On the Depth of the Associated Graded Ring of an Ideal

Let R be a local Cohen–Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G fails to be Cohen–Macaulay. We assume that I has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers. The main theorem unifies and generalizes several known results. We also give conditions that imply the Serre properties of the blow-up rings.