THE EQUALITY I 2 = QI IN BUCHSBAUM RINGS WITH MULTIPLICITY TWO

Abstract. Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denotethe multiplicity of A. Let Q be a parameter ideal in A and put I = Q : m. Then the equalityI 2 = QI holds true, if e(A) = 2 and depth A > 0. The assertion is no longer true, unlesse(A) = 2. Counterexamples are given. 1. Introduction.Let Abe a Noetherian local ring with the maximal ideal m and d= dimA. Let Qbea parameter ideal in Aand let I= Q: m. In this paper we are interested in the problemof when the equality I 2 = QI holds true. This problem was completely solved by A.Corso, C. Huneke, C. Polini, and W. Vasconcelos [CHV, CP, CPV] in the case where Ais a Cohen-Macaulay ring. When Ais a Buchsbaum ring, partial answers only recentlyappeared in the authors’ paper [GSa], supplying [Y1, Y2] and [GN] with ample examplesof ideals I, for which the Rees algebras R(I) =L n≥0 I n , the associated graded ringsG(I) = R(I)/IR(I), and the fiber cones F(I) = R(I)/mR(I) are all Buchsbaum ringswith certain specific graded local cohomology modules.This research is a succession of [GSa] and the present purpose is to prove the following,in which e(A) = e