The Hilbert function H(M, n) measures the dimension of the n-th homogeneous piece of a graded module M. In the first section of this chapter we study the Hilbert function of modules over homogeneous rings, prove that it is a polynomial for large values of n, and introduce the Hilbert series and multiplicity of a graded module. The next section is devoted to the proof of Macaulay's theorem which describes the possible Hilbert functions. The third section complements these results by Gotzmann's regularity and persistence theorem.

The Hilbert function behaves quite regular, even for graded, non-homogeneous rings. Such rings will be considered in the fourth section, where we will also investigate the Hilbert function of the canonical module.

The passage to the associated graded ring with respect to a filtration allows us to extend some concepts for graded rings like ‘Hilbert function’ or ‘multiplicity’ to non-graded rings, and leads to the Hilbert–Samuel function and the multiplicity of a finite module with respect to an ideal of definition. We shall study basic properties of filtrations and their associated Rees rings and modules, and sketch the theory of reduction ideals. Finally we prove Serre's theorem which interprets multiplicity as the Euler characteristic of a certain Koszul homology.

Hilbert functions over homogeneous rings

We begin by studying numerical properties of finite graded modules over a graded ring R. Our standard assumption in this section will be that R0 is an Artinian local ring, and that R is finitely generated over R0. Notice that for each finite graded R-module M, the homogeneous components Mn of M are finite R0-modules, and hence have finite length.