On associated graded rings of normal ideals (Q1969477)

Let \(R\) be a commutative ring and let \(I\) be an ideal of \(R\). The associated graded ring of \(R\) with respect to \(I^n\) is denoted by \(\mathbb G(I^n)\). Huckaba and Huneke proved that if \((R,\mathfrak m)\) is a two-dimensional Cohen-Macaulay local ring and \(I\) is an \(\mathfrak m\)-primary normal ideal (i.e., all powers of \(I\) are integrally closed), then \(\mathbb G(I^n)\) is Cohen-Macaulay for large \(n\) [\textit{S. Huckaba} and \textit{C. Huneke}, J. Reine Angew. Math. 510, 63-82 (1999; Zbl 0923.13005)]. In this paper the authors generalize the above theorem by dropping the condition \(\mathfrak m\)-primary for \(I\). The techniques they use are completely different from those of Huckaba and Huneke. Among other results they prove the following: ``Let \(R\) be a two-dimensional Cohen-Macaulay local ring and let \(I\) be an ideal of \(R\) such that \(\text{Proj } R[It]\) is normal. Suppose that either \(\text{ht } I>0\) or \(R\) is analytically unramified. Then \(\mathbb G(I^n)\) is Cohen-Macaulay for all \(n\) sufficiently large''.