A note on the Hilbert-Samuel function in a two-dimensional local ring (Q1373098)

Let \((A,m)\) be a local CM-ring with infinite residue field, and let \(I\) be an \(m\)-primary ideal. Let \(H_I(n)\) (resp. \(P_I(n))\) be the Hilbert function of \(I\) (resp. Hilbert polynomial of \(I\)). Denote by \(\widetilde I\) the Ratliff-Rush closure of \(I\), i.e., the largest ideal containing \(I\) with the same Hilbert polynomial as \(I\). The main result of this paper is: If \(\dim A=2\) and \(\widetilde I=I\), the following conditions are equivalent: (1) \(H_I(n) =P_I(n)\), \(n=1,2\). (2) \(H_I(n) =P_I(n)\), \(n\geq 1\). (3) \(\text{grade} G(I)_+\geq 1\) and \(r_J(I) \leq 2\) for any minimal reduction \(J\) of \(I\). Here \(G(I)\) denotes the associated graded ideal with respect to \(I\), and \(r_J(I)\) is the reduction number.