At the beginning of Chapter 16, we explained why we are interested in bounding the regularity at and above level 2 of a non-zero graded prime ideal p in the polynomial ring K [X0, …, Xr] in r + 1 indeterminates X0, …, Xr (where r ∈ ℕ) over an algebraically closed field K: when the vanishing ideal of a projective variety, then reg2(p) provides an upper bound on the degrees of homogeneous polynomials needed to define V.

Suppose that is positively graded and R0 is an Artinian ring. In Chapter 16, we were concerned with a priori bounds of diagonal type for reg2, and the bounds we obtained are always available for finitely generated graded R-modules. In this chapter, we are going to establish, in the particular case when R is a polynomial ring over R0 and R0 is local, a bound on this regularity which applies to all graded submodules of a given finitely generated graded free K-module; the bound is expressed in terms of numerical invariants defined by means of the characteristic function of such a module.

In more detail, let M be a non-zero finitely generated graded R-module of dimension d.