) (See [SV, Lemma 1.5] .) If b = h y , . . . , y m i is another ideal of R with a ⊆ b , then for each i thereis a commutative diagram H i ( y , . . . , y m , M ) / / (cid:15) (cid:15) H i ( x, M ) (cid:15) (cid:15) H i b ( M ) / / H i a ( M ) . Next, we recall the definitions of some sequences which play key roles in the study of the modulesintroduced in Definition 1.1.
Definition 2.2.
Let a be an ideal of R and M a finitely generated R -module. (i) Following [CST] , a sequence x , . . . , x r of elements of R is called an a -filter regular sequence on M if h x , . . . , x i − i M : M x i ⊆ [ t ∈ N h x , . . . , x i − i M : M a t for all i = 1 , . . . , r . An a -filter regular sequence x , . . . , x r on M is called an unconditioned a -filterregular sequence on M , if it is an a -filter regular sequence on M in any order. (ii) Following [T1, page 39] , a sequence x , . . . , x r of elements of R is called an a -weak sequence on M if h x , . . . , x i − i M : M x i ⊆ h x , . . . , x i − i M : M a for all i = 1 , , . . . , r . It is an unconditioned a -weak sequence on M if x α , . . . , x α r r is an a -weaksequence on M in any order for all positive integers α , . . . , α r . (iii) The theory of d -sequences was introduced by Huneke in [H] and Trung in [T1] . A sequence x , . . . , x r of elements of R is called a d -sequence on M if h x , x , . . . , x i − i M : M x i x j = h x , x , . . . , x i − i M : M x j for any ≤ i ≤ j ≤ r (this is actually a slight weakening of Huneke’s definition). When x n , x n , . . . , x n r r form a d -sequence on M for all integers n , n , . . . , n r ∈ N , then x is called astrong d -sequence on M . A d -sequence on M is termed unconditioned, u.s.d -sequence, when itforms a strong d -sequence on M in any order. THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 5
In the following remark, we collect some properties of the sequences that are already defined above.
Remark 2.3.
Let a be an ideal of R and M a finitely generated R -module. (A) We review the relationship between the three types of sequences introduced in Definition 2.2. (i)
Clearly, every a -weak sequence on M is an a -filter regular sequence on M . By [T1, Theorem1.1(vi)] a d -sequence x , . . . , x r on M is an a -weak sequence on M if a ⊆ h x , . . . , x r i . (ii) By [T1, Theorem 1.1(vii)] , a d -sequence x , . . . , x r on M is an a -filter regular sequence on M if a ⊆ Rad (cid:0) h x , . . . , x r i (cid:1) . Also by [T1, Proposition 2.1] , if x , . . . , x r ∈ a is an a -filterregular sequence on M , then there exist natural integers s ≤ · · · ≤ s r such that x s , . . . , x s r r is a d -sequence on M . (iii) By [T1, Proposition 2.2] , x , . . . , x r ∈ a is a d -sequence on M if one of the following condi-tions is satisfied: (a) x , . . . , x i − , x i is an a -weak sequence on M for all i = 1 , . . . , r . (b) x , . . . , x r is an a -weak sequence on M in a . (B) Below, we record three essential properties of filter regular sequences. (i)
Let x = x , . . . , x r ∈ a be an a -filter regular sequence on M . Then by [AS, Proposition 2.3] ,there are the natural isomorphisms H i a ( M ) ∼ = H i h x i ( M ) for ≤ i < r H i − r a (cid:16) H r h x i ( M ) (cid:17) for i ≥ r. (ii) Let a = h x , . . . , x r i . By [TZ, Proposition 1.2] , a has generators y , . . . , y r which form anunconditioned a -filter regular sequence on M . We call a such set of generators for a , anf-generating set of a with respect to M . Note that for any positive integer ℓ , we may find anf-generating set of a with respect to M such that it has more than ℓ elements. (iii) A sequence x , . . . , x r of elements of R is an a -filter regular sequence on M if and onlyif for every ≤ i ≤ r , the element x i doesn’t belong to the union of the elements of Ass R (cid:16) M h x ,...,x i − i M (cid:17) \ V ( a ) . In the next result, we consider the situation in which a given filter regular sequence on M forms aweak sequence on M . Lemma 2.4.
Let a be an ideal of R and M a finitely generated R -module. Let x := x , . . . , x n be asequence of elements of a which forms an unconditioned a -filter regular sequence on M . Assume that thenatural map λ iM : H i ( x, M ) −→ H i h x i ( M ) is surjective for all ≤ i ≤ n − . Then x is an unconditioned h x i -weak sequence on M . In particular, x is an u.s.d -sequence on M . Proof.
Set b := h x i . By induction on 0 < i ≤ n , we show that for any permutation δ of the set { , , ..., n } and all positive integers t , . . . , t n , the sequence x t δ (1) , . . . , x t i δ ( i ) is a b -weak sequence on M .Since λ M is surjective, (0 : M b ) = H b ( M ). By Remark 2.3(B)(i), one has:H h x t δ (1) i ( M ) = H a ( M ) = H b ( M ) , and so (cid:16) M x t δ (1) (cid:17) ⊆ H h x t δ (1) i ( M ) = H b ( M ) = (0 : M b ) . Hence, the case i = 1 holds. K. DIVAANI-AAZAR, A. GHANBARI DOUST, M. TOUSI AND HOSSEIN ZAKERI
Now, let 2 ≤ i ≤ n and the result has been proved for i −
1. Let σ be a permutation of the set { , , ..., n } and u , . . . , u n be positive integers. Set z i := x u i σ ( i ) for every 1 ≤ i ≤ n . By the inductionhypothesis, z , . . . , z i − is a b -weak sequence on M , and it remains to show that h z , . . . , z i − i M : M z i ⊆ h z , . . . , z i − i M : M b . By [T2, Lemma 3.5], one deduces that h z k , . . . , z ki − i M : M (cid:0) z . . . z i − (cid:1) k − = h z , . . . , z i − i M + i − X j =1 (cid:0) h z , . . . , ˆ z j , . . . , z i − i M : M b (cid:1) for all k ≥
2. On the other hand, we have the following obvious containment h z , . . . , z i − i M + i − X j =1 (cid:0) h z , . . . , ˆ z j , . . . , z i − i M : M b (cid:1) ⊆ h z , . . . , z i − i M : M b . So the argument will be complete, if we show that for each m i ∈ h z , . . . , z i − i M : M z i , there exist n i ∈ h z , . . . , z i − i M : M b and l ≥ m i − n i ∈ h z l , . . . , z li − i M : M (cid:0) z . . . z i − (cid:1) l − . Let m i ∈ h z , . . . , z i − i M : M z i . Then z i m i = − P i − j =1 z j m j for some elements m , . . . , m i − in M . Clearly t := P ij =1 m j e j ∈ ker d , where d : K ( z , . . . , z i , M ) −→ K ( z , . . . , z i , M ) is the firstdifferential map of the Koszul complex K • ( z , . . . , z i , M ). By Definition and Remark 2.1(v), the followingdiagram is commutative: H i − ( x, M ) θ / / λ i − M (cid:15) (cid:15) H i − ( z , . . . , z i , M ) g (cid:15) (cid:15) H i − b ( M ) h / / H i − h z ,...,z i i ( M )By Remark 2.3(B)(i), h is an isomorphism. Also by the assumption, λ i − M is surjective. So, there is z ∈ H i − ( x, M ) such that g (cid:0) θ ( z ) (cid:1) = g ( t + im d ). Suppose that θ ( z ) = f +im d , where f = P ij =1 n j e j .Since b H i − ( x, M ) = 0, it follows that b f ⊆ im d . It implies that x j n i ∈ h z , . . . , z i − i M for all j = 1 , . . . , n , and so n i ∈ h z , . . . , z i − i M : M b . Now, one has0 = g (cid:0) ( t − f ) + im d (cid:1) ∈ H i − h z ,...,z i i ( M ) ∼ = lim −→ α H (cid:0) z α , . . . , z αi , M (cid:1) . So, there exists an integer l ≥ ψ l (cid:0) ( t − f ) + im d (cid:1) = 0, where ψ l : H ( z , . . . , z i , M ) −→ H (cid:16) z l , . . . , z li , M (cid:17) is the natural map induced by the map ψ l given in Definition and Remark 2.1(iv). Thus, we have (cid:0) z . . . z i − (cid:1) l − ( m i − n i ) ∈ h z l , . . . , z li − i M. For the second assertion, note that by Remark 2.3(A)(iii), every unconditioned h x i -weak sequence on M is an u.s.d -sequence on M . So, it is obvious by the first assertion. (cid:3) The following lemma together with Theorem 2.6 are required to prove Theorem 2.7.
THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 7
Lemma 2.5.
Let x := x , . . . , x s be a sequence of elements of R and set a := h x i . Let M be a finitelygenerated R -module and n ≤ s a natural integer. Suppose that every n elements of { x , . . . , x s } forms a d -sequence on M . Then the natural R -homomorphism H i ( x, M a ) −→ H i ( x, M ) induced by the inclusionmap M a ֒ → M is injective for every nonnegative integer i that is satisfying ≤ s − i ≤ n . Proof.
Let i be a nonnegative integer such that 0 ≤ s − i ≤ n and set K i := K i ( x, M ) and K i := K i ( x, M a ). Let H i and H i denote the i th homology modules of these complexes; respectively. Since x j (0 : M a ) = 0 for all j = 1 , . . . , s , one has H i = K i . We have to show that the natural R -homomorphism K i −→ H i ( x, M ) z z + im d i +1 is injective. That is K i T im d i +1 = 0 . Let x ∈ K i T im d i +1 . Then x = X l ∈I ( i,s ) ´ m l ...l i e l . . . e l i ∈ K i and there is y = X j ∈I ( i +1 ,s ) m j j ...j i +1 e j e j . . . e j i +1 ∈ K i +1 such that d i +1 ( y ) = x . So, X l ∈I ( i,s ) ´ m l l ...l i e l e l . . . e l i = X j ∈I ( i +1 ,s ) i +1 X p =1 ( − p − x j p m j j ...j i +1 e j . . . ˆ e j p . . . e j i +1 . Assume that k ∈ I ( i, s ) and set B k := { ℓ ∈ { , . . . , s }| k can be induced by deleting ℓ from an element of I ( i + 1 , s ) } . So, | B k | = s − i . Hence ´ m k k ...k i ∈ x j M + x j M + . . . + x j s − i M, where j , j , . . . , j s − i ∈ B k . As x j ´ m k k ...k i = 0, [T1, Theorem 1.1] yields that´ m k k ...k i ∈ (cid:0) M x j (cid:1) \ h x j , x j , . . . , x j s − i i M = 0 . (Note that, by [G, Lemma 2.2], if a , . . . , a n is a d -sequence on M , then, for each 1 ≤ t ≤ n , a , a , . . . , a t is also a d -sequence on M .) (cid:3) The next result connects the notion of weak sequences to that of local cohomology.
Theorem 2.6.
Let a be an ideal of R and M a finitely generated R -module. Let ℓ be a natural integerand x , . . . , x n ∈ a ℓ . Then the following are equivalent: (i) x , . . . , x n is an a ℓ -weak sequence on M . (ii) a ℓ H i a ( M ) = 0 for all ≤ i < n and x , . . . , x n is an a -filter regular sequence on M . Proof. (i)= ⇒ (ii) Suppose that x , . . . , x n is an a ℓ -weak sequence on M . So, x , . . . , x n is an a -filterregular sequence on M . Note that by Remark 2.3(A)(iii), x , . . . , x n is d -sequence on M . Hence, byRemark 2.3(B)(i), Γ a ( M ) = Γ h x i ( M ) = 0 : M x = 0 : M a ℓ . K. DIVAANI-AAZAR, A. GHANBARI DOUST, M. TOUSI AND HOSSEIN ZAKERI
We use induction on n . For n = 1, we are done. Let n > n − x , . . . , x n − is an a ℓ -weak sequence on M , the induction hypothesis implies that a ℓ H i a ( M ) = 0 forall 0 ≤ i < n −
1. Thus, it remains to show that a ℓ H n − a ( M ) = 0. We have the following two exactsequences 0 −→ M x ֒ → M ρ −→ x M −→ −→ x M λ −→ M ։ M/x M −→ , in which all maps are natural. Since 0 : M x = Γ a ( M ), we get H i a (0 : M x ) = 0 for all i ≥
1. Hence,H n − a ( ρ ) is an isomorphism. It induces the given isomorphism in the following display:ker (cid:16) H n − a ( λ ) (cid:17) ∼ = ker (cid:16) H n − a ( λ ) H n − a ( ρ ) (cid:17) = ker (cid:16) H n − a ( λρ ) (cid:17) = ker (cid:16) H n − a ( x M ) (cid:17) = 0 : H n − a ( M ) x . As x , . . . , x n is an a ℓ -weak sequence on M/x M , the induction hypothesis yields that a ℓ H n − a (cid:0) M/x M (cid:1) = 0 . The exact sequence H n − a (cid:0) M/x M (cid:1) −→ H n − a ( x M ) H n − a ( λ ) −→ H n − a ( M )implies the exact sequence H n − a (cid:0) M/x M (cid:1) −→ ker (cid:16) H n − a ( λ ) (cid:17) −→ , and consequently a ℓ ker (cid:16) H n − a ( λ ) (cid:17) = 0. So, a ℓ (cid:18) H n − a ( M ) x (cid:19) = 0. Now, one has0 : H n − a ( M ) x ⊆ H n − a ( M ) a ℓ ⊆ H n − a ( M ) a ℓ ⊆ H n − a ( M ) x . Hence, 0 : H n − a ( M ) a ℓ = 0 : H n − a ( M ) a ℓ . It implies thatH n − a ( M ) = Γ a (cid:16) H n − a ( M ) (cid:17) = Γ a ℓ (cid:16) H n − a ( M ) (cid:17) = [ t ≥ H n − a ( M ) a tℓ = 0 : H n − a ( M ) a ℓ , and so a ℓ H n − a ( M ) = 0.(ii)= ⇒ (i) We use induction on n . Let n = 1. As x is an a -filter regular sequence on M , Remark2.3(B)(i) yields that 0 : M x ⊆ Γ h x i ( M ) = Γ a ( M ) . Hence, a ℓ (0 : M x ) ⊆ a ℓ Γ a ( M ) = 0 . Thus, x is an a ℓ -weak sequence on M . THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 9
Next, let n > n −
1. As x is an a -filter regular sequence on M in a ℓ , [CQ, Theorem 1.1] implies thatH i a (cid:0) M/x M (cid:1) ∼ = H i a ( M ) M H i +1 a ( M )for all i < n −
1. So, our assumption implies that a ℓ H i a (cid:0) M/x M (cid:1) = 0 for all i < n −
1. Thus by theinduction hypothesis, x , . . . , x n is an a ℓ -weak sequence on M/x M . As x is an a ℓ -weak sequence on M ,we deduce that x , . . . , x n is an a ℓ -weak sequence on M . (cid:3) Theorem 2.7.
Let a be an ideal of R , M a finitely generated R -module and n a natural integer. Thenthe following are equivalent: (i) Any n elements of every f-generating set { x , . . . , x s } of a with respect to M with s ≥ n form anunconditioned a -weak sequence on M . (ii) There exists an f-generating set { x , . . . , x s } of a with respect to M such that s ≥ n and any n elements in it form an unconditioned a -weak sequence on M . (iii) There exists an f-generating set { x , . . . , x s } of a with respect to M with s ≥ n such that thenatural map λ iM : H i ( x , . . . , x s , M ) −→ H i a ( M ) is surjective for all ≤ i ≤ n − . Proof. (i)= ⇒ (ii) is obvious.(ii)= ⇒ (iii) Assume that { x , . . . , x s } satisfies the given assumptions in (ii) and set x := x , . . . , x s . Let1 ≤ i ≤ s . Since 0 : M a ⊆ M x i ⊆ M a ⊆ M a , we have 0 : M a = 0 : M a . It implies thatH a ( M ) = 0 : M a = 0 : M h x i = H ( x, M ) . Thus, the natural map λ M is an isomorphism.We use induction on n . For n = 1, we are done. Let n > n − λ iM is surjective for all 1 ≤ i ≤ n − a , M ) ≥
1. In particular, Γ a ( M ) = 0. Since x , . . . , x n is an a -weak sequenceon M in a , Theorem 2.6 implies that a H i a ( M ) = 0 for all 0 ≤ i ≤ n −
1. In particular, x H i a ( M ) = 0for all i < n . Also, we have x H i ( x, M ) = 0 for all i ≥
0. Since x is an a -filter regular sequence on M , x is not zero divisor on M . The exact sequence0 −→ M x −→ M −→ M/x M −→ / / H i − ( x, M ) / / (cid:15) (cid:15) H i − (cid:0) x, M/x M (cid:1) / / (cid:15) (cid:15) H i ( x, M ) / / (cid:15) (cid:15) / / H i − a ( M ) / / H i − a (cid:0) M/x M (cid:1) / / H i a ( M ) / / i < n , where the vertical maps are the canonical homomorphisms. Set R := R/x R , a := a R and M := M/x M . We write x for the image of x ∈ R in R . Since x , . . . , x s is an unconditioned a -filterregular sequence on M and every subset of { x , . . . , x s } with n − a -weaksequence on M , by induction hypothesis, the natural map H i (cid:16) x , . . . , x s , M (cid:17) −→ H i a (cid:16) M (cid:17) is surjectivefor all i ≤ n −
2. The R -modules H i (cid:16) x , . . . , x s , M (cid:17) and H i (cid:16) x , . . . , x s , M (cid:17) are isomorphic. Also, by the Independence Theorem for local cohomology the two R -modules H i h x ,...,x s i (cid:16) M (cid:17) and H i h x ,...,x s i (cid:16) M (cid:17) are isomorphic. Therefore, the corresponding natural homomorphismsH i (cid:16) x , . . . , x s , M (cid:17) −→ H i h x ,...,x s i (cid:16) M (cid:17) are surjective for all i ≤ n −
2. By [SV, Corollary 1.7], one has the following commutative diagram withexact rows H i (cid:16) x, M (cid:17) / / (cid:15) (cid:15) H i (cid:16) x , . . . , x s , M (cid:17) / / (cid:15) (cid:15) i h x i (cid:16) M (cid:17) ∼ = / / H i h x ,...,x s i (cid:16) M (cid:17) / / i ≥
1. (Note that the assumption R being local is not needed in the proof [SV, Corollary 1.7].) Itfollows that H i (cid:16) x, M (cid:17) −→ H i a (cid:16) M (cid:17) is surjective for all i ≤ n − i ( x, M ) −→ H i a ( M )is surjective for all i ≤ n − a , M ) = 0. Since x is an unconditioned a -filter regular sequence on M , itfollows that x is an unconditioned a -filter regular sequence on M/ Γ a ( M ). We show that every subsetof { x , . . . , x s } with n elements is an unconditioned a -weak sequence on M/ Γ a ( M ). Let { y , . . . , y n } ⊆{ x , . . . , x s } and let t , . . . , t n be positive integers. One has to show that (cid:16) h y t , . . . , y t i − i − i M + Γ a ( M ) (cid:17) : M y t i i ⊆ (cid:16) h y t , . . . , y t i − i − i M + Γ a ( M ) (cid:17) : M a for every 1 ≤ i ≤ n . Let m ∈ (cid:16) h y t , . . . , y t i − i − i M + Γ a ( M ) (cid:17) : M y t i i . Then, there exists ℓ ≥ a ℓ (cid:16) y t i i m (cid:17) ⊆ h y t , . . . , y t i − i − i M . So, a ℓ m ⊆ h y t , . . . , y t i − i − i M : M y t i i ⊆ h y t , . . . , y t i − i − i M : M a . Hence, m ∈ h y t , . . . , y t i − i − i M : M a ℓ +1 . On the other hand, we have h y t , . . . , y t i − i − i M : M a ℓ +1 ⊆ h y t , . . . , y t i − i − i M : M y ℓ +1 i ⊆ h y t , . . . , y t i − i − i M : M a ⊆ (cid:16) h y t , . . . , y t i − i − i M + Γ a ( M ) (cid:17) : M a . As grade (cid:0) a , M/ Γ a ( M ) (cid:1) >
0, the natural map λ i M Γ a ( M ) is surjective for all i ≤ n − . The exact sequence0 → Γ a ( M ) → M → M/ Γ a ( M ) → i ( x, M ) π i / / λ iM (cid:15) (cid:15) H i (cid:0) x, M/ Γ a ( M ) (cid:1) / / λ i M Γ a ( M ) (cid:15) (cid:15) H i +1 (cid:0) x, Γ a ( M ) (cid:1) µ i +1 / / H i +1 ( x, M )H i a ( M ) ∼ = / / H i a (cid:0) M/ Γ a ( M ) (cid:1) THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 11 with exact top row for all i ≥
1. By Remark 2.3(A)(iii) every unconditioned a -weak sequence is d -sequence.By Lemma 2.5, µ i is injective for all i ≤ n . So, π i is surjective for i ≤ n −
1. Hence, from the abovecommutative diagram, we get that λ iM is surjective for all 1 ≤ i ≤ n −
1. Since λ M is an isomorphism, λ iM is surjective for all 0 ≤ i ≤ n − ⇒ (i) Assume that { w , . . . , w t } satisfies the given assumptions in (iii). Let { x , . . . , x s } be anarbitrary f-generating set of a with respect to M with s ≥ n and set x := x , . . . , x s . By Definition andRemark 2.1(v), for each j ≥
0, we have the following commutative diagram in which all maps are naturalH j ( w , . . . , w t , M ) / / (cid:15) (cid:15) H j ( x, M ) (cid:15) (cid:15) H j h w ,...,w t i ( M ) ∼ = / / H j h x i ( M ) . It yields that the natural map λ jM : H j ( x, M ) −→ H j a ( M ) is surjective for all 0 ≤ j ≤ n − { z , . . . , z n } ⊆ { x , . . . , x s } and t , . . . , t n be positive integers. Set y i := z t i i for all 1 ≤ i ≤ n . Let1 ≤ i ≤ n and set X := { x , . . . , x s } \ { y , . . . , y i − } . We choose α ∈ X . As above for each j ≥
0, wehave the following commutative diagramH j ( x, M ) / / (cid:15) (cid:15) H j (cid:0) y , . . . , y i − , α, M (cid:1) (cid:15) (cid:15) H j h x i ( M ) / / H j h y ,...,y i − ,α i ( M ) . By Remark 2.3(B)(i), H j h x i ( M ) ∼ = H j a ( M ) ∼ = H j h y ,...,y i − ,α i ( M )for all 0 ≤ j ≤ i −
1. Thus from the above diagram, we see that the natural mapH j (cid:0) y , . . . , y i − , α, M (cid:1) −→ H j h y ,...,y i − ,α i ( M )is surjective for all 0 ≤ j ≤ i −
1. Hence by Lemma 2.4, y , . . . , y i − , α is an u.s.d -sequence on M . Set M := M/ h y , . . . , y i − i M . By [G, Lemma 2.2(iv)], h y , . . . , y i − i M : M α = h y , . . . , y i − i M : M α . So 0 : M α = 0 : M α . Thus, as α is an a -filter regular sequence on M , one hasΓ a (cid:16) M (cid:17) = Γ h α i (cid:16) M (cid:17) = 0 : M α. Now, we have 0 : M y i ⊆ Γ h y i i (cid:16) M (cid:17) = Γ a (cid:16) M (cid:17) = \ α ∈ X (cid:16) M α (cid:17) = 0 : M a , and so h y , . . . , y i − i M : M y i ⊆ h y , . . . , y i − i M : M a . (cid:3)
3. Relative generalized Cohen-Macaulay modules
The class of generalized Cohen-Macaulay modules contains the class of Cohen-Macaulay modules.Indeed generalized Cohen-Macaulay modules enjoy many interesting properties similar to the ones ofCohen-Macaulay modules. As a generalization of the notion of Cohen-Macaulay modules, in [RZ] and[DGTZ], relative Cohen-Macaulay modules were studied. It could be of interest to establish a theory ofrelative generalized Cohen-Macaulay modules. To this end, first we specify some notation and facts.
Definition and Remark 3.1.
Let a be an ideal of R and M a finitely generated R -module. (i) If a M = M , then ht M a ≤ cd ( a , M ) , otherwise one has ht M a = + ∞ and cd ( a , M ) = −∞ .(Recall that, by convention, inf ∅ = + ∞ and sup ∅ = −∞ .) (ii) Following [BS, Definition 9.1.3] , the a -finiteness dimension of M , f a ( M ) , is defined byf a ( M ) := inf { i ∈ N | H i a ( M ) is not finitely generated } † = inf { i ∈ N | a * Rad (cid:18)
Ann R (cid:16) H i a ( M ) (cid:17)(cid:19) } ! . (The equality † holds by Faltings’ Local-global Principle Theorem [F, Satz 1] .) (iii) If c := cd( a , M ) > , then by [DV, Corollary 3.3(i)] , the R -module H c a ( M ) is not finitely generated.So in this case, one has f a ( M ) ≤ cd( a , M ) . From this, we can also deduce that if ht M a > , thenf a ( M ) ≤ ht M a . (iv) Following [BS, Definitions 9.2.2] , we set λ a ( M ) := inf { depth M p + ht (cid:18) a + pp (cid:19) | p ∈ Spec R \ V ( a ) } . By [BS, Theorem 9.3.7] , one has f a ( M ) ≤ λ a ( M ) . The next lemma is needed in the proof of Theorem 3.3. Theorem 3.3 is one of the main results ofthe paper which provides some properties of relative system of parameters whenever the module underconsideration in relative generalized Cohen-Macaulay.
Lemma 3.2.
Let a be an ideal of R which is contained in the Jacobson radical of R and M an a -relative generalized Cohen-Macaulay R -module with c := cd ( a , M ) > . Then f a ( M ) = λ a ( M ) and cd (cid:0) a , R/ p (cid:1) = c for every p ∈ Ass R M \ V ( a ) . Proof.
First of all note that the assumption c >
0, implies that Ass R M * V ( a ). Let p ∈ Ass R M \ V ( a ).Then depth R p M p = 0. Using the assumption, [BS, Theorem 9.3.7] and [DNT, Theorem 2.2] yields thatcd ( a , M ) = f a ( M ) ≤ λ a ( M ) ≤ depth R p M p + ht (cid:16) a + pp (cid:17) = ht (cid:16) a + pp (cid:17) ≤ cd (cid:16) a + pp , R/ p (cid:17) = cd (cid:0) a , R/ p (cid:1) ≤ cd ( a , M ) . (cid:3) THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 13
Theorem 3.3.
Let a be an ideal of R which is contained in the Jacobson radical of R and M an a -relativegeneralized Cohen-Macaulay R -module such that c := cd ( a , M ) = ara ( a , M ) . Assume that x := x , . . . , x c is an a -Rs.o.p of M. Then (i) x , . . . , x c is an unconditioned a -filter regular sequence on M . (ii) M/ h x , . . . , x i i M is an a -relative generalized Cohen-Macaulay R -module and cd (cid:0) a , M/ h x , . . . , x i i M (cid:1) = c − i for all i = 0 , . . . , c . Proof.
Clearly if z , z , . . . , z c ∈ a is an a -Rs.o.p of M , then for all t , . . . , t c ∈ N , every permutation of z t , . . . , z t c c is also an a -Rs.o.p of M . So to prove (i), it is enough to show that x , . . . , x c is an a -filterregular sequence on M . In case c = 0, there is nothing to prove. Hence, we may assume that c >
0. It isenough to show that the claim holds for i = 1. Set x := x . (i) To contrary, assume that there is p ∈ Ass R M \ V ( a ) such that x ∈ p . Note thatRad (cid:16) Ann R (cid:0) M/ p M (cid:1)(cid:17) = Rad ( p + Ann R M ) = Rad p = p . Thus Supp R (cid:0) M/ p M (cid:1) = Supp R (cid:0) R/ p (cid:1) , and so by [DNT, Theorem 2.2], cd (cid:0) a , R/ p (cid:1) = cd (cid:0) a , M/ p M (cid:1) . Soby Lemma 3.2, cd (cid:0) a , M/ p M (cid:1) = c . As x ∈ p , one has a natural R -epimorphism M/xM −→ M/ p M , andhence Supp R (cid:0) M/ p M (cid:1) ⊆ Supp R (cid:0) M/xM (cid:1) , by employing [DNT, Theorem 2.2] again. Therefore, c = cd (cid:0) a , M/ p M (cid:1) ≤ cd (cid:0) a , M/xM (cid:1) . On the other hand, by [DGTZ, Lemma 2.4], cd (cid:0) a , M/xM (cid:1) = c −
1, which yields our desired contradiction.(ii) Now, we prove that
M/xM is an a -relative generalized Cohen-Macaulay R -module. One hascd (cid:0) a , M/xM (cid:1) = c −
1. So if c = 1, the claim holds. Hence, we may assume that c >
1. Note thatf a (cid:0) M/xM (cid:1) ≤ cd (cid:0) a , M/xM (cid:1) = c − . Thus, to complete the proof, we need to show that H i a (cid:0) M/xM (cid:1) is finitely generated for all 0 ≤ i ≤ c − a (cid:0) M/xM (cid:1) is finitely generated, we can assume that 1 ≤ i ≤ c −
2. Since x is an a -filter regularsequence on M , x is non-zero divisor on M/ Γ a ( M ). Applying the functor H i a ( − ) on the exact sequence0 −→ M/ Γ a ( M ) x −→ M/ Γ a ( M ) −→ M/ (cid:0) xM + Γ a ( M ) (cid:1) −→ , yields the following exact sequenceH i a (cid:0) M/ Γ a ( M ) (cid:1) −→ H i a (cid:16) M/ (cid:0) xM + Γ a ( M ) (cid:1)(cid:17) −→ H i +1 a (cid:0) M/ Γ a ( M ) (cid:1) . Since H j a (cid:0) M/ Γ a ( M ) (cid:1) ∼ = H j a ( M ) is finitely generated for j = i, i + 1, the above exact sequence impliesthat H i a (cid:16) M/ (cid:0) xM + Γ a ( M ) (cid:1)(cid:17) is finitely generated. From the exact sequence0 −→ (cid:0) xM + Γ a ( M ) (cid:1) /xM −→ M/xM −→ M/ (cid:0) xM + Γ a ( M ) (cid:1) −→ , we obtain the following exact sequenceH i a (cid:16)(cid:0) xM + Γ a ( M ) (cid:1) /xM (cid:17) −→ H i a (cid:0) M/xM (cid:1) −→ H i a (cid:16) M/ (cid:0) xM + Γ a ( M ) (cid:1)(cid:17) As (cid:0) xM + Γ a ( M ) (cid:1) /xM is a -torsion, it follows that H i a (cid:16)(cid:0) xM + Γ a ( M ) (cid:1) /xM (cid:17) = 0. So, the above exactsequence implies that H i a (cid:0) M/xM (cid:1) is finitely generated. (cid:3)
Part (ii) of the following example provides an ideal a and an a -relative generalized Cohen-Macaulay R -module M such that a contains no a -Rs.o.p of M . Example 3.4. (i)
Let a be an ideal of R . Any finitely generated R -module with cd ( a , M ) = 1 is an a -relative generalized Cohen-Macaulay R -module. (ii) Let K be a field and S := K [[ X, Y, Z, W ]] . Consider the elements f := XW − Y Z , g := Y − X Z and h := Z − Y W of S and set R := S/ h f i and a := h f, g, h i / h f i . Then R is a Noetherian localring of dimension , cd ( a , R ) = 1 and ara ( a , R ) ≥ ; see [HS, Remark 2.1(ii)] . As ara ( a , R ) =cd ( a , R ) , it follows that a contains no a -Rs.o.p of R . But, by (i) R is an a -relative generalizedCohen-Macaulay. The following result provides a characterization of a -relative generalized Cohen-Macaulay modules byusing a -weak sequences. Proposition 3.5.
Let a be an ideal of R which is contained in the Jacobson radical of R and M a finitelygenerated R -module. Suppose that cd ( a , M ) = ara ( a , M ) . Then the following condition are equivalent: (i) M is a -relative generalized Cohen-Macaulay. (ii) There is an integer ℓ such that every a -Rs.o.p of M is an a ℓ -weak sequence on M . Proof. (i)= ⇒ (ii) By Theorem 3.3, every a -Rs.o.p of M is an a -filter regular sequence on M . So, theassertion follows by [KS, Theorem (i) ⇒ (iii)].(ii)= ⇒ (i) It follows by [KS, Theorem (iv) ⇒ (i)]. (cid:3) Proposition 3.6.
Let a be an ideal of R and M be an a -relative generalized Cohen-Macaulay R -modulewith cd ( a , M ) > . Then M p is Cohen-Macaulay for every p ∈ Supp R M with p $ a . Proof.
Let p ∈ Supp R M such that p $ a . Then p ∈ Supp R M \ V ( a ). One hascd ( a , M ) = f a ( M ) ≤ λ a ( M ) ≤ depth R p M p + ht (cid:0) ( a + p ) / p (cid:1) = depth R p M p + ht (cid:0) a / p (cid:1) ≤ dim R p M p + ht (cid:0) a / p (cid:1) ≤ ht M a ≤ cd ( a , M ) . Thus depth R p M p = dim R p M p , as required. (cid:3)
4. Relative Buchsbaumness
We start this section by the following characterization of relative quasi Buchsbaum modules.
Theorem 4.1.
Let a be an ideal of R and M a finitely generated R -module with cd ( a , M ) = ara ( a , M ) .Consider the following conditions: (i) M is a -relative quasi Buchsbaum. (ii) There is an a -Rs.o.p of M contained in a which is an a -weak sequence on M . (iii) a H i a ( M ) = 0 for all ≤ i < cd ( a , M ) . THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 15
Then ( i ) implies ( ii ) and ( ii ) implies ( iii ) . Furthermore if a is contained in the Jacobson radical of R ,then ( iii ) implies ( i ) . Proof. (i)= ⇒ (ii) holds by the definition.(ii)= ⇒ (iii) follows by Theorem 2.6.(iii)= ⇒ (i) Assumption (iii) implies that M is a -relative generalized Cohen-Macaulay. Let x , . . . , x c ∈ a be an a -Rs.o.p of M . By Theorem 3.3, the sequence x , . . . , x c is an a -filter regular sequence on M .So, by Theorem 2.6, it turns out that x , . . . , x c is an a -weak sequence on M . (cid:3) Goto in [G, Corollary 2.8] proved that a finitely generated R -module M is Buchsbaum if and only ifevery system of parameters of M forms a d -sequence on M . Accordingly, the authors tried to show thatthe conditions (iii) and (iv) in the following result are equivalent, but so far without success. Theorem 4.2.
Let a be an ideal of R and M a finitely generated R -module. Assume that c := cd ( a , M ) =ara ( a , M ) > . Consider the following conditions: (i) For every generating set { b , . . . , b t } of a , the natural map λ iM : H i ( b , . . . , b t , M ) −→ H i a ( M ) issurjective for all ≤ i ≤ c − . (ii) There exists an f -generating set { a , . . . , a s } of a with respect to M such that the natural map λ iM : H i ( a , . . . , a s , M ) −→ H i a ( M ) is surjective for all ≤ i ≤ c − . (iii) M is a -relative Buchsbaum. (iv) Every a -Rs.o.p of M is an u.s.d -sequence on M .Then ( i ) and ( ii ) are equivalent and ( iii ) implies ( iv ) . Furthermore if a is contained in the Jacobsonradical of R , then ( i ) implies ( iii ) . Proof. (i)= ⇒ (ii) is clear by Remark 2.3(B)(ii).(ii)= ⇒ (i) Let { b , . . . , b t } be a generating set of a . By Definition and Remark 2.1(v), one has thefollowing commutative diagramH i ( a , . . . , a s , M ) / / (cid:15) (cid:15) H i ( b , . . . , b t , M ) (cid:15) (cid:15) H i a ( M ) ∼ = / / H i a ( M )for all 0 ≤ i ≤ c −
1. It implies that H i ( b , . . . , b t , M ) −→ H i a ( M ) is surjective for all 0 ≤ i ≤ c − ⇒ (iv) follows by Remark 2.3(A)(iii).(i)= ⇒ (iii) Let x , . . . , x c ∈ a be an a -Rs.o.p of M . The assumption implies that a H i a ( M ) = 0 for0 ≤ i ≤ c −
1. Hence by Theorem 3.3, x , . . . , x c is an unconditioned a -filter regular sequence on M . Wecan find z , . . . , z l in a such that a = h x , . . . , x c , z , . . . , z l i . As in the proof of [TZ, Proposition 1.2], thereare w , . . . , w l in a such that a = h x , . . . , x c , w , . . . , w l i and x , . . . , x c , w , . . . , w l is an unconditioned a -filter regular sequence on M . By the hypothesis, the natural map H i ( x , . . . , x c , w , . . . , w l ) −→ H i a ( M )is surjective for all 0 ≤ i ≤ c −
1. The claim follows from Theorem 2.7. (cid:3)
Next, we provide the comparison between the classes of modules which are introduction in Definition1.1.
Corollary 4.3. (i)
Any a -relative Cohen-Macaulay R -module is a -relative surjective Buchsbaum. (ii) If the ideal a is contained in the Jacobson radical of R and M is an a -relative surjective Buchsbaum R -module with cd ( a , M ) = ara ( a , M ) , then M is a -relative Buchsbaum. (iii) Any a -relative Buchsbaum R -module is a -relative quasi Buchsbaum. (iv) Any a -relative quasi Buchsbaum R -module is a -relative generalized Cohen-Macaulay. Proof. (i) and (iii) are obvious by the definitions.(ii) Let a = h x , . . . , x s i . By [SV, Lemma 1.5], we have the following commutative diagramExt iR (cid:0) R/ a , M (cid:1) ϕ iM / / ψ iM (cid:15) (cid:15) H i a ( M )H i ( x , . . . , x s , M ) λ iM ♦♦♦♦♦♦♦♦♦♦♦ Since ϕ iM is surjective for all i < c := cd( a , M ) and λ iM ψ iM = ϕ iM , we deduce that λ iM is surjective forall i < c . Thus the claim follows by Theorem 4.2.(iv) Theorem 4.1 implies that a H i a ( M ) = 0 for all i < cd ( a , M ). Then by Faltings’ Local-globalPrinciple Theorem [F, Satz 1], H i a ( M ) is finitely generated for all i < cd ( a , M ). Thus, either cd ( a , M ) ≤ a , M ) = f a ( M ). (cid:3) Below, we record another corollary of Theorem 4.2.
Corollary 4.4.
Let ( R, m ) be a Buchsbaum ring. Let a be an ideal of R which is generated by a part ofa system of parameters of R . Then R is an a -relative Buchsbaum R -module. Proof.
Let d := dim R . Suppose that a := h x , x , . . . , x n i , where x , . . . , x n , . . . , x d is a system ofparameters of R . By the definition of Buchsbaum rings, it turns out that x , . . . , x n is an unconditioned m -weak sequence on R . In view of Remark 2.3(B)(i) and Grothendieck’s Non-vanishing Theorem, wehave H d − n m (cid:0) H n a ( R ) (cid:1) ∼ = H d − n m (cid:16) H n h x ,...,x n i ( R ) (cid:17) ∼ = H d m ( R ) = 0 . Hence, H n a ( R ) = 0, and so cd( a , R ) = n = ara( a , R ).On the other hand, Remark 2.3(A)(iii) implies that x , . . . , x n is an u.s.d -sequence on R . So, by [G,Theorem 2.6], the canonical map H i ( x , . . . , x n , R ) −→ H i h x ,...,x n i ( R )is surjective for all 0 ≤ i ≤ n −
1. Also, Remark 2.3(A)(ii) yields that x , . . . , x n is an unconditioned a -filter regular sequence on R . Thus, by Theorem 4.2, R is a -relative Buchsbaum. (cid:3) The following example shows that the condition a is contained in the Jacobson radical of R is necessaryin Corollary 4.3(ii). Example 4.5.
Let k be a field and u , v , w be independent variables. Consider the polynomial ring R := k [ u, v, w ] . Set a := v (1 − u ) , a := w (1 − u ) , a := u and a := h a , a , a i . Note that a is notcontained in the Jacobson radical of R and a = h u, v, w i . We have H i a ( R ) = 0 for every ≤ i < and H a ( R ) = 0 . So, cd ( a , R ) = ara ( a , R ) = 3 . Hence, R is a -relative Cohen-Macaulay. If R is a -relative Buchsbaum, then by Theorem 4.2 the sequence a , a , a is a d -sequence. On the other hand, v ∈ h a i : R a a , but v / ∈ h a i : R a . So, a , a , a is not a d -sequence on R . THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 17
Proposition 4.6.
Let a be an ideal of R which is contained in the Jacobson radical of R and M a finitelygenerated R -module with cd ( a , M ) = ara ( a , M ) . Assume that r := grade ( a , M ) < c := cd ( a , M ) and H i a ( M ) = 0 for all i = r, c . Then the following are equivalent: (i) M is a -relative surjective Buchsbaum. (ii) M is a -relative Buchsbaum. (iii) M is a -relative quasi Buchsbaum. Proof. (i)= ⇒ (ii) and (ii)= ⇒ (iii) follow by Corollary 4.3.(iii)= ⇒ (i) Since grade ( a , M ) = r , [AS, Lemma 2.5(b)] implies the following natural R -isomorphismExt rR (cid:0) R/ a , M (cid:1) ∼ = Hom R ( R/ a , H r a ( M )) . As M is a -relative quasi Buchsbaum, Theorem 4.1 implies that a H r a ( M ) = 0, and so one deduces thenatural R -isomorphism Ext rR (cid:0) R/ a , M (cid:1) ∼ = H r a ( M ). Thus, M is a -relative surjective Buchsbaum. (cid:3) The following lemma is needed in the proof of Theorem 4.8.
Lemma 4.7.
Let a be an ideal of R and M a finitely generated R -module. Let n be a natural integer and x , . . . , x n ∈ a . If a Γ h x k +1 i (cid:0) M/ h x , . . . , x k i M (cid:1) = 0 for all ≤ k ≤ n − , then x , . . . , x n is both a -weaksequence on M and d -sequence on M . Proof.
Let 0 ≤ k ≤ n − N := M h x ,...,x k i M . We have0 : N x k +1 ⊆ N x k +1 ⊆ Γ h x k +1 i ( N )= 0 : N a ⊆ N h x , . . . , x n i⊆ N x k +1 . Thus 0 : N x k +1 = 0 : N a , and so x , . . . , x n is an a -weak sequence on M .Since 0 : N x k +1 = 0 : N h x , . . . , x n i , by [T1, Theorem 1.1(v)], one deduces that x , . . . , x n is a d -sequence on M . (cid:3) The notion of relative system of parameters plays an important role in the proof of the next resultwhich establishes a characterization of relative Buchsbaum modules in terms of relative quasi Buchsbaummodules.
Theorem 4.8.
Let a be an ideal of R which is contained in the Jacobson radical of R and let M be afinitely generated R -module with c := cd ( a , M ) = ara ( a , M ) . Then the following are equivalent: (i) For every a -Rs.o.p x , . . . , x c of M , a Γ h x k +1 i (cid:0) M/ h x , . . . , x k i M (cid:1) = 0 for all ≤ k ≤ c − . (ii) M is a -relative Buchsbaum. (iii) For every a -Rs.o.p x , . . . , x c ∈ a and every ≤ k ≤ c − , the R -module M/ h x , . . . , x k i M is a -relative quasi Buchsbaum. Proof. (i)= ⇒ (ii) By Lemma 4.7, every a -Rs.o.p of M is an a -weak sequence on M . So, M is a -relativeBuchsbaum.(ii)= ⇒ (iii) Let M be an a -relative Buchsbaum R -module and x , . . . , x c be an a -Rs.o.p of M . Let0 ≤ k ≤ c −
1. Note that ´ x := x , . . . , x k , x k +1 , . . . , x c is also an a -Rs.o.p of M . Hence, ´ x is an a -weak sequence on M . So, x k +1 , . . . , x c is an a -weak sequence on M/ h x , . . . , x k i M . Thus, by Theorem 2.6, a H i a (cid:0) M/ h x , . . . , x k i M (cid:1) = 0 for all 0 ≤ i < c − k . Also, by [DGTZ, Lemma 2.4], it turns out thatcd( a , M/ h x , . . . , x k i M ) = ara( a , M/ h x , . . . , x k i M ) = c − k. Therefore, by Theorem 4.1, M/ h x , . . . , x k i M is a -relative quasi-Buchsbaum.(iii)= ⇒ (i) Let x , . . . , x c be an a -Rs.o.p of M and let 0 ≤ k ≤ c −
1. By the assumption andCorollary 4.3, M/ h x , . . . , x k i M is a -relative generalized Cohen-Macaulay. Since by [DGTZ, Lemma 2.4], x k +1 , . . . , x c is an a -Rs.o.p of M/ h x , . . . , x k i M , Theorem 3.3(i) implies that x k +1 , . . . , x c is an a -filterregular sequence on M/ h x , . . . , x k i M . Therefore, by Remark 2.3(B)(i)Γ a (cid:0) M/ h x , . . . , x k i M (cid:1) = Γ h x k +1 i (cid:0) M/ h x , . . . , x k i M (cid:1) , and so Theorem 4.1 yields that a Γ h x k +1 i (cid:0) M/ h x , . . . , x k i M (cid:1) = 0. (cid:3) References [AS] J. Asadollahi and P. Schenzel,
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E-mail address : [email protected] THEORY OF RELATIVE GENERALIZED COHEN-MACAULAY MODULES 19
A. Ghanbari Doust, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.
E-mail address : [email protected] M. Tousi, Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran-and-School of Math-ematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.
E-mail address : [email protected] H. Zakeri, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.
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