The equality of Elias and Valla and Buchsbaumness of ...

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THE EQUALITY OF ELIAS AND VALLA AND BUCHSBAUMNESS OF ASSOCIATED GRADED RINGS KAZUHO OZEKI

1. Introduction and the statement of main results The purpose of this paper is to study the Buchsbaumness of the associated graded ring of ideals in a Buchsbaum local ring satisfying the equality of Elias and Valla [3], without assuming that the base local ring is Cohen-Macaulay. Throughout this paper let A denote a Noetherian local ring with the maximal ideal m and d = dim A > 0. For simplicity, we assume the residue class ﬁeld A/m is inﬁnite. Let I be an m-primary ideal in A and suppose that I contains a parameter ideal Q = (a1 , a2 , · · · , ad ) of A as a reduction, that is Q ⊆ I and the equality I n+1 = QI n holds true for some (and hence any) integer n ≫ 0. Let ℓA (M ) denote, for an A-module M , the length of M . Then we have integers {ei (I)}0≤i≤d such that the equality ) ) ( ( n+d−1 n+d n+1 + · · · + (−1)d ed (I) − e1 (I) ℓA (A/I ) = e0 (I) d−1 d holds true for all integers n ≫ 0, which we call the Hilbert coeﬃcients of A with respect to I. Let R = R(I) := A[It] and T = R(Q) := A[Qt] ⊆ A[t] denote, respectively, the Rees algebras of I and Q. Let F = T /IT , R′ = R′ (I) := A[It, t−1 ] ⊆ A[t, t−1 ]

and G = G(I) := R′ /t−1 R′ ∼ =

⊕

I n /I n+1 .

n≥0

In the case where A is a Cohen-Macaulay local ring, we have the inequality 2e0 (I) − e1 (I) ≤ 2ℓA (A/I) + ℓA (I/I 2 + Q) which is given in [3] and [7], and they showed that the equality 2e0 (I) − e1 (I) = 2ℓA (A/I) + ℓA (I/I 2 + Q) holds true if and only if I 3 = QI 2 and Q ∩ I 2 = QI. When this is the case, the associated graded ring G of I is Cohen-Macaulay. Thus the ideal I with 2e0 (I) − e1 (I) = 2ℓA (A/I) + ℓA (I/I 2 + Q) enjoys nice properties. Key words and phrases: Buchsbaum local ring, associated graded ring, Hilbert function, Hilbert coeﬃcient 2010 Mathematics Subject Classiﬁcation: 13D40, 13A30, 13H10, 13H15.

The purpose of this paper is to extend their results without assuming that A is a Cohen-Macaulay ring. In an arbitrary Noetherian local ring A the inequality 2e0 (I) − e1 (I) + e1 (Q) ≤ 2ℓA (A/I) + ℓA (I/I 2 + Q) holds true ([8, Theorem 2.4], [1, Theorem 3.1]). It seems natural to ask, what happens on the ideals I satisfying the equality 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q). To sate the results of the present paper, let us consider the following three conditions: (C1 ) The sequence a1 , a2 , · · · , ad is a d+ -sequence in A, that is for all integers n1 , n2 , · · · , nd ≥ 1 the sequence an1 1 , an2 2 , · · · , and d forms a d-sequence in any order. (C2 ) (a1 , a2 , · · · , aˇi , · · · , ad ) :A ai ⊆ I for all 1 ≤ i ≤ d. (C3 ) depth A > 0. These conditions (C1 ), (C2 ), and (C3 ) are naturally satisﬁed, when A is a CohenMacaulay local ring. Condition (C1 ) (resp. condition (C2 )) is always satisﬁed, if A is a Buchsbaum local ring (resp. I = m). Here we notice that condition (C1 ) is equivalent to saying that our local ring A is a generalized Cohen-Macaulay ring, that is all the local cohomology modules Him (A) (i ̸= d) of A with respect to the maximal ideal m are ﬁnitely generated and the parameter ideal Q is standard, that is the equality ) d−1 ( ∑ d−1 ℓA (A/Q) − e0 (Q) = ·ℓA (Him (A)) i i=0 holds true. Hence condition (C1 ) is independent of the choice of a minimal system {ai }1≤i≤d of generators of the parameter ideal Q. We note here that condition (C2 ) is also independent of the choice of a minimal system {ai }1≤i≤d of generators of Q. Let us now state our own result. The main result of this paper is the following Theorem 1.1, which generalizes the result of [3] and [7] given in the case where A is a Cohen-Macaulay local ring, because ei (Q) = 0 for all 1 ≤ i ≤ d. We notice that, thanks to condition (C1 ), the Hilbert coeﬃcients ei (Q) of Q are given by the formula  if i = 0,  e0 (Q) 0 i (A)) if i = d, ℓ (H (−1) ei (Q) = A m  ∑d−i (d−i−1) j if 1 ≤ i ≤ d − 1 ℓA (Hm (A)) j=1 j−1 (n+d−i) ∑d i for all n ≥ 0 ([9, and one has the equality ℓA (A/Qn+1 ) = i=0 (−1) ei (Q) d−i Korollar 3.2]), so that {ei (Q)}1≤i≤d are independent of the choice of the reduction Q of I and so, are invariants of A. Here W = H0m (A) denotes the 0-th local cohomology modules of A with respect to m and HiM (G) the i-th local cohomology modules of G with respect to the graded maximal ideal M = mT + T+ of T .

Theorem 1.1. Suppose that conditions (C1 ) and (C2 ) are satisfied. Then the following two conditions are equivalent to each other. (1) 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q). (2) I 3 ⊆ QI 2 + W , (Q + W ) ∩ (I 2 + W ) = QI + W , and (a1 , a2 , · · · , aˇi , · · · , ad ) :A ai ⊆ I 2 + Q for all 1 ≤ i ≤ d. When this is the case, we have I 2 ⊇ W and the following assertions also hold true. (i) For all n ∈ Z,

 if n = 2,  W/I 3 ∩ W 0 n n+1 ∼ I ∩ W/I ∩W if n ≥ 3, [HM (G)]n =  (0) otherwise. Hence [H0M (G)]2 ∼ = I 3 ∩ W , and [H0M (G)]n = (0) for all = W/I 3 ∩ W , [H0M (G)]3 ∼ n ̸= 2, 3 if A is a Buchsbaum local ring.

(ii)

HiM (G) = [HiM (G)]2−i ∼ = Him (A)

for all 1 ≤ i ≤ d − 1, (iii) the a-invariant a(G) = max{n ∈ Z | [HdM (G)]n ̸= (0)} of G is at most 2 − d, (iv) e2 (I) = e1 (Q) + e2 (Q) − e0 (I) + e1 (I) + ℓA (A/I), (v) ei (I) = ei−2 (Q) + 2ei−1 (Q) + ei (Q) for all 3 ≤ i ≤ d, and (vi) the associated graded ring G is a Buchsbaum ring with the Buchsbaum invariant I(G) = I(A) if A is a Buchsbaum local ring. The key of the proof of Theorem 1.1 is the use of the Sally module S of I with respect to Q. We are now in a position to brieﬂy explain how we organized this paper. In Section 2 we will summarize some auxiliary results on Sally modules for the later use in this paper. We will give in Section 3 an outline of a proof of the implication (1) ⇒ (2) and the last assertions of Theorem 1.1. In Section 3 we will also introduce some techniques of Sally modules which is the key for the proof of Theorem 1.1. In Section 4 we will give one example of an m-primary ideal I with 2e0 (I)−e1 (I)+e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q) in a Buchsbaum local ring (A, m). In what follows, unless otherwise speciﬁed, let (A, m) be a Noetherian local ring with maximal ideal m and d = dim A > 0. Assume that the residue class ﬁeld A/m of A is inﬁnite. Let I be an m-primary ideal in A and put Q = (a1 , a2 , . . . , ad ) be a parameter ideal of A which forms a reduction of I. We put R = A[It], T = A[Qt], R′ = A[It, t−1 ], G = R′ /t−1 R′ , and F = T /IT.

We denote by Him (∗) (i ∈ Z) the i-th local cohomology functor of A with respect ∑ (d−1) to m. Let I(A) = d−1 ℓA (Him (A)) denotes the Buchsbaum invariant of A. Let i=0 i M = mT +T+ be the unique graded maximal ideal in T . We denote by HiM (∗) (i ∈ Z) the ∑ (d−1) i-th local cohomology functor of T with respect to M and I(G) = d−1 ℓG (HiM (G)) i=0 i the Buchsbaum invariant of G. Let L be a graded T -module. We denote by L(α), for each α ∈ Z, the graded T -module whose grading is given by [L(α)]n = [L]α+n for all n ∈ Z. Let µA (I) denotes the number of a minimal system of generators of I. 2. The structure of Sally modules In our proof of Theorem 1.1 we need some structure theorems of Sally modules. Following Vasconcelos [11], we deﬁne ⊕ S = SQ (I) := IR/IT ∼ I n+1 /Qn I = n≥1

and call it the Sally module of I with respect to Q. The purpose of this section is to summarize some auxiliary results on Sally modules, which we need throughout this paper. See [5, 6, 11] for the detailed proofs. Remark 2.1 (cf. [5, 6, 11]). We notice that S is a ﬁnitely generated graded T -module and mℓ ·S = (0) for some integer ℓ ≫ 0, since R is a module ﬁnite extension of the √ graded ring T and m = Q, so that dimT S ≤ d. Lemma 2.2. Suppose that conditions (C1 ) and (C2 ) are satisfied. Then F = T /IT ∼ = (A/I)[X1 , X2 , · · · , Xd ] as graded A-algebras, where (A/I)[X1 , X2 , · · · , Xd ] denotes the polynomial ring with d indeterminate over the Artinian local ring A/I. Hence F is a Cohen-Macaulay ring with dim F = d. Proof. See [6, Proposition 2.2].

Let us note the following lemma. Lemma 2.3. Suppose that conditions (C1 ), (C2 ), and (C3 ) are satisfied. Then AssT S ⊆ {mT }, whence dimT S = d if S ̸= (0). Proof. See [6, Lemma 2.3]. Proposition 2.4. Suppose that conditions (C1 ) and (C2 ) are satisfied. Then ( ) ( ) n+d n+d−1 n+1 ℓA (A/I ) = e0 (I) − {e0 (I) + e1 (Q) − ℓA (A/I)} d d−1 ( ) d ∑ n+d−i + (−1)i {ei−1 (Q) + ei (Q)} − ℓA (Sn ) d−i i=2

for all n ≥ 0.

Proof. See [6, Proposition 2.4]. Put s = dimT S ≤ d. Then we write ( ) ( ) n+s−1 n+s−2 ℓA (Sn ) = e0 (S) − e1 (S) + · · · + (−1)s−1 es−1 (S) s−1 s−2

for all n ≫ 0 with integers {ei (S)}0≤i≤s−1 . Then by Proposition 2.4 we get the following result, which is also given in [8, Proposition 6.2]. Corollary 2.5. Suppose that conditions (C1 ) and (C2 ) are satisfied. Then we have the following. (1) e1 (I) = e0 (I) + e1 (Q) − ℓA (A/I) + e0 (S) and (2) ei (I) = ei−1 (Q) + ei (Q) + ei−1 (S) for all 2 ≤ i ≤ d. 3. Proof of our main theorem In this section let us introduce some techniques, being inspired by [1, 2], which plays a crucial role throughout this paper. Let us begin with the following. Lemma 3.1. Assume that I ) Q and put µ = µA (I/Q). Then there exists an exact sequence ϕ

T (−1)µ → R/T → S(−1) → 0 as graded T -modules. Proof. Let us write I = Q + (x1 , x2 , · · · , xµ ) and put ϕ : T (−1)µ → R/T

∑ denotes a homomorphism of graded T -modules with ϕ(α1 , α2 , · · · , αµ ) = µi=1 αi xi t ∈ R/T for αi ∈ T and 1 ≤ i ≤ µ, where αi xi t denotes the image of αi xi t in R/T . Then we have Cokerϕ = R/[It·T + T ] ∼ = R+ /It·T as graded T -modules. Then two isomorphisms −1 td

−1 td

R+ → IR(−1) and It·T → IT (−1) of graded T -modules induce the isomorphism R+ /It·T ∼ = (IR/IT )(−1) of graded T modules. Therefore Cokerϕ ∼ = S(−1) as graded T -modules, whence we get a required exact sequence.

Tensoring the exact sequence of Lemma 3.1 with A/I, we get exact sequence ϕ

F (−1)µ → R/IR + T → (S/IS)(−1) → 0 (∗) of graded T -modules, where ϕ = A/I ⊗ ϕ. Then thanks to this exact sequence (∗) and Corollary 2.5, we get the following inequality which is originally proved by [8, Theorem 4.1] and [1, Theorem 3.1]. Proposition 3.2. Suppose that d > 0. Then we have 2e0 (I) − e1 (I) + e1 (Q) ≤ 2ℓA (A/I) + ℓA (I/I 2 + Q). The following result plays a key role in our proof of Theorem 1.1. Proposition 3.3. Suppose that conditions (C1 ), (C2 ), and (C3 ) are satisfied. Assume that I ) Q. Then the following two conditions are equivalent to each other. (1) 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q), (2) there exist exact sequences 0 → ((I/I 2 + Q) ⊗ F )(−1) → R/IR + T → S(−1) → 0 and 0 → F → G → R/IR + T → 0 of graded T -modules. We get the following corollary by Proposition 3.3. Corollary 3.4. Suppose that conditions (C1 ), (C2 ), and (C3 ) are satisfied. Assume that 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q). Then we have the following. (1) (2) (3) (4)

I n+2 ⊆ Qn I for all n ≥ 0, Qn ∩ I n+1 = Qn I for all n ≥ 0, depthG > 0, (a1 , a2 , · · · , aˇi , · · · , ad ) : ad ⊆ I 2 + Q for all 1 ≤ i ≤ d.

In the rest of Section 3, let us introduce an outline of our proof of the implication (1) ⇒ (2) and the last assertions of Theorem 1.1. Let us note the following lemma. Lemma 3.5. Put C = A/W with W = H0m (A). Then 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q) if and only if 2e0 (IC) − e1 (IC) + e1 (QC) = 2ℓA (C/IC) + ℓA (IC/I 2 C + QC) and I 2 + Q ⊇ W . Suppose that condition (C1 ) and (C2 ) are satisﬁed and assume that 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q).

Put C = A/W then we have 2e0 (IC) − e1 (IC) + e1 (QC) = 2ℓA (C/IC) + ℓA (IC/I 2 C + QC) and W ⊆ I 2 + Q by Lemma 3.5. Therefore, passing to the ring C, we get (Q + W ) ∩ (I 2 + W ) = QI + W and [(a1 , a2 , · · · , aˇi , · · · , ad ) : ai ] ⊆ I 2 + Q for all 1 ≤ i ≤ d by Corollary 3.4 (2) and (4). Thus the implication (1) ⇒ (2) in Theorem 1.1 has been proven modulo the following Theorem 3.6. Theorem 3.6. Suppose that conditions (C1 ) and (C2 ) are satisfied and assume that 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q). Then we have I 3 ⊆ QI 2 + W and I2 ⊇ W . Taking the local cohomology functor HiM (∗) to the exact sequences of Proposition 3.3 (2), we may calculate the local cohomology modules HiM (G) of the associated graded ring G. Thus we may prove that our assertions (i), (ii), and (iii) are satisﬁed. Our assertion (iv) and (v) are also satisﬁed by the exact sequences of Proposition 3.3 (2) and Corollary 2.5. To prove our assertion (vi) of Theorem 1.1, we need to compute the Koszul cohomology of the associated graded ring G of I. 4. An example In this section we construct one example of m-primary ideal I which satisﬁes 2e1 (I)− e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q) in a Buchsbaum local ring. Our goal is the following. Theorem 4.1. Let ℓ > 0 and d ≥ 2 be integers. Then there exists an m-primary ideal I in a Buchsbaum local ring (A, m) such that d = dim A, Him (A) = (0) for i ̸= 1, d, ℓA (H1m (A)) = ℓ, and 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q) for some reduction Q = (a1 , a2 , · · · , ad ) of I. To construct necessary examples we need some techniques which due to [4, Section 6]. Let us begin with the following. Let m, ℓ > 0 and d ≥ 2 be integers. Let U = k[{Xi }1≤i≤m , {Yj }1≤j≤ℓ , {Vjk }1≤j≤ℓ,1≤k≤d , {Zk }1≤k≤d ] be the polynomial ring with m + ℓ + ℓd + d indeterminates over an inﬁnite ﬁeld k and let a = ({Xi }1≤i≤m , {Yj }1≤j≤ℓ , {Vjk }1≤j≤ℓ,1≤k≤d )·({Xi }1≤i≤m , {Yj }1≤j≤ℓ ) +(Vjk Vj ′ k′ | 1 ≤ j, j ′ ≤ ℓ, 1 ≤ k, k ′ ≤ d, j ̸= j ′ or k ̸= k ′ ) +(Vj2k − Yj Zk | 1 ≤ j ≤ ℓ, 1 ≤ k ≤ d).

We put C = U/a and denote the images of Xi , Yj , Vjk , and Zk in C by xi , yj , vjk , and √ ak , respectively. Then dim C = d, since a = (Xi , Yj , Vjk | 1 ≤ i ≤ m, 1 ≤ j ≤ ℓ, 1 ≤ k ≤ d). Let M = C+ := (xi , yj , vjk , ak | 1 ≤ i ≤ m, 1 ≤ j ≤ ℓ, 1 ≤ k ≤ d) be the graded maximal ideal in C. Let Λ be a subset of {1, 2, · · · , m}. We put q = (ai | 1 ≤ i ≤ d) and JΛ = q + (xα | α ∈ Λ) + (vjk | 1 ≤ j ≤ ℓ, 1 ≤ k ≤ d). Then M2 = qM, JΛ2 = qJΛ + q(y1 , y2 , · · · , yℓ ), and JΛ3 = qJΛ2 , whence q is a reduction of both M and JΛ , and a1 , a2 , · · · , ad is a homogeneous system of parameters for the graded ring C. Let B = CM and put n = MB denotes the maximal ideal of B. We then have the following. Theorem 4.2. The following assertions hold true. (1) (2) (3) (4) (5)

B is a Cohen-Macaulay local ring with dim B = d. e0 (qB) = e0 (JΛ B) = m + ℓd + ℓ + 1. e1 (JΛ B) = e0 (JΛ B) − ℓB (B/JΛ B) + ℓ = ♯Λ + ℓd + ℓ. ei (JΛ B) = 0 for all 2 ≤ i ≤ d. G(JΛ B) is a Buchsbaum ring with depth G(JΛ B) = 0 and I(G(JΛ B)) = ℓd.

Let us now consider the following. Put J = J{1,2,··· ,m} B and A = k + J. Then A is a local k-subalgebra of B with maximal ideal m = J and B is a module ﬁnite extension of A, because ℓA (B/A) = ℓA (B/J) − 1 = ℓ. Hence A is a Noetherian local ring with dim A = dim B = d by Eakin-Ngata’s Theorem. We ﬁx a subset Λ of {1, 2, · · · , m} and put I = JΛ B and Q = (a1 , a2 , · · · , ad )A. Then I is an m-primary ideal in A and Q is a parameter ideal in A and a reduction of I. We then have the following. Theorem 4.3. The following assertions hold true. (1) A is a Buchsbaum local ring with Him (A) = (0) for all i ̸= 1, d and H1m (A) ∼ = B/A, 1 whence h (A) = ℓ, (2) e0 (I) = m + ℓd + ℓ + 1, (3) e1 (I) = ♯Λ + ℓd + ℓ, (4) ei (I) = 0 for 2 ≤ i ≤ d − 1 and ed (I) = (−1)d+1 ℓ, (5) 2e0 (I) − e1 (I) + e1 (Q) = 2ℓA (A/I) + ℓA (I/I 2 + Q), and (6) G(I) is a Buchsbaum ring with HiM (G(I)) = (0) for all i ̸= 1, d and H1M (G(I)) = [H1M (G(I))]1 ∼ = B/A. = H1m (A) ∼

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