Depth properties of Rees algebras and associated graded rings (Q1312874)

Let \(R\) be a Cohen-Macaulay local ring of dimension \(\geq 1\) with maximal ideal \(M\) and \(I\) an \(M\)-primary ideal. This paper is interested in determining the depth of the associated graded ring \(gr_ I(R)\) and in connecting \(\text{depth} gr_ I(R)\) with the depth of the Rees algebra \(R[It]\). The main result of this paper is that if \(gr_ I(R)\) is not Cohen-Macaulay, then \(\text{depth} R[It] = \text{depth} gr_ I(R) + 1\). Moreover, if \(R\) is a regular local ring, then this equality holds even if \(gr_ I (R)\) is Cohen-Macaulay. Several results are also given about when the reduction number of an ideal is independent of the choice of minimal reduction.