Ideals of reduction number two

Abstract

In a local Cohen-Macaulay ring $(A,\mathrm{m})$, we study the Hilbert function of an $\mathrm{m}$-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 ${\mathrm{e}}_1(I)\ge {\mathrm{e}}_0(I)-{\ell }_A(A/I)+{\mathrm{e}}_2(I)$ 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.

arXiv comment11 pages, comments are welcome