ABSTRACT
Chapter 5 Dimension Theoretic Properties of Rings of Invariants In the previous chapters we have learned some of the basic finiteness prop erties of rings of polynomial invariants of finite groups: the finiteness of the number of generators and bounds for their number and degrees. In addition to the number of generators there are other measures of finiteness. For exam ple, in a Noetherian ring A every ascending chain of ideals must eventually stabilize: but for a given ring A is there an upper bound on the length of such a sequence? For arbitrary ideals the answer must be no, as for x € A one has for each integer k the chain of ideals
so if x is not nilpotent this yields an ascending chain of arbitrary length. By restricting our attention to prime ideals however we receive a useful invariant called the Krull dimension of A. This is the first of the further finiteness properties we investigate.
