Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals

Abstract

Abstract. The conjecture of Wolmer Vasconcelos [V] on the vanishing of the firstHilbert coefficient e 1 (Q) is solved affirmatively, where Q is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which e 1 (Q) vanishes are derived.The invariance of e 1 (Q) for parameter ideals Q and its relationship to Buchsbaum rings are studied.

On the Buchsbaum–Rim function of a parameter module

“…Moreover, as a direct consequence of (1), we have the non-positivity of the first Buchsbaum-Rim coefficient of a parameter module. Mandal, Singh and Verma have recently proved that e 1 (A/Q ) 0 for any parameter ideal Q in A (see [14], and also [8]). Corollary 1.2 can be viewed as the module version of this fact.…”

“…(ii) there exists an integer r > 0 and a parameter module N of rank r in F = A r such that the equality for all ν 0 (see [8],…”

Generalized Hilbert coefficients and Northcott's inequality

“…It is also called the Chern number by W. V. Vasconcelos for its tracking position in distinguishing Noetherian filtrations with the same Hilbert multiplicity [31]. The first Hilbert coefficient e 1 (Q), where Q is a parameter ideal, was used to characterize the CohenMacaulay property for large classes of rings [10]. Moreover, G. Colomé-Nin, C. Polini, B. Ulrich and Y. Xie used the generalized first Hilbert coefficient j 1 (I) to bound the number of steps in a process of normalization of ideals of maximal analytic spread [6].…”

Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

“…Mandal and Verma ([MV,Theorem 8]; see also [GhGHOPV,Corollary 2.5]) showed that e 1 Q (A) ≤ 0 for every parameter ideal Q in arbitrary Noetherian local rings A with dim A > 0. The authors of [GhGHOPV] thereafter started the study of the next extreme case, that is, the case where the set…”

“…Because e 1 Q (A) is constant if A is a Buchsbaum local ring (see [Sch,Korollar 3.2]), it seems now very natural to conjecture that A is a Buchsbaum local ring if A is unmixed and the value e 1 Q (A) is constant, which our Theorem 1.1 settles affirmatively. (See [GhGHOPV,Theorems 4.8,4.10] for partial answers for the cases where Λ(A) = {−1} and Λ(A) = {−2}, respectively. )…”