Cohen-Macaulay graded rings associated to ideals

Let G(I) be the associated graded ring of an ideal I in a Cohen-Macaulay local ring A. We give a sufficient condition for G(I) to be a Cohen-Macaulay ring. It is described in terms of the depths of A/Iⁿ for finitely many n, the reduction numbers of I_Q for certain prime ideals Q and the Artin-Nagata property of I in the sense of Ulrich (Contemp. Math. vol. 159, pp. 373-400). The main theorem unifies several results which are already known in this aspect of the theory. Although the statement of the main theorem is rather complicated, we restate the conditions under special situations, so that they are practical and simple.