I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
- The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
- The Krull dimension, based on chains of prime ideals.
- The transcendence degree of the fraction field of A over F.
According to Artin, GK dimension is the most robust notion because it applies to certain noncommutative algebras. And in the noncommutative setting one can't form fraction fields, so the transcendence degree is out of the question (right?).
But Krull dimension has the advantage that it applies to arbitrary rings. As far as I can tell, the definition of GK dimension only applies to algebras. So: as a matter of pedagogy, which notion of dimension is most appropriate for what applications? Which is easiest to prove things about when?