## The depth of the associated graded ring of ideals with any reduction number

##### Abstract

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 is not necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r+ℓ powers of I, where r denotes the Castelnuovo regularity of G and ℓ denotes the analytic spread of I.