Buchsbaumness of the associated graded rings of filtration
aa r X i v : . [ m a t h . A C ] A p r BUCHSBAUMNESS OF THE ASSOCIATED GRADED RINGS OFFILTRATION
KUMARI SALONI
Abstract.
Let ( A, m ) be a Noetherian local ring of dimension d > I an m -primaryideal of A . In this paper, we discuss a sufficient condition, for the Buchsbaumness of thelocal ring A to be passed onto the associated graded ring of filtration. Let I denote an I -good filtration. We prove that if A is Buchsbaum and the I -invariant, I ( A ) and I ( G ( I )),coincide then the associated graded ring G ( I ) is Buchsbaum. As an application of our result,we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions forHilbert coefficients. Introduction
The purpose of this paper is to examine the Buchsbaum property of associated gradedrings of filtration. Let ( A, m ) be a Noetherian local ring of dimension d > m and M a finite A -module. We say that M is a Buchsbaum module if theinvariant ℓ A ( M/QM ) − e ( Q, M ) is independent of the choice of parameter ideals Q of M , where ℓ A ( M/QM ) and e ( Q, M ) denote the length of
M/QM and the multiplicity of M relative to Q , respectively. Note that a Cohen-Macaulay module is Buchsbaum with ℓ A ( M/QM ) − e ( Q, M ) = 0 for every parameter ideal Q and the converse also holds. Let I be an m -primary ideal of A and R ( I ) = ⊕ n ≥ I n the Rees algebra of I . The associated gradedring G ( I ) = ⊕ n ≥ I n /I n +1 is called Buchsbaum if the localization G ( I ) N is Buchsbaum as an R ( I ) N -module where N is the unique homogeneous maximal ideal of R ( I ).The structural properties of the local ring and the associated graded ring are deeply con-nected to each other. Many important properties, such as reducedness, normality, Cohen-Macaulayness etc., are inherited by the local ring A from the associated graded ring (however,this is not true for Buchsbaum rings, see [6, Example 4.10] for a counter example). We areinterested in the inheritance of such properties while passing from A to G ( I ) which is impor-tant in the theory of blowing-up rings. In this process, many good properties are lost. A greatdeal of research has been focused on studying the cases where this loss could be avoided. Forexample, if A is Cohen-Macaulay and the reduction number of I is at most one, then G ( I )is Cohen-Macaulay. Since the notion of Buchsbaum modules is a generalization of Cohen-Macaulay modules, an interesting problem is to study the cases when the associated gradedring of a Buchsbaum local ring is Buchsbaum.Now suppose that A is a Buchsbaum local ring. Goto proved the following results: (i) if A hasmaximal embedding dimension, then G ( m ) is Buchsbaum [3], (ii) if dim A ≥
2, depth
A > A has multiplicity 2, then G ( m ) is Buchsbaum [4] and (iii) G ( Q ) is Buchsbaum for a parameterideal Q [5]. Certain boundary conditions on the Hilbert coefficients e i ( I ) are also expectedto force Buchsbaumness of G ( I ) which we discuss briefly at the end of the paper, see [1, 7]. Date : July 24, 2020.2010
Mathematics Subject Classification.
Primary: 13H10, 13A30, Secondary: 13D45.
Key words and phrases.
Buchsbaum modules, I -invariant, local cohomology, d-sequence, Koszul homology.The author was supported by NBHM-DAE funded by Govt. of India and Chennai Mathematical Institute. Furthermore, the m -primary ideals with reduction number one exhibit nice properties. Manyauthors have investigated the Buchsbaum properties of associated graded rings of such ideals,see [6, 7, 10]. Nakamura in [10] proved the following result which generalizes Goto’s result,mentioned above, for parameter ideals. If I = QI for some minimal reduction Q = ( a , . . . , a d )of I , then G ( I ) is Buchsbaum if and only if the equality ( a , . . . , a d ) ∩ I n = ( a , . . . , a d ) I n − holds for 3 ≤ n ≤ d + 1 . In most of the cases, when G ( I ) is Buchsbaum, the I -invariant of G ( I )coincides with that of A . Recall that the I -invariant of an A -module M of dimension s is I ( M ) = s − X i =0 (cid:18) s − i (cid:19) h i ( M ) , (1)where h i ( M ) = ℓ A (H i m ( M )) and H i m ( M ) is the local cohomology module of M with supportin the maximal ideal m . Indeed, Yamagishi [19, 20] removed the condition on the reductionnumber of I and proved that G ( I ) is Buchsbaum and h i ( G ( I )) = h i ( A ) for 0 ≤ i < d if and onlyif I ( G ( I )) = I ( A ). Here, h i ( G ( I )) = ℓ (H i N ( G ( I )) is the length of the local cohomology moduleof G ( I ) with support in N and I ( G ( I )) = s − P i =0 (cid:0) d − i (cid:1) h i ( G ( I )). See Section 2 for notations.We extend the above theory to the framework of good filtration. An I -good filtration of A is a sequence of ideals I = { I n } n ≥ such that I n +1 ⊆ I n , II n ⊆ I n +1 for all n > I n +1 = II n for all n ≫
0, see [14]. If A is analytically unramified then the filtration ofintegral closures and the tight closures (provided A contains a field) of powers of I are I -goodfiltration. Thus the study of good filtration is in itself interesting. Moreover, many results inliterature concerning I -adic filtration { I n } n ≥ require non adic-filtration in their proofs such asRatliff-Rush filtration and the filtration { QI n } with Q a reduction of I etc. For instance, seethe method of proof for the bound on reduction number by Rossi in [13]. Another interestingreason to consider this general set up of I -good filtration is its application in the study of othergraded algebras such as Sally modules, fiber cones and symmetric algebras. For instance, Rossiand Valla in [14], have developed very elegant general methods for the above graded algebrasusing the theory of I -good filtration.Let I = { I n } be an I -good filtration of A . A reduction of I is an ideal Q ⊆ I such that I n +1 = QI n for n ≫
0. Let R ( I ) = ⊕ n ≥ I n and G ( I ) = ⊕ n ≥ I n /I n +1 denote the Rees algebraand the associated graded ring of I . We write N for the unique homogeneous maximal ideal m R ( I ) + R ( I ) + of R ( I ). As earlier, h i ( A ) (resp. h i ( G ( I ))) denotes the length of the localcohomology modules H i m ( A ) of A with support in m (resp. H i N ( G ( I )) of G ( I ) with supportin N ). We say that G ( I ) is Buchsbaum (resp. quasi-Buchsbaum) over R ( I ) if G ( I ) N isBuchsbaum (resp. quasi-Buchsbaum) over R ( I ) N . The I -invariant of G ( I ) can be definedanalogous to (1). With this setting, we prove the following results in this paper. Theorem 1.1.
Let A be a quasi-Buchsbaum local ring and suppose that the equality I ( G ( I )) = I ( A ) holds. Then G ( I ) is quasi-Buchsbaum over R ( I ) . Theorem 1.2.
Let A be a Buchsbaum local ring of dimension d > . Let Q = ( a , . . . , a d ) ⊆ I be a reduction of I and r ≥ be an integer such that I n +1 = QI n for all n ≥ r . Then thefollowing statements are equivalent.(1) G ( I ) is a Buchsbaum R ( I ) -module and h i ( G ( I )) = h i ( A ) for ≤ i < d .(2) The equality I ( G ( I )) = I ( A ) holds.(3) The equality ( a , . . . , a d ) ∩ I n = ( a , . . . , a d ) I n − holds for all < n ≤ d + r . This paper is organized as follows. Section 2 is devoted to recalling the basic definitions andextend certain facts about the I -invariant and d-sequences to the case of I -good filtration. InSection 3, we consider the equi- I -invariant case, i.e., when the equality I ( G ( I )) = I ( A ) holds. UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 3
Theorem 1.1 and most of Theorem 1.2 are proved in this section. The methods of our proofsare inspired by the work of Yamagishi in [19] and [20]. However, obtaining the Buchsbaum-ness of G ( I ) from the equality I ( G ( I )) = I ( A ) is difficult because the generators of uniquehomogeneous maximal ideals of R ( I ) or G ( I ) are not necessarily of degree one. Therefore,Yamagishi’s methods, which are heavily dependent on the existence of a certain generating setof N in degree one, can not be applied in its present form for filtration. In Section 4, we discussthe existence of a suitable generating set of N which allows to generalize the methods of [19]to the case of filtration. In Section 5, we complete the proof of Theorem 1.2 by establishingthe Buchsbaumness of G ( I ). As an application of our results, we indicate an alternative proofof a conjecture, stated by Corso in [1], on Hilbert coefficients and Buchsbaumness of G ( I ).In this paper, let M denote a finite A -module of dimension s >
0. We write ℓ A ( M ) for thelength of M and µ ( M ) for the minimal number of generators of M . We set ν := µ ( m ). Let U = H m ( A ). For a quotient ring A ′ = A/J , we write I A ′ for the filtration { ( I n + J ) /J } n ≥ ofthe ring A ′ . It is easy to see that I A ′ is an IA ′ -good filtration. The multiplicity of A relative tothe filtration I is written as e ( I ) instead of e ( I , A ). If needed, we write an element a ∈ R ( I ) n of degree n as at n since R ( I ) ⊆ R [ t ]. For a graded module L , the n -th graded component isdenote by [ L ] n . We assume that the residue field A/ m is infinite.2. Preliminaries
In this section, we discuss some results on generalized Cohen-Macaulay modules and their I -invariant. The notion of generalized Cohen-Macaulay modules was first discussed in [2].For a parameter ideal Q of M , we define I ( Q ; M ) := ℓ A ( M/QM ) − e ( Q, M ). If M is Cohen-Macaulay, I ( Q ; M ) = 0 for any parameter ideal Q . The ideas discussed in [2] and [15], led to thestudy of modules for which I ( Q, M )), a measure of non Cohen-Macaulayness, is independentof the parameter ideals Q of M . Such modules have many properties similar to those ofCohen-Macaulay modules. We refer to [16] for details. The I -invariant, also known as theBuchsbaum-invariant, of M is defined as I ( M ) := sup { I ( Q ; M ) : Q is a parameter ideal of M. } In general, 0 ≤ I ( M ) ≤ ∞ and I ( M ) is finite if and only if M is generalized Cohen-Macaulay.Equivalently, M is generalized Cohen-Macaulay if ℓ A (H i m ( M )) < ∞ for 0 ≤ i < s . In this case, I ( M ) = s − X i =0 (cid:18) s − i (cid:19) ℓ A (H i m ( M )) . A system of parameters a , . . . , a s ∈ m of M is called standard if I ( M ) = I ( Q ; M ) where Q = ( a , . . . , a s ). An ideal I with ℓ ( M/IM ) < ∞ is called a standard ideal for M if everysystem of parameters of M contained in I is standard. Every system of parameters of M is standard, i.e., m is a standard ideal for M , if and only if M is Buchsbaum. We say that M is quasi-Buchsbaum if m is a standard ideal for M . Clearly, Buchsbaum modules arequasi-Buchsbaum and I ( M ) < ∞ whenever M is quasi-Buchsbaum. We say that G ( I ) isquasi-Buchsbaum (respectively generalized Cohen-Macaulay) if G ( I ) N is quasi-Buchsbaum(respectively generalized Cohen-Macaulay) R ( I ) N -module. Note that H i N R ( I ) N ( G ( I ) N ) ∼ =H i N ( G ( I )) for each i . We have that I ( G ( I )) = I ( G ( I ) N ) = s − X i =0 (cid:18) s − i (cid:19) ℓ R ( I ) (H i N ( G ( I ))) . For I -adic filtration, i.e., I n = I n for all n , it is known that I ( G ( I )) ≥ I ( A ). The followingresult is a generalization of [18, Lemma 5.1]. KUMARI SALONI
Proposition 2.1.
Let p ≥ be an integer and let a , . . . , a d ⊆ I p a sequence of elements suchthat a t p , . . . , a d t p ∈ R ( I ) p is a system of parameters of G ( I ) . Then I (( a t p , . . . , a d t p ); G ( I )) ≥ I (( a , . . . , a d ); A ) and equality holds if and only if ( a , . . . , a d ) ∩ I k = ( a , . . . , a d ) I k − p for all k ≥ .Proof. For n ≥
0, we have ℓ ( A/ ( a , . . . , a d ) n ) ≥ ℓ ( A/I np ). Therefore e (( a , . . . , a d ) , A ) ≥ e ( I ) p d . (2)Let k ≥ I k +1 = ( a , . . . , a d ) I k +1 − p for all k ≥ k . Then for n ≥ k ≥ k + np , I k +1 = ( a , . . . , a d ) n I k +1 − np . Put k ( n ) = k + np + 1, Then, for all n ≥ ℓ ( A/I k ( n ) ) = k ( n ) − X k =0 ℓ ( I k /I k +1 ) ≥ ℓ ( G ( I ) / ( a t p , . . . , a d t p ) n G ( I ))= ∞ X k =0 ℓ ( I k / (( a , . . . , a d ) n I k − np + I k +1 ))= k ( n ) − X k =0 ℓ ( I k / (( a , . . . , a d ) n I k − np + I k +1 )) ≥ k ( n ) − X k =0 ℓ (cid:0) I k / (( a , . . . , a d ) n ∩ I k ) + I k +1 ) (cid:1) (3) ≥ k ( n ) − X k =0 ℓ (( I k + ( a , . . . , a d ) n ) / ( I k +1 + ( a , . . . , a d ) n ))= ℓ ( A/ (( a , . . . , a d ) n + I k ( n ) )) = ℓ ( A/ ( a , . . . , a d ) n ) . (4)Thus e ( I ) p d ≥ e (( a t p , . . . , a d t p ) , G ( I )) ≥ e (( a , . . . , a d ) , A ). Now using (2), we get that e (( a t p , . . . , a d t p ) , G ( I )) = e (( a , . . . , a d ) , A ) . (5)Further, putting n = 1 in (4), we get ℓ ( G ( I ) / ( a t p , . . . , a d t p ) G ( I )) − e (( a t p , . . . , a d t p ) , G ( I )) ≥ ℓ ( A/ ( a , . . . , a d )) − e (( a , . . . , a d ) , A ) . Thus I (( a t p , . . . , a d t p ); G ( I )) ≥ I (( a , . . . , a d ); A ) . Suppose equality holds. Then ℓ ( G ( I ) / ( a t p , . . . , a d t p ) G ( I )) = ℓ ( A/ ( a , . . . , a d )). For k > k ,( a , . . . , a d ) ∩ I k = ( a , . . . , a d ) ∩ ( a , . . . , a d ) I k − p = ( a , . . . , a d ) I k − p . Suppose k ≤ k . Then wehave ( a , . . . , a d ) ∩ I k ⊆ ( a , . . . , a d ) I k − p + I k +1 from the equality in (3). Therefore,( a , . . . , a d ) ∩ I k ⊆ ( a , . . . , a d ) I k − p + (( a , . . . , a d ) ∩ I k +1 ) ⊆ ( a , . . . , a d ) I k − p + (( a , . . . , a d ) I k +1 − p + (( a , . . . , a d ) ∩ I k +2 )... ⊆ ( a , . . . , a d ) I k − p + ( a , . . . , a d ) I k − p = ( a , . . . , a d ) I k − p . Clearly, if the above equality holds for all k >
0, then ℓ ( G ( I ) / ( a t p , . . . , a d t p ) G ( I )) = ℓ ( A/ ( a , . . . , a d )) . (cid:3) UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 5
The role played by regular sequences in the study of Cohen-Macaulay modules is broadlyreplaced by d-sequences in case of generalized Cohen-Macaulay modules. A sequence a , . . . , a s of elements of A is said to be a d-sequence on M if the equality q i − M : a i a j = q i − : a j holds for 1 ≤ i ≤ j ≤ s where q i − = ( a , . . . , a i − ) and q = (0). It is said to be an unconditioned d-sequence if it is a d-sequence in any order. Moreover, we say that a , . . . , a s isan unconditioned strong d-sequence (u.s.d-sequence) on M if a n , . . . , a n s s is an unconditionedd-sequence on M for all integers n , . . . , n s >
0. Recall that a standard system of parametersis a d-sequence and a module is Buchsbaum if and only if every system of parameters is u-s-d-sequence. Indeed, we have the following characterization for standard system of parameters.
Remark 2.2. [15, Theorem A][18, Theorem 2.1]
For any system of parameters a , . . . , a s of M , the following conditions are equivalent:(1) a , . . . , a s is an u.s.d-sequence on M ;(2) a , . . . , a s is a standard system of parameters of M , i.e., I (( a , . . . , a s ); M ) = I ( M ) ;(3) the equality I (( a , . . . , a s ); M ) = I (( a , . . . , a s ); M ) holds. The following proposition is discussed in [20, Proposition 2.2] except the last part.
Proposition 2.3.
Let a , . . . , a d be a sequence of elements in I . Let m > be an integersuch that a n , . . . , a n d d is an unconditioned d-sequence for all n i ≥ m and ≤ i ≤ d . Then thefollowing conditions are equivalent:(1) The equality ( a n i i | i ∈ Λ) ∩ I n = X i ∈ Λ a n i i I n − n i holds for all Λ ⊆ [1 , d ] , n ∈ Z and n i ≥ m ;(2) The equality ( a m , . . . , a md ) ∩ I n = ( a m , . . . , a md ) I n − m holds for all n ∈ Z .When this is the case, G ( I ) / ( a i t ) n i G ( I ) ∼ = G ( I A/ ( a n i i )) as graded R ( I ) -modules for all n i ≥ m and ≤ i ≤ d .Proof. Equivalence of (1) and (2) follows from [20, Proposition 2.2]. We only prove the lastpart. We have that, for all k ≥ , [ G ( I ) / ( a i t ) n i G ( I )] k ∼ = I k a n i i I k − n i + I k +1 ∼ = I k ( a n i i ∩ I k ) + I k +1 ∼ = I k + ( a n i i ) I k +1 + ( a n i i ) ∼ = [ G ( I A/ ( a n i i ))] k . (cid:3) It is often useful to consider rings of positive depth. Particularly, it provides a way to reducea problem to lower dimensional cases and apply method of induction. We now briefly relatethe properties of I -invariant in the ring A and A/U where U = H m ( A ). Consider the exactsequence 0 → U ∗ → G ( I ) φ −→ G ( I A/U ) → R ( I )-modules where φ : G ( I ) → G ( I A/U ) is the canonical epimorphism of gradedmodules induced from the projection map A → A/U and U ∗ = ker φ. Since [ G ( I A/U )] n =( I n + U ) / ( I n +1 + U ) ∼ = I n / (( U ∩ I n ) + I n +1 ), we have[ U ∗ ] n = (( U ∩ I n ) + I n +1 ) /I n +1 ∼ = ( U ∩ I n ) / ( U ∩ I n +1 ) KUMARI SALONI for n ∈ Z . Thus ℓ A ( U ∗ ) = ℓ A ( U ) = ℓ A (H m ( A )). Then, applying the local cohomology functorto the above sequence, we get the short exact sequence0 → U ∗ → H N ( G ( I )) → H N ( G ( I A/U )) → i N ( G ( I )) ∼ = H i N ( G ( I A/U )) for all i ≥ . Lemma 2.4.
With the notations as above(1) h ( G ( I A/U )) = h ( G ( I )) − ℓ ( U ∗ ) = h ( G ( I )) − h ( A ) and h i ( G ( I /U )) = h i ( G ( I )) for i ≥ .(2) The following conditions are equivalent:(a) h ( G ( I A/U )) = 0 ;(b) h ( G ( I )) = h ( A ) ;(c) H N ( G ( I )) = U ∗ .(3) Suppose A is generalized Cohen-Macaulay and I ( G ( I )) = I ( A ) . Then I ( G ( I A/U )) = I ( A/U ) .Proof. The assertions (1) and (2) are discussed above. For (3), suppose I ( G ( I )) = I ( A ). Then,by assertion (1), I ( G ( I A/U )) = I ( G ( I )) − h ( A ) = I ( A ) − h ( A ) = I ( A/U ) (cid:3) Equi- I -invariant case and quasi-Buchsbaumness of G ( I )Let ( a , . . . , a d ) ⊆ I be a reduction of I and r ≥ I n +1 = ( a , . . . , a d ) I n for all n ≥ r . We write Q for the ideal ( a , . . . , a d ). For this section, we assume that A isgeneralized Cohen-Macaulay. We prove several equivalent conditions for the equi- I -invariantcase, i.e., when the equality I ( G ( I )) = I ( A ) holds. The results of this section are discussed in[20] for I -adic filtration. Similar methods extend the results to filtration. Proposition 3.1.
Let m > be an integer such that a n , . . . , a n d d is an unconditioned d-sequence for all n i ≥ m , ≤ i ≤ d . Then the following conditions are equivalent:(1) I ( G ( I )) = I ( A ) ;(2) The equality ( a m , . . . , a md ) ∩ I n = ( a m , . . . , a md ) I n − m holds for m < n ≤ d (2 m −
1) + r ;(3) The equality ( a m , . . . , a md ) ∩ I n = ( a m , . . . , a md ) I n − m holds for all n ∈ Z .When this is the case, ( a t ) n , ( a t ) n , . . . , ( a d t ) n d is an unconditioned d-sequence for G ( I ) forall n i ≥ m , ≤ i ≤ d .Proof. Note that a m , . . . , a md forms an u-s-d-sequence. By Proposition 2.1, we have that, forall l ≥ I ( G ( I )) ≥ I ((( a t ) lm , . . . , ( a d t ) lm ); G ( I )) ≥ I (( a lm , . . . , a lmd ); A ) = I ( A ) . If I ( G ( I )) = I ( A ), then Proposition 2.1 gives that ( a lm , . . . , a lmd ) ∩ I n = ( a lm , . . . , a lmd ) I n − lm for n ∈ Z and for any l ≥ a lm , . . . , a lmd ) ∩ I n = ( a lm , . . . , a lmd ) I n − lm holds for n ∈ Z and for all l ≥
1. Consequently, Proposition 2.1 gives the equality I ((( a t ) lm , . . . , ( a d t ) lm ); G ( I )) = I (( a lm , . . . , a lmd ); A ) = I ( A ) for all l ≥
1. Thus I ( G ( I )) = I ((( a t ) lm , . . . , ( a d t ) lm ); G ( I )) = I ( A ) for all l ≥ a t ) m , . . . , ( a d t ) m ) forms a u.s.d-sequence and by Remark 2.2. This establishes (1) ⇔ (3). UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 7
The statements (2) and (3) are equivalent since for all n > d (2 m −
1) + r , we have that I n = Q n − r I r ⊆ ( a m , . . . , a md ) Q n − r − m I r ⊆ ( a m , . . . , a md ) I n − m . The last statement follows from Remark 2.2 and the fact that for n i ≥ m , I ( G ( I )) ≥ I ((( a t ) n , . . . , ( a d t ) n d ); G ( I )) ≥ I ((( a t ) m , . . . , ( a d t ) m ); G ( I )) = I ( G ( I )) . (cid:3) We now prove the quasi-Buchsbaumness of the associated graded ring. Recall that an A -module M is quasi-Buchsbaum if and only at least one (equivalently every) system of pa-rameters contained in m is a weak M -sequence. Recall, from [17], that a sequence a , . . . , a s of elements of A is said to be a weak M -sequence if the equality q i − M : a i = q i − M : m holdsfor 1 ≤ i ≤ s where q i − = ( a , . . . , a i − ) and q = (0). The following result is a generalizationof [20, Theorem 1.2]. See [20] for the converse statement in I -adic case. Theorem 3.2.
Suppose that A is a quasi-Buchsbaum local ring and I ( G ( I )) = I ( A ) . Then G ( I ) is quasi-Buchsbaum over R ( I ) .Proof. Suppose A is quasi-Buchsbaum and I ( G ( I )) = I ( A ). Then a n , . . . , a n d d is a weaksequence and an unconditioned d-sequence in A for all n i ≥
2. Let f i = a i t ∈ R ( I ) . We showthat f , . . . , f d is a weak G ( I ) N -sequence, i.e.,( f , . . . , f i − ) G ( I ) : f i ⊆ ( f , . . . , f i − ) G ( I ) : N for every 1 ≤ i ≤ d. For this, we fix an i and let g be a homogeneous element of G ( I ) withdeg g = m such that gf i ⊆ ( f , . . . , f i − ) G ( I ). Let g = x + I m +1 for some x ∈ I m . Thenthere exist an element y ∈ ( a , . . . , a i − ) I m and z ∈ I m +3 such that a i x = y + z . Since a n , . . . , a n d d is an unconditioned d-sequence in A for all n i ≥ I ( G ( I )) = I ( A ), wehave z ∈ ( a , . . . , a i ) ∩ I m +3 = ( a , . . . , a i ) I m +1 by Proposition 3.1 and Proposition 2.3. Let v ∈ I m +1 such that a i ( x − v ) ∈ ( a , . . . , a i − ) . Now x − v ∈ ( a , . . . , a i − ) : a i = ( a , . . . , a i − ) : m as a , . . . , a d is a weak A -sequence.Now we have that m ( x − v ) ⊆ ( a , . . . , a i − ) ∩ I m ⊆ ( a , . . . , a i − ) I m − and I n ( x − v ) ⊆ ( a , . . . , a i − ) ∩ I m + n ⊆ ( a , . . . , a i − ) I m + n − for all n ≥ N g ⊆ ( f , . . . , f i − ) G ( I ). Thus G ( I ) is a quasi-Buchsbaummodule over R ( I ). (cid:3) Proposition 3.3.
The following conditions are equivalent:(1) I ( G ( I )) = I ( A ) ;(2) h i ( G ( I )) = h i ( A ) for ≤ i < d. When this is the case,
A/U satisfies the above equivalent conditions. Furthermore, we havethat(i) the sequence → H N ( G ( I )) → G ( I ) → G ( I A/U ) → of graded R ( I ) -modules is exact, [H N ( G ( I ))] n ∼ = ( U ∩ I n ) / ( U ∩ I n +1 ) for all n ∈ Z and(ii) if m H m ( A ) = 0 , then N H N ( G ( I )) = 0 . Proof.
The last part is clear from Lemma 2.4.(1) ⇒ (2) We apply induction on d . For d = 1, it is clear. Suppose d ≥
2. By passing to
A/U and using Lemma 2.4, we may assume that depth
A >
0. Then I ( G ( I )) < ∞ , so G ( I )is generalized Cohen-Macaulay. Let a t, . . . , a d t ∈ R ( I ) be a system of parameters of G ( I )(see Lemma 4.6 for existence of such a , . . . , a d ∈ I ). Let m > KUMARI SALONI a n , a n , . . . , a n d d forms a standard system of parameters of A for each n i ≥ m . By Proposition3.1 and Proposition 2.3, ( a m ) ∩ I k = a m I k − m for all k ∈ Z and G ( I ) / ( a t ) m G ( I ) ∼ = G ( I A/ ( a m ))as graded R ( I )-modules. Since depth A >
0, ( a t ) m is a non-zero-divisor on G ( I ). Thereforeh ( G ( I )) = 0 . Now, consider the short exact sequences0 → A a m −−→ A → A/ ( a m ) → → G ( I )( − m ) ( a t ) m −−−−→ G ( I ) → G ( I A/ ( a m )) → . By Proposition 3.1, ( a t ) m , . . . , ( a d t ) m is an u.s.d-sequence on G ( I ). We can choose m largeenough so that a m (resp. ( a t ) m ) annihilates the local cohomology modules H i m ( A ) (resp.H i N ( G ( I ))) for 0 ≤ i ≤ s −
1. Then, for 0 ≤ i < s −
1, the following sequences are exact:0 → H i m ( A ) → H i m ( A/ ( a m )) → H i +1 m ( A ) → , → H i N ( G ( I )) → H i N ( G ( I A/ ( a m ))) → H i +1 N ( G ( I ))( − m ) → . Consequently, for 0 ≤ i < s −
1, we haveh i ( A/ ( a m )) = h i ( A ) + h i +1 ( A ) and h i ( G ( I A/ ( a m ))) = h i ( G ( I )) + h i +1 ( G ( I )) (6)which gives that I ( G ( I / ( a m ))) = I ( A/ ( a m )). By induction hypothesis, h i ( G ( I A/ ( a m ))) =h i ( A/ ( a m )) for 0 ≤ i < s −
1. Now using (6) inductively, we get that h i ( G ( I )) = h i ( A ) for0 ≤ i < s. (2) ⇒ (1) It follows from the definition of I -invariant. (cid:3) A generating set of N In this section, we discuss the existence of a generating set of N consisting of homogeneouselements, not necessarily in the same degree, which possesses the properties of a G ( I )-basis of N . We first recall the following definition from [16]. Definition 4.1. [16, Definition 1.7]
Let J be an ideal such that dim M/J M = 0 . A system ofelements a , . . . , a t of A is called an M -basis of J if ( i ) a , . . . , a t is a minimal generating setof J and ( ii ) for every system i , . . . , i s of integers such that ≤ i < i < . . . < i s ≤ t , theelements a i , . . . , a i s form a system of parameters of M . Let R = ⊕ n ≥ R n be a Noetherian graded ring and N a Noetherian graded R -module. Fora homogeneous ideal J of R with dim N/ J N = 0, an N -basis of J consisting of homogeneouselements can be defined in a similar way. In local case, such a basis always exists, see [16,Proposition 1.9, Chap I]. For k -algebras, the existence of N -bases of homogeneous ideals isdiscussed in [16, Section 3, Chap I]. The main goal of this section is to prove Lemma 4.6 whichgives the existence of a G ( I )-basis of N such that its degree zero elements form an A -basisof m . First, we briefly discuss preliminary facts. In the rest of this section, we assume that R = ⊕ n ≥ R n is a Noetherian graded ring where R is a local ring with maximal ideal m andinfinite residue field R / m . Let M = m ⊕ ⊕ n ≥ R n . Lemma 4.2.
Let m , . . . , m p ∈ M be a generating set of M and { M i : 1 ≤ i ≤ k } a finitecollection of proper submodules of M . Then there exists an element y = α m + . . . + α p m p ∈ M \ ( M ∪ . . . ∪ M k ) where α j is either zero or a unit in A for ≤ j ≤ p . UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 9
Proof.
Since M i + m M m M is a proper subspace of M m M for each i , we have that M + m M m M ∪ . . . ∪ M k + m M m M M m M . (cid:3)
Lemma 4.3.
Let a , . . . , a p ∈ R l be homogeneous elements of degree l > and J , . . . , J k homogeneous ideals of R such that ( a , . . . , a p ) R * J i for all ≤ i ≤ k . Then there exists anelement α a + . . . + . . . α p a p / ∈ J ∪ . . . ∪ J k where α j is either zero or a unit in R for ≤ j ≤ p .Proof. Let M be the R -submodule of R l generated by a , . . . , a p and M i = M ∩ J i for i =1 , . . . , k . Since ( a , . . . , a p ) R * J i , there exists a i j for each J i such that a i j / ∈ J i . So, M i is aproper submodule of M . Now the conclusion follows from Lemma 4.2. (cid:3) For a Noetherian graded R -module N , we now prove the existence of an N -basis of ahomogeneous ideal of R generated in degree g . Our result is a generalization of [16, Proposition1.9, Chap I]. Proposition 4.4.
Let J ⊆ R be an ideal generated by homogeneous elements of degree g . Let N , . . . , N k be Noetherian graded R -modules with dim N i / J N i = 0 for ≤ i ≤ k. Then thereexists a system of homogeneous elements of degree g forming N i -basis of J for all ≤ i ≤ k. Proof.
Let J = ( b , . . . , b t ) R with t = µ ( J M ) and deg b i = g for 1 ≤ i ≤ t. We may assumethat dim N i = s i > t ≥ s i for each i . Claim 1.
For each 0 ≤ m ≤ t , there exist homogeneous elements a , . . . , a m ∈ J such thateither a i = 0 or deg a i = g for 1 ≤ i ≤ m and the following conditions hold:(1) J = ( a , . . . , a m , b m +1 , . . . , b t ) R and(2) for all l = 1 , . . . , k , for all j = 0 , . . . , min( s l , m ) and all 1 ≤ i < . . . < i j ≤ m , a i , . . . , a i j is a part of a system of parameters of N l . Proof of Claim 1.
We prove by induction on m . For m = 0, this is trivial. Now let 0 < m ≤ t and there exist elements a , . . . , a m − as stated in Claim 1 such that (1) and (2) are satisfied.Define L = { p ∈ Spec ( R ) : p is homogeneous and there are l, j, i , . . . , i j , ≤ l ≤ k, ≤ j ≤ min( s l , m ) , ≤ i < . . . < i j < m with p ∈ Supp N l / ( a i , . . . , a i j ) N l ) and dim R/ p = s l − j } . Clearly, J * p for all p ∈ L and J * ( a , . . . , a m − ) R . By Lemma 4.3, there exists an element,say a m , such that a m = α a + . . . + α m − a m − + α m b m + . . . α t b t / ∈ ( a , . . . , a m − ) R ∪ ∪ p ∈ L p with α j is either zero or a unit in R for 1 ≤ j ≤ t . Then α j = 0 for some m ≤ j ≤ t . We mayassume that j = m after rearrangements if needed. Then b m ∈ ( a , . . . , a m , b m +1 , . . . , b t ) R ⇒ J = ( a , . . . , a m , b m +1 , . . . , b t ) R . In this case, deg a m = g and (2) holds for the choice of a m . (cid:3) Now, { a , . . . , a t } is the desired basis of J . (cid:3) Proposition 4.5.
Let N be a Noetherian graded R -module of dimension s > . Let J , . . . , J m be ideals of R generated by homogeneous elements of degrees g , . . . , g m respectively and dim N/ J i N = 0 for ≤ i ≤ m. Let B be an N -basis of J consisting of homogeneous el-ements of degree g . Then there exist N -bases B i of J i consisting of homogeneous elements of degree g i for ≤ i ≤ m such that any s elements of the set m S i =1 B i form a system of parametersof N .Proof. We apply induction on m . The case m = 1 is trivial. Suppose that the result holdsfor m −
1. Consider the family F of quotient modules N/J N where J is the ideal generatedby any k elements of the set m − S i =1 B i for 0 ≤ k ≤ s . Now by Proposition 4.4, there exists asystem of homogeneous elements of degree g m , say B m , which forms N/J N -basis of J m for all N/J N ∈ F . This completes the proof. (cid:3) We now use the above results for R ( I ) and G ( I ) to obtain a generating set of N with theproperties (2)-(4) described in next lemma. We regard the Rees algebra R ( I ) = ⊕ n ≥ I n as the A -subalgebra of the polynomial ring A [ t ]. Let R ≥ c = ⊕ n ≥ c I n , c ≥ R ( I ) and β be an integer such that I n +1 = I I n for all n ≥ β. Recall that ν := µ ( m ). Lemma 4.6.
There exist elements x , . . . , x ν ∈ m and a ij ∈ I i , ≤ j ≤ u i for some integers u i ≥ and for ≤ i ≤ β which satisfy the following conditions:(1) x , . . . , x ν , a t, . . . , a u t, . . . , a ij t i , . . . , a βu β t β is a minimal generating set of N ,(2) any d elements from the set { a ij t i : 1 ≤ j ≤ u i , ≤ i ≤ β } is a system of parametersof G ( I ) ,(3) x , . . . , x ν is a minimal generating set of m and(4) any d elements from { x , . . . , x ν , a ij : 1 ≤ j ≤ u i , ≤ i ≤ β } is a system of parametersof A .Proof. Since R ≥ β has basis in degree β , there exists a G ( I )-basis, say B β = { a β t β , . . . , a βw β t β } , of R ≥ β by Proposition 4.4. Let b i , . . . , b iv i be a minimal generatingset of I i for 1 ≤ i ≤ β −
1. Consider the ideal J i = ( b i t i , . . . , b iv i t i ) R ( I ) for 1 ≤ i ≤ β − J i ) n = I n t n for n ≫ G ( I ) / J i G ( I ) = 0 for 1 ≤ i ≤ β − G ( I )-bases, say B i = { a i t i , . . . , a iw i t i } of J i , consisting ofhomogeneous elements of degree i for all 1 ≤ i ≤ β − d elements from the set β S i =1 B i = { a ij t i | ≤ j ≤ w i , ≤ i ≤ β } is a system of parameters of G ( I ). Clearly, the set β S i =1 B i generates R + . We choose B ′ ⊆ t S i =1 B i a minimal generating set of R + . Let B ′ = { a t, . . . , a u t, . . . , a ij t i , . . . , a βu β t β } . Then (2) holds. Further, any d elements from { a ij ; 1 ≤ j ≤ u i , ≤ i ≤ β } is a systemof parameters of A . Let F be the family of the quotient modules A/J where J is an idealgenerated by any k elements of the set { a ij : 1 ≤ j ≤ u i , ≤ i ≤ β } for 0 ≤ k ≤ d . By [16,Proposition 1.9, Chap I], there exists an A/J -basis x , . . . , x ν of m for each A/J ∈ F . Thus weobtain (1)-(4). (cid:3) Buchsbaumness of G ( I )In this section, we discuss the proof of Theorem 1.2. Suppose A is a Buchsbaum local ringand Q = ( a , . . . , a d ) is a reduction of I with I n +1 = QI n for n ≥ r . Then (1) implies (2) isobvious and the equivalence of (2) and (3) follows from Proposition 3.1 since a n , . . . , a n d d is anunconditioned d-sequence for all n i ≥
1, see Remark 2.2. The aim of this section is to proveTheorem 5.1 stated below which, along with Proposition 3.3, completes the proof of (2) ⇒ (1). UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 11
An important ingredient of our proof is Lemma 4.6 which provides a desired generating set of N for defining Koszul complex on N . We use [16, Theorem 2.15, Chap I] for our proof whichprovides a sufficient condition for the Buchsbaumness of G ( I ) using canonical maps betweenthe Koszul (co)homology modules and the local cohomology modules of G ( I ).We first fix some notations. Let K · ( N ; G ( I )) denote the Koszul (co)complex on a minimalgenerating set of the ideal N . Let x , . . . , x ν ∈ m and a ij t i ∈ R ( I ) i for 1 ≤ j ≤ u i , 1 ≤ i ≤ β be a minimal generating set of N such that all the conditions in Lemma 4.6 are satisfied. Then K · ( N ; G ( I )) is the Koszul complex on the above system of elements up to isomorphism andwe have K · ( N ; G ( I )) = M Γ ⊆ [1 ,v ] , Λ j ⊆ [1 ,u j ] , ≤ j ≤ β G ( I ) e Λ ... Λ β Γ with K i ( N ; G ( I )) = M | Γ | + | Λ | + ... + | Λ β | = i, Γ ⊆ [1 ,v ] , Λ j ⊆ [1 ,u j ] , ≤ j ≤ β G ( I ) e Λ Λ ... Λ β Γ where { e Λ Λ ... Λ β Γ | Γ ⊆ [1 , v ] and Λ j ⊆ [1 , u j ] for 1 ≤ j ≤ β } is the graded free basis withdeg e Λ Λ ... Λ β Γ = − ( | Λ | + 2 | Λ | + . . . + β | Λ β | ) . We write H i (cid:0) N ; G ( I ) (cid:1) for the homology modulesH i (cid:0) K · ( N ; G ( I )) (cid:1) . The notations x ( n ) and a i t i ( n ) are used for the sequences x n , . . . , x nν and( a i t i ) n , . . . , ( a iu i t i ) n respectively for 1 ≤ i ≤ β . Theorem 5.1.
Suppose that A is a Buchsbaum local ring and the equality I ( G ( I )) = I ( A ) holds. Then the associated graded ring G ( I ) is Buchsbaum over R ( I ) .Proof. Suppose A is Buchsbaum and I ( G ( I )) = I ( A ). Then G ( I ) is quasi-Buchsbaum byTheorem 3.2. We prove that G ( I ) is Buchsbaum by induction on d . We may assume that d ≥
2. By [16, Theorem 2.15, Chap I], it is enough to prove that the canonical map φ iG ( I ) : H i (cid:0) N ; G ( I ) (cid:1) −→ lim −→ H i (cid:0) x ( n ) , a i t i ( n ); G ( I ) (cid:1) ∼ = H i N ( G ( I ))is surjective for all 0 ≤ i < d .Since any system of parameters is a d-sequence in A , using Proposition 2.3 and Proposition3.1, we get that G ( I A ′ ) ∼ = G ( I ) /f G ( I ) (7)where A ′ = A/ ( a ) , f = a t and I A ′ denote the filtration { I n + ( a ) / ( a ) } n ≥ of A ′ . Since A and G ( I ) are quasi-Buchsbaum, m H i m ( A ) = 0 and N H i N ( G ( I )) = 0 for 0 ≤ i < d . Thenit is easy to see that the equality I ( A ′ ) = I ( G ( I ) /f G ( I )) = I ( G ( I A ′ )) holds, see [18, Lemma1.7]. Therefore, by induction hypothesis, G ( I A ′ ) is Buchsbaum.Now suppose depth A >
0. Then a is a non-zero-divisor. In view of (7), f is a non-zero-divisor on G ( I ). The exact sequence0 −→ G ( I )( − f −→ G ( I ) → G ( I A ′ ) −→ R ( I )-modules gives the following commutative diagram H i − ( N ; G ( I A ′ )) H i ( N ; G ( I ))( −
1) 0H i − N ( G ( I A ′ )) H i N ( G ( I ))( −
1) 0 φ i − G ( I A ′ ) fφ iG ( I )( − f where the rows are exact and φ i − G ( I A ′ ) is surjective for 0 ≤ i − < d −
1. This implies thesurjectivity of φ iG ( I ) for 0 < i < d . Thus G ( I ) is a Buchsbaum module over R ( I ).Now suppose depth A = 0 . Then, by Proposition 3.3, we have that H N ( G ( I )) = 0, I ( A/U ) = I ( G ( I A/U )) and the following is a short exact sequence0 −→ H N ( G ( I )) −→ G ( I ) −→ G ( I A/U ) −→ of graded R ( I )-modules. Since A/U is Buchsbaum and depth
A/U > G ( I A/U ) is Buchs-baum by the previous case. So, φ iG ( I A/U ) is surjective for 0 ≤ i < d . In order to prove thesurjectivity of φ iG ( I ) , it is enough to show that the canonical maps τ iA : H i ( N ; H N ( G ( I ))) −→ H i ( N ; G ( I ))are injective for 0 ≤ i ≤ d , see proof of [16, Theorem 2.15, Chap I]. First, we discuss the case0 ≤ i ≤ d −
1. Consider the following commutative diagram with canonical maps:H N ( G ( I )) G ( I )H N ( G ( I A ′ )) G ( I A ′ ) τ A σ τ A ′ It follows from Remark 2.2 and Proposition 3.1 that a t, . . . , a d t is a d-sequence on G ( I )and I ( G ( I )) = I (( a t, . . . , a d t ); G ( I )) . Therefore H N ( G ( I )) ∩ f G ( I ) = 0 which implies that τ A ′ ◦ σ is injective. Since H N ( G ( I )) and H N ( G ( I A ′ )) are vector spaces over R ( I ) / N , thevertical map σ splits. Therefore, σ i splits in the following commutative diagram: H i ( N ; H N ( G ( I ))) H i ( N ; G ( I )) H i ( N ; H N ( G ( I A ′ ))) H i ( N ; G ( I A ′ )) τ iA σ i τ iA ′ By induction hypothesis, the map τ iA ′ is injective for 0 ≤ i ≤ d −
1. Hence τ iA is injective for0 ≤ i ≤ d −
1. Now we prove the injectivity of τ dA : H d ( N ; H N ( G ( I ))) −→ H d ( N ; G ( I )) . By (8), we have the following exact sequence of graded Koszul complexes0 → K · ( N ; H N ( G ( I ))) → K · ( N ; G ( I )) → K · ( N ; G ( I A/U )) → . (9)Since N H N ( G ( I )) = 0, all the differentials of the Koszul complex K · ( N ; H N ( G ( I ))) are zeromaps. We get the following commutative diagram:0 K d − ( N ; H N ( G ( I ))) K d − ( N ; G ( I ))0 K d ( N ; H N ( G ( I ))) K d ( N ; G ( I ))) ∂ Let ξ ∈ K d ( N ; H N ( G ( I ))) be a homogeneous element with deg ξ = n such that there existsa homogeneous element η ∈ K d − ( N ; G ( I )) with ξ = ∂ ( η ) ∈ K d ( N ; G ( I ))). We show that ξ = 0. Then it follows that τ dA is injective. Write ξ and η as follows: ξ = X | Γ | + | Λ | + ... + | Λ β | = d, Γ ⊆ [1 ,ν ] , Λ j ⊆ [1 ,u j ] , ≤ j ≤ β ξ Λ ... Λ β Γ e Λ ... Λ β Γ and η = X | Q | + | P | + ... + | P β | = d − ,Q ⊆ [1 ,ν ] ,P j ⊆ [1 ,u j ] , ≤ j ≤ β η P ...P β Q e P ...P β Q where ξ Λ ... Λ β Γ ∈ [ G ( I )] n + | Λ | +2 | Λ | + ... + β | Λ β | and η P ...P β Q ∈ [ G ( I )] n + | P | +2 | P | + ... + β | P β | . Since ξ = ∂ ( η ), we get ξ Λ ... Λ β Γ = X j ∈ Γ ( − Γ( j ) x j η Λ ... Λ β Γ \ j + β X k =1 X i ∈ Λ k ( − | Γ | + | Λ | + ... + | Λ k − | +Λ k ( i ) a ki t k η Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ (10) UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 13 where Γ( j ) := |{ j ′ ∈ Γ | j ′ < j }| and Λ k ( i ) := |{ i ′ ∈ Λ k | i ′ < i }| . Note that ξ Λ ... Λ β Γ ∈ H N ( G ( I )).We show that ξ Λ ... Λ β Γ = 0 for all Γ , Λ , . . . , Λ β with | Γ | + | Λ | + . . . + | Λ β | = d .Let c P ,...,P β Q ∈ I n + | P | +2 | P | + ... + β | P β | be a representative of η P ,...,P β Q , i.e., c P ,...,P β Q = η P ,...,P β Q in [ G ( I )] n + | P | +2 | P | + ... + β | P β | . Then the element b Λ ... Λ β Γ := X j ∈ Γ ( − Γ( j ) x j c Λ ... Λ β Γ \{ j } + β X k =1 X i ∈ Λ k ( − | Γ | + | Λ | + ... + | Λ k − | +Λ k ( i ) a ki c Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ (11) ∈ I n + | Λ | +2 | Λ | + ... + β | Λ β | and b Λ ... Λ β Γ = ξ Λ ... Λ β Γ in [ G ( I )] n + | Λ | +2 | Λ | + ... + β | Λ β | . This representation b Λ ... Λ β Γ of ξ Λ ... Λ β Γ depends on the choice of representations c P ,...,P β Q of η P ...P β Q in (10). Using the following lemma,we will conclude that b Λ ... Λ β Γ = 0 after a suitable change of representations c P ...P β Q . Thus ξ Λ ... Λ β Γ = 0. Lemma 5.2.
Let Γ ⊆ [1 , v ] , Λ j ⊆ [1 , u j ] , ≤ j ≤ β with | Γ | + | Λ | + . . . + | Λ β | = d and b Λ ... Λ β Γ ∈ I n + | Λ | +2 | Λ | + ... + β | Λ β | be same as described in (11) such that b Λ ... Λ β Γ = ξ Λ ... Λ β Γ .Suppose that there exists a subset Λ ′ of [1 , u ] such that the following conditions are satisfied:(1) Λ ⊆ Λ ′ and | Λ ′ | = | Λ | + 1 , (2) b Λ ′ Λ ... Λ k Γ ′ = 0 for all Γ ′ ⊆ Γ such that | Γ ′ | = | Γ | − and(3) b Λ ′ Λ ... Λ k − Λ ′ k Λ k +1 Λ β Γ = 0 for all Λ ′ k ⊆ Λ k such that | Λ ′ k | = | Λ k | − for ≤ k ≤ β .Then, b Λ Λ ... Λ β Γ = 0 after a suitable change of representations c Λ \{ i } Λ ... Λ β Γ for i ∈ Λ in (11) .Proof of Lemma. First consider the case when | Λ | = d . Then Γ = Λ k = φ for 2 ≤ k ≤ β and ξ Λ ... Λ β Γ = X i ∈ Λ ( − Λ ( i ) a i tη Λ \{ i } Λ ... Λ β Γ ∈ ( a i t : i ∈ Λ ) G ( I ) ∩ H N ( G ( I )) = 0since { a i t : i ∈ Λ } is a d-sequence on G ( I ) by Proposition 3.1. Thus b Λ ... Λ β Γ = X i ∈ Λ ( − Λ ( i ) a i c Λ \{ i } Λ ... Λ β Γ ∈ ( a i : i ∈ Λ ) ∩ I n + | Λ | +1 = ( a i : i ∈ Λ ) I n + | Λ | where the last equality follows from Proposition 2.1 and the fact that I ( A ) = I ( G ( I )) ≥ I (( a t, . . . , a d t ); G ( I )) ≥ I (( a , . . . , a d ); A ) = I ( A ) . Let g i ∈ I n + | Λ | such that b Λ ,..., Λ β Γ = P i ∈ Λ a i g i . Then, for each i ∈ Λ , η Λ \{ i } Λ ... Λ β Γ = c Λ \{ i } Λ ... Λ β Γ = c Λ \{ i } Λ ... Λ β Γ − g i in G ( I ) n + | Λ |− . Therefore the assertion follows after replacing c Λ \{ i } Λ ... Λ β Γ by c Λ \{ i } Λ ... Λ β Γ − g i , for each i ∈ Λ , for the representative of η Λ \{ i } Λ ... Λ β Γ in the expression (11) of b Λ ,..., Λ β Γ .Now, suppose | Λ | < d . Then Γ = φ or Λ k = φ for some 2 ≤ k ≤ β . It is enough to considerthe case when Γ = φ . Similar arguments will hold when Λ k = φ for some k . For an integer k ,we write S ( k ) for the sum | Λ | + . . . + | Λ k | . Now, for each j ∈ Γ , we have that0 = b Λ ′ Λ ... Λ β Γ \{ j } = X j ′ ∈ Γ \{ j } ( − Γ \{ j } ( j ′ ) x j ′ c Λ ′ Λ ... Λ β Γ \{ j,j ′ } + X i ∈ Λ ′ ( − | Γ |− ′ ( i ) a i c Λ ′ \{ i } Λ ... Λ β Γ \{ j } + β X k =2 X i ∈ Λ k ( − | Γ |− S ( k − k ( i ) a ki c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ \{ j } where the notation Γ \ { j } ( j ′ ) is used for the cardinality |{ j ′′ ∈ Γ \ { j }| j ′′ < j ′ }| = |{ j ′′ ∈ Γ | j = j ′′ < j ′ }| . Multiplying by ( − Γ( j ) x j and taking the sum P j ∈ Γ , we get that0 = X j ∈ Γ X j ′ ∈ Γ \{ j } ( − Γ( j ) ( − Γ \{ j } ( j ′ ) x j x j ′ c Λ ′ Λ ... Λ β Γ \{ j,j ′ } + X i ∈ Λ ′ ( − | Γ |− ′ ( i ) a i X j ∈ Γ ( − Γ( j ) x j c Λ ′ \{ i } Λ ... Λ β Γ \{ j } + β X k =2 (cid:16) X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki X j ∈ Γ ( − Γ( j ) x j c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ \{ j } (cid:17) = X i ∈ Λ ′ ( − | Γ |− ′ ( i ) a i X j ∈ Γ ( − Γ( j ) x j c Λ ′ \{ i } Λ ... Λ β Γ \{ j } + β X k =2 (cid:16) X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki X j ∈ Γ ( − Γ( j ) x j c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ \{ j } (cid:17) (12)as the first term P j ∈ Γ P j ′ ∈ Γ \{ j } ( − Γ( j ) ( − Γ \{ j } ( j ′ ) x j x j ′ c Λ ′ Λ ... Λ β Γ \{ j,j ′ } = 0. Further, for each i ∈ Λ k ,2 ≤ k ≤ β , we have that0 = b Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ = X j ∈ Γ ( − Γ( j ) x j c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ \{ j } + X i ′ ∈ Λ ′ ( − | Γ | +Λ ′ ( i ′ ) a i ′ c Λ ′ \{ i ′ } Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ + k − X k ′ =2 X i ′ ∈ Λ k ′ ( − | Γ | + S ( k ′ − k ′ ( i ′ ) a k ′ i ′ c Λ ′ Λ ... Λ k ′− Λ k ′ \{ i ′ } Λ k ′ +1 ... Λ k \{ i } Λ k +1 ... Λ β Γ + X i ′ ∈ Λ k \{ i } ( − | Γ | + S ( k − k \{ i } ( i ′ ) a ki ′ c Λ ′ Λ ... Λ k − Λ k \{ i,i ′ } Λ k +1 ... Λ β Γ + β X k ′ = k +1 X i ′ ∈ Λ k ′ ( − | Γ | + S ( k ′ − k ′ ( i ′ ) a k ′ i ′ c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ k ′− Λ k ′ \{ i ′ } Λ k ′ +1 ... Λ β Γ . (13)Again, we multiply the last three terms by P i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki and take the sum P βk =2 to obtain the following expression which is zero. Claim 2.
We have that β X k =2 (cid:16) X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki (cid:16) ( − | Γ | X i ′ ∈ Λ k \{ i } ( − S ( k − k \{ i } ( i ′ ) a ki ′ c Λ ′ Λ ... Λ k − Λ k \{ i,i ′ } Λ k +1 ... Λ β Γ + k − X k ′ =2 X i ′ ∈ Λ k ′ ( − S ( k ′ − k ′ ( i ′ ) a k ′ i ′ c Λ ′ Λ ... Λ k ′− Λ k ′ \{ i ′ } Λ k ′ +1 ... Λ k \{ i } Λ k +1 ... Λ β ΓUCHSBAUMNESS OF THE ASSOCIATED GRADED RING 15 + β X k ′ = k +1 X i ′ ∈ Λ k ′ ( − S ( k ′ − k ′ ( i ′ ) a k ′ i ′ c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ k ′− Λ k ′ \{ i ′ } Λ k ′ +1 ... Λ β Γ (cid:17)(cid:17) = 0 . Proof of Claim 2.
First note that, X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki X i ′ ∈ Λ k \{ i } ( − | Γ | + S ( k − k \{ i } ( i ′ ) a ki ′ c Λ ′ Λ ... Λ k − Λ k \{ i,i ′ } Λ k +1 ... Λ β Γ = X i ∈ Λ k X i ′ ∈ Λ k \{ i } ( − Λ k ( i )+1+Λ k \{ i } ( i ′ ) a ki a ki ′ c Λ ′ Λ ... Λ k − Λ k \{ i,i ′ } Λ k +1 ... Λ β Γ = 0for each 2 ≤ k ≤ β . Further, let 2 ≤ l < m ≤ β and i ∈ Λ l , j ∈ Λ m . Then the coefficients of a li a mj in the expression of Claim 2 is(( − | Γ | + S ( l − l ( i )+ S ( m − m ( j ) + ( − | Γ | + S ( m − m ( j )+ S ( l − l ( i ) ) c = 0where c = c Λ ′ Λ ... Λ l − Λ l \{ i } Λ l +1 ... Λ m − Λ m \{ j } Λ m +1 ... Λ β Γ . This completes the proof of Claim 2. (cid:3) Suppose i ∈ Λ ′ such that Λ = Λ ′ \ { i } . Putting the value of P j ∈ Γ ( − Γ( j ) x j c Λ ′ Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ \{ j } from (13) into (12) for each 2 ≤ k ≤ β and usingClaim 2, we get that( − | Γ |− ′ ( i ) a i (cid:16) X j ∈ Γ ( − Γ( j ) x j c Λ ... Λ β Γ \{ j } + β X k =2 X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki c Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ (cid:17) = β X k =2 (cid:16) X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki X i ′ ∈ Λ ′ ,i ′ = i ( − | Γ | +Λ ′ ( i ′ ) a i ′ c Λ ′ \{ i ′ } Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ (cid:17) − X i ∈ Λ ′ ,i = i ( − | Γ |− ′ ( i ) a i X j ∈ Γ ( − Γ( j ) x j c Λ ′ \{ i } Λ ... Λ β Γ \{ j } ∈ ( a i : i ∈ Λ ) A. (14)Since A is Buchsbaum, ( a i : i ∈ Λ ) : a i = ( a i : i ∈ Λ ) : m ⊆ ( a i : i ∈ Λ ) : x j ′ for j ′ ∈ Γ.We rewrite the expression in (11) and use (14) to get b Λ ..., Λ β Γ − X i ∈ Λ ( − | Γ | +Λ ( i ) a i c Λ \{ i } Λ ... Λ β Γ = X j ∈ Γ ( − Γ( j ) x j c Λ Λ ... Λ β Γ \{ j } + β X k =2 X i ∈ Λ k ( − | Γ | + S ( k − k ( i ) a ki c Λ ... Λ k − Λ k \{ i } Λ k +1 ... Λ β Γ ∈ (( a i : i ∈ Λ ) : a i ) ∩ ( x j , a ki | j ∈ Γ , i ∈ Λ k , ≤ k ≤ β ) ⊆ (( a i : i ∈ Λ ) : x j ′ ) ∩ ( x j , a ki | j ∈ Γ , i ∈ Λ k , ≤ k ≤ β )= ( a i : i ∈ Λ )where the equality holds since { x j , a ki : j ∈ Γ , i ∈ Λ k , ≤ k ≤ β } is a d-sequence in A , see [9,Proposition 2.1]. Now, b Λ ..., Λ β Γ − X i ∈ Λ ( − | Γ | +Λ ( i ) a i c Λ \{ i } Λ ... Λ β Γ ∈ ( a i : i ∈ Λ ) ∩ I n + | Λ | +2 | Λ | + ... + β | Λ β | = ( a i : i ∈ Λ ) I n + | Λ | +2 | Λ | + ... + β | Λ β |−
16 KUMARI SALONI using Proposition 2.1. This implies that b Λ ,..., Λ β Γ = ξ Λ ,..., Λ β Γ ∈ ( a i t : i ∈ Λ ) G ( I ) ∩ H N ( G ( I )) = 0 since { a i t : i ∈ Λ } is a d-sequence on G ( I ) by Proposition 3.1. Hence b Λ ,..., Λ β Γ ∈ ( a i : i ∈ Λ ) ∩ I n + | Λ | +2 | Λ | + ... + β | Λ β | +1 = ( a i : i ∈ Λ ) I n + | Λ | +2 | Λ | + ... + β | Λ β | using Proposition 2.1 again. Let g i ∈ I n + | Λ | +2 | Λ | + ... + β | Λ β | such that b Λ ,..., Λ β Γ = P i ∈ Λ a i g i . Then, for each i ∈ Λ , η Λ \{ i } Λ ... Λ β Γ = c Λ \{ i } Λ ... Λ β Γ = c Λ \{ i } Λ ... Λ β Γ − g i in G ( I ) n + | Λ | +2 | Λ | + ... + β | Λ β |− . Therefore, for each i ∈ Λ , we may replace c Λ \{ i } Λ ... Λ β Γ by c Λ \{ i } Λ ... Λ β Γ − g i for the representative of η Λ \{ i } Λ ... Λ β Γ in the expression (11) of b Λ ,..., Λ β Γ whichcompletes the proof. (cid:3) (Proof of Theorem 5.1 continued.) Let us fix Γ ⊆ [1 , ν ], Λ ⊆ [1 , u ] , . . . , Λ β ⊆ [1 , u β ]such that | Γ | + | Λ | + . . . + | Λ β | = d. Let | Λ | = p . We may assume that Λ = [1 , p ] afterrearrangement of { a , . . . , a u } . Claim 3. b [1 ,l ]Λ ′ ,... Λ ′ β Γ ′ = 0 for all Γ ′ ⊆ Γ, Λ ′ k ⊆ Λ k , ≤ k ≤ β with | Γ ′ | + | Λ ′ | + . . . + | Λ ′ β | = d − l and for all p ≤ l ≤ d after a suitable change of representations c [1 ,l ] \{ i } Λ ′ ... Λ ′ β Γ ′ for i ∈ [1 , l ] . Proof of Claim 3.
We use induction on p . If p = d , then l = d and Γ ′ = Λ ′ = . . . = Λ ′ β = φ .By Lemma 5.2, we have a choice of representatives c [1 ,d ] \{ i } Λ ′ ... Λ ′ β Γ ′ in the expression (11) suchthat b [1 ,d ]Λ ′ ,... Λ ′ β Γ ′ = 0. Suppose 0 ≤ p < d and the assertion holds for p + 1. Then it is enough toconsider the case when l = p . Let Γ ′ ⊆ Γ, Λ ′ k ⊆ Λ k , ≤ k ≤ β with | Γ ′ | + | Λ ′ | + . . . + | Λ ′ β | = d − l be fixed. Put Λ ′ = [1 , p + 1], then by induction hypothesis we have that b Λ ′ Λ ′ ... Λ ′ β Γ ′′ = 0 whenever Γ ′′ ⊆ Γ ′ with | Γ ′′ | = | Γ ′ | − b Λ ′ Λ ′ ... Λ ′ k − Λ ′′ k Λ ′ k +1 ... Λ ′ β Γ ′ = 0 whenever Λ ′′ k ⊆ Λ ′ k with | Λ ′′ k | = | Λ ′ k | − ≤ k ≤ β up to change of representations. Then, by Lemma 5.2, b [1 ,l ]Λ ′ ,... Λ ′ β Γ ′ = 0 after suitable change ofrepresentations c [1 ,l ] \{ i } Λ ′ ... Λ ′ β Γ ′ for i ∈ [1 , l ] . This complete the proof of Claim 3. (cid:3)
Now, put l = p , Γ ′ = Γ and Λ ′ k = Λ k for 2 ≤ k ≤ β in Claim 3 to complete the proof ofTheorem 5.1. (cid:3) Application.
In a Noetherian local ring A , the Hilbert coefficients of an m -primary ideal I satisfies the following relation, see [14, Theorem 2.4], e ( I ) − e ( Q ) ≥ e ( I ) − ℓ A ( A/I )) − ℓ A ( I/I + Q ) (15)where Q ⊆ I is a minimal reduction of I . If A is Cohen-Macaulay, then the above inequalityfollows from the famous bounds of Huckaba-Marley [8]. Corso in [1] conjectured that if A isBuchsbaum and equality holds in (15) for I = m , then G ( m ) is Buchsbaum. Corso’s conjectureholds more generally for m -primary ideals, which is proved in [11] and [12]. An alternativeproof can be given using Theorem 1.2. In fact, one can show that if A is Buchsbaum andequality holds in (15) then I ( G ( I )) = I ( A ), see [12, Theorem 1.1]. Then, by Theorem 1.2,we conclude that G ( I ) is Buchsbaum. Further discussions concerning Corso’s conjecture issubject of a subsequent paper. UCHSBAUMNESS OF THE ASSOCIATED GRADED RING 17
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