This chapter has for aim the development of techniques to determine the most basic measure of a symmetric algebra—its Krull dimension. It requires an exposition of the classical Fitting ideals of modules, with estimates of codimensions of determinantal ideals.

The Krull dimension of a symmetric algebra S(E) of a module E turns out to be connected to the invariant b(E) of the module E introduced by Forster [71], a quarter of a century ago, that bounds the number of generators of E. This was shown by Huneke and Rossi [150]; furthermore, it was accomplished in a manner that makes the search for dimension formulas for S(E) much easier. Based on slightly different ideas, [261] gives another proof of that result in terms of the heights of the Fitting ideals of E. It makes for an often effective way of determining the Krull dimension of S(E).

The Fitting ideals are introduced from various directions, and their divisorial properties are noted. It is to be expected that these ideals play such an important role since they code the symmetric algebra of the module. In later chapters, their primary components will be used in expressing the divisor class group of normal algebras.

One topic which is central to this study, that of ascertaining when a symmetric algebra is an integral domain will have a very brief development here but will recur under various circumstances in several other chapters.