Castelnuovo–Mumford regularity, postulation numbers, and reduction numbers

Abstract

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 = gr m ( 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 dim G − grade G + ⩽ 1 , a precise description is known as to how these integers are related. We consider the case where dim G − grade G + ⩾ 2 , and prove that if dim G − grade G + = 2 , then reg G = max { p ( G ) + dim G − 1 , r ( m ) } , where p ( G ) is the postulation number of G and r ( m ) is the reduction number of m.