Integrally Closed Ideals of Reduction Number Three

Abstract

In a Cohen-Macaulay local ring [Formula: see text], we study the Hilbert function of an integrally closed [Formula: see text]-primary ideal [Formula: see text] whose reduction number is three. With a mild assumption we give an inequalityi [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th Hilbert coefficient and [Formula: see text] denotes a minimal reduction of [Formula: see text]. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore, this inequality becomes an equality if and only if the depth of the associated graded ring of [Formula: see text] is larger than or equal to [Formula: see text]. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.