Gelfand–Kirillov dimension


In algebra, the Gelfand–Kirillov dimension of a right module M over a k-algebra A is:
where the sup is taken over all finite-dimensional subspaces and.
An algebra is said to have polynomial growth if its Gelfand–Kirillov dimension is finite.

Basic facts

Given a right module M over the Weyl algebra, the Gelfand–Kirillov dimension of M over the Weyl algebra coincides with the dimension of M, which is by definition the degree of the Hilbert polynomial of M. This enables to prove additivity in short exact sequences for the Gelfand–Kirillov dimension and finally to prove Bernstein's inequality, which states that the dimension of M must be at least n. This leads to the definition of holonomic D-Modules as those with the minimal dimension n, and these modules play a great role in the geometric Langlands program.