Coefficient ideals and the Cohen-Macaulay property of Rees algebras (Q2701621)

Let \(A\) be a local ring of dimension \(d\) having an infinite residue field, and let \(I\subset A\) be an ideal of positive height \(h\). If \(J\subset I\) is a reduction of \(I\), the coefficient ideal \(a(I,J)\) is defined as the largest ideal \(a\) such that \(Ia=Ja\). The purpose of this article is to investigate the coefficient ideal \(a(I,J)\) under the assumption that the Rees algebra \({\mathcal R}(I)\) of \(I\) is Cohen-Macaulay. The main result of the paper, theorem 3.4, says that for any minimal reduction \(J\subset I\) the coefficient ideal \(a(I,J)\) coincides with the ideal \(\Gamma(X,I^{h-1}\omega_X)\subset A\), where \(X=\text{Proj}\) \({\mathcal R}(I)\), under the assumption that the ring \(A\) is Gorenstein, the ideal \(I\) satisfies the condition \(G_l\) (i. e. \(\mu(I_p)\leq\text{ht }p\) for every \(p\in\text{V}(I)\) with \(\text{ht} p\leq l-1\)), and \(\text{depth}\) \(A/I^n\geq d-h-n+1\) for \(n=1,\ldots,l-h\). In fact, when \(h>1\), this means that \(a(I,J)\) can be identified with the homogeneous component \([\omega_{{\mathcal R}(I)}]_{h-1}\) of the canonical module \(\omega_{{\mathcal R}(I)}\). In particular, it shows that \(a(I,J)\) is independent of the minimal reduction \(J\). -- As a corollary of this theorem, the author shows that if \(A\) is a regular local ring essentially of finite type over a field of characteristic zero and \(I\) is an equimultiple ideal such that \({\mathcal R}(I)\) is normal and Cohen-Macaulay, then \(a(I,J)=\text{adj} (I^{h-1})\) if and only if \({\mathcal R}(I)\) has rational singularities.