Hilbert Series and Dimension

Abstract

An affine algebra A of positive Krull dimension is always infinite-dimensional as a vector space (see Theorem 5.11). The goal of introducing the Hilbert series is nevertheless to measure the size in some way. The trick is to break up A into finite-dimensional pieces, given by the degrees. The Hilbert series then is the power series whose coefficients are the dimensions of the pieces. So instead of measuring the dimension by a number, we measure its growth as the degrees rise, and encode that information into a power series. In the first section of this chapter, we prove the surprising fact that the Hilbert serie can always be written as a rational function, and almost all its coefficients are given by a polynomial, the Hilbert polynomial. We also learn how the Hilbert series can be computed algorithmically. In fact, as in the last chapter, algorithms and theory go hand in hand here. In the second section we show that the degree of the Hilbert polynomial is equal to the Krull dimension of A. Apart from being an interesting result in itself, this leads to a new and better algorithm for computing the dimension. The result also plays an important role in Chap. 12.