Castelnuovo-Mumford regularity, postulation numbers and relation types

We establish a bound for the Castelnuovo-Mumford regularity of the associated graded ring GI(A) of an m-primary ideal I of a local Noetherian ring (A,m) in terms of the dimension of A, the relation type and the number of generators of I. As a consequence, we obtain that the existence of uniform bounds for the regularity of the associated graded ring, and the relation type of parameter ideals in A, are equivalent conditions. In addition, we establish an equation for the postulation number and the Castelnuovo-Mumford regularity of the associated graded ring Gq(A) of a parameter ideal q, which holds under certain conditions on the depths of the occurring rings. We also show, that the regularity of the ring Gq(A) is bounded in terms of the dimension of A, the length of A/q and the postulation number of Gq(A).