Castelnuovo–Mumford regularity, postulation numbers, and reduction numbers

Suppose G is a standard graded ring over an infinite field. We obtain a sharp lower bound for the regularity of G in terms of the postulation number, the depth, and the dimension of G. We present a class of examples in dimension 1 where the postulation number is 0 and the regularity of G can take on any value between 1 and the embedding codimension of G.Suppose G=grm(R) is the associated graded ring of a Cohen–Macaulay local ring (R,m). We compute the regularity, the reduction number and the postulation number of G and consider the relationship among these invariants. In the case where dimG−gradeG+⩽1, a precise description is known as to how these integers are related. We consider the case where dimG−gradeG+⩾2, and prove that if dimG−gradeG+=2, then regG=max{p(G)+dimG−1,r(m)}, where p(G) is the postulation number of G and r(m) is the reduction number of m.