Bounds on the normal Hilbert coefficients (Q2790910)

Let \((R,m)\) be an analytically unramified Cohen-Macaulay local ring of dimension \(d>0\). Assume that the residue field of \(R\) is infinite. Let \(I\) be an \(m\)-primary ideal. This paper studies the normal filtration \(\{\overline{I^n}\}\), where \(\overline{I^n}\) is the integral closure of \(I^n\). Let \(\overline{e}_i\) be the corresponding normal Hilbert coefficients and \(\overline{\mathcal{G}}\) the associated graded ring of the normal filtration.NEWLINENEWLINEFirst, the authors prove that if \(\overline{e}_1\leq e_0-\lambda(R/\overline{I})+1\), then \(\text{depth}(\overline{\mathcal{G}})\geq d-1\).NEWLINENEWLINENext, assume \(d\geq 3\), \(\overline{I}=m\), \(\overline{e}_3=0\) and let \(J\) be a minimal reduction of \(I\). The authors prove that if type \(t(R)\leq \max\{2,1+\lambda(\overline{I^2}/J\overline{I})\}\), then \(\overline{\mathcal{G}}\) is Cohen-Macaulay.