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Let (R, m) be a local ring and I an m-primary ideal. A result of Trung shows that if the local cohomology of gr1(R) satisfies certain conditions, then the reduction number of I is independent of the minimal reduction chosen. These conditions consist of t = dim R grade gr1 (R)+ inequalities. We show that if R is Cohen-Macaulay, then one of these inequalities is always satisied, while another can often be easily checked. Applications are then given in two-dimensional Cohen-Macaulay rings. For instance, we show that if the Hilbert function of I equals the Hilbert polynomial of I for all integers greater than 1, then the reduction number is independent of the choice of minimal reduction.

The reduction number of an ideal and the local cohomology of the associated graded ring