Chapter 8 - DIMENSION

Summary

In this chapter our rings will mostly be noetherian and the problems local. We begin with the basic theorem in noetherian dimension theory: the Principal Ideal Theorem of Krull, or in pithy German, the Hauptidealsatz. Later on in this chapter we discuss two natural generalizations, the Eisenbud-Evans-Bruns result on heights of order ideals, and the Homological Height Theorem of Hochster. The latter has several consequences which, in turn, suggest fresh questions. First though we introduce systems of parameters and develop some dimension theory. Parameters are compared to regular sequences or heights to depths, and this gives rise to a natural question concerning parameters: Hochster's Monomial Conjecture. In this work we use the theory developed in Chapter 5, sections 1 and 2. As part of the material in this chapter is well known and can be found in most books on commutative algebra, we state a number of standard facts in the form of excercises. At other points our treatment may present distinctive features.

KRULL'S HAUPTIDEALSATZ

We shall first introduce the Krull dimension.

DEFINITIONS. Let A be a ring and W a subset of Spec A. The Krull dimension of W (notation dim W) is defined to be the supremum of the lengths of all chains of prime ideals ⊂…⊂, where the to, i - O, …,n, are in W and the length of such a chain is counted as n. If this supremum is finite it is of course achieved in a saturated chain: between two successive primes no third prime ideal can be inserted.