Ideals of reduction number two

Abstract

In a local Cohen-Macaulay ring (A, m), we study the Hilbert function of an m-primary ideal I whose reduction number is two. It is a continuous work of the papers of Huneke, Ooishi, Sally, and Goto-Nishida-Ozeki. With some conditions, we show the inequality e1(I) ≥ e0(I) − ℓA(A/I) + e2(I) of the Hilbert coefficients, which is the converse inequality of Sally and Itoh. We also study relations between the Hilbert coefficients and the depth of the associated graded ring.