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

My talk aims at a Buchsbaumness of the associated graded rings of m-primary ideals in terms of the equality which was originally given by J. Elias and G. Valla [EV]. To state the results of my talk I need some notation. In what follows, let A be a Noetherian local ring with the maximal ideal m and d = dim A > 0. Let I be an m-primary ideal in A and suppose that our ideal I contains a parameter ideal Q = (a1 , a2 , · · · , ad ) of A as a reduction. Let M I n /I n+1 R = R(I) := A[It], T = R(Q) := A[Qt] ⊆ A[t], and G = G(I) := R/IR ∼ = n≥0

denote the Rees algebras of I, Q, and the associated graded ring of I, respectively, where t stands for an indeterminate over A. Put ei (∗) denotes the Hilbert coefficients. 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 [EV] and [GR], 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. The purpose of my talk 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 ([RV, Theorem 2.4], [C, 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 my talk, let us consider the following two 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. These two conditions are naturally satisfied when the local ring A is Cohen-Macaulay. (Notice that condition (C1 ) is satisfied if A is a Buchsbaum local ring.) The main result of my talk is the following Theorem 1, which generalizes the result of [EV] and [GR] given in the case where A is a Cohen-Macaulay local ring, because

ei (Q) = 0 for all 1 ≤ i ≤ d. 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. 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 ∼ I n ∩ W/I n+1 ∩ W if n ≥ 3, [HM (G)]n =  (0) otherwise. Hence [H0M (G)]2 ∼ = W/I 3 ∩ W , [H0M (G)]3 ∼ = I 3 ∩ W , and [H0M (G)]n = (0) for all n 6= 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 6= (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. References [C] A. Corso, Sally modules of m-primary ideals in local rings, Comm. Algebra, 37 (2009) 4503–4515. [EV] J. Elias and G. Valla, Rigid the Hilbert functions, J. Pure and Appl. Algebra 71 (1991), 19–41. [GR] A. Guerrieri and M. E. Rossi, Hilbert coefficients of Hilbert filtrations, J. Algebra 199 (1998) 40–61. [RV] M. E. Rossi and G. Valla Hilbert functions of filtered modules, UMI Lecture Notes 9, Springer (2010). Organisation for the Strategic Coordination of Research and Intellectual Property, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan E-mail address: [email protected]