Algebraic Aspects of Dimension

Abstract

The basic invariant of a vector space is its dimension, and in this context the class of finite-dimensional vector spaces is distinguished. For modules, which are a direct generalisation of vector spaces, there are analogous notions, which play the same fundamental role. On the other hand, we have considered algebraic curves, surfaces, and so on, and have ‘coordinatised’ each such object C by assigning to it the coordinate ring K[C] or the rational function field K(C). The intuitive notion of dimension (1 for an algebraic curve, 2 for a surface, and so on) is reflected in algebraic properties of the ring K[C] or of the field K(C), and these properties are meaningful and important for more general types of rings and fields. As one might expect, the situation becomes more complicated in comparison with the simplest examples: we will see that there exist various ways of expressing the ‘dimension’ of rings or modules as a number, and various analogues of finite dimensionality.