Quasi-socle ideals in Buchsbaum rings

Abstract

Abstract. Quasi-socle ideals, that is, ideals of the form I = Q : m q (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A, m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) = n≥0 I n /I n+1 of I is Buchsbaum for all n 0, where A ( * ) denotes the length of the module. With this notation, the purpose of this article is to prove the following. The associated graded ring G(I) = n≥0 I n /I n+1 of I is a Buchsbaum ring withas A-modules for all i < d anddenotes the homogeneous component with degree n in the ith graded local cohomology module H i M (G(I)) of G(I) with respect to M . Thus, the quasi-socle ideals I = Q : m q behave very well, inside Buchsbaum rings also, under the conditions stated in Theorem 1.1. Notice that, because A is a Buchsbaum ring, the Hilbert coefficients e i Q (A) of the parameter ideal Q are given by the formulaand one has the equalityfor all n ≥ 0 (see [Sch, Korollar 3.2]), so that {e i Q (A)} 1≤i≤d are independent of the choice of Q and are invariants of A. The crucial point in Theorem 1.1 is the equality I 2 = QI; assertions (1), (2), and (3) a d = ab for some a ∈ m q and b ∈ m is rather technical, but at this moment, we do not know whether this additional condition is superfluous.We now briefly explain the background of Theorem 1.1. Our research dates back to works of A. Corso, C. Polini, C. Huneke, W. V. Vasconcelos, and the first author in which the socle ideals Q : m for parameter ideals Q in Cohen-Macaulay rings A were explored with the following result. provided that depth G(m) ≥ 2.Since Buchsbaum rings are very akin to Cohen-Macaulay rings, it seems quite natural to expect that similar results of the Cohen-Macaulay case, such as Theorem 1.3, should be true also in the Buchsbaum case after suitable modifications of the corresponding conditions, which we now report in Theorem 1.1.The proof of Theorem 1.1 is given in Section 2, which we divide into two parts. The first part shows that m q I = m q Q. The second part proves that I 2 = QI. Since A is not necessarily a Cohen-Macaulay ring, the equality I 2 = QI does not readily follow from the fact that m q I = m q Q. We carefully analyze this phenomenon in Section 2. A similar but more restricted result also holds true in the case where G(m) is a Buchsbaum ring with depth G(m) = 1, which we discuss in Section 3. In Section 4, we give exam- Let F denote the set of all the productswhere the family {f i } 1≤i≤q of elements in m \ m 2 is assumed to satisfy the conditions stated above. Let us writeWe put g j = 1≤k≤q,k =j f k for each 1 ≤ j ≤ q, and we choose g ∈ m \ m 2 so that g * , f * j is a regular sequence in G(m) for all 1 ≤ j ≤ q. Letwhence, for all 1 ≤ i ≤ d and 1 ≤ j ≤ q, we get Hence,and soWe may assume that k > 1 and that our assertion holds true for k − 1. Hence,Let y ∈ A, and assume thatThanks to Claim 1, we getWe need the following to show the equality I 2 = QI. Proof. The former assertion is proved exactly in the same way as [GSa3, Lemma 2.3], wher...

Stability of quasi-socle ideals

“…, a d of parameters that a d = ab for some a ∈ m q and b ∈ m. This is a technical but crucial condition in order to use the result of Goto and Sakurai [16, Lemma2.3], and thanks to the condition, they were able to get the equality I 2 = QI by induction on dimension d, where I = Q : m q and Q ⊆ m q+ +1 . The present proof of Theorem 1.1 and Corollary 1.2 is substantially different from the one in [7]. It is based on Proposition 2.2 and valid for every parameter ideal Q contained in m t , choosing an integer t such that t ≥ q + + 1.…”

“…with respect to M. In [7] Goto, Sakurai and the author proved Theorem 1.1 and Corollary 1.2, assuming the extra condition on systems a 1 , a 2 , . .…”

“…To prove Theorem 1.1 we need the following result of [7], in which we make use of the assumption that depth G(m) ≥ 2. In [7, Proposition 2.2] the integer q is assumed to be q ≥ 2.…”

“…Since the proof of the case q = 1 is quite different from that of the case q ≥ 2, we have included it here. Proposition 2.4 [7,Proposition 2.2]. Let A be a generalized Cohen-Macaulay ring with depth G(m) ≥ 2.…”