The Gelfand-Kirillov dimension and the Hilbert polynomial

Abstract

In contrast with the commutative case, for non-commutative algebras the classical Krull dimension is usually not a very useful tool. Indeed, as this notion is defined by using chains of (two-sided) prime ideals, it only makes sense for rings having “many” two-sided ideals. Fortunately, for finitely generated k-algebras R, we may define the so-called Gelfand-Kirillov dimension GKdim(R), which is a far better invariant and which, moreover, coincides with the Krull dimension in the commutative case. Of course, nothing (and nobody) is perfect [125], even the Gelfand-Kirillov dimension: in general, GKdim(R) is extremely hard to calculate. In fact, in view of its rather technical definition, it is even surprising that the Gelfand-Kirillov dimension takes integer values for the algebras we consider!