# Gelfand–Kirillov dimension

In algebra, the **Gelfand–Kirillov dimension** (or **GK 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

- The Gelfand–Kirillov dimension of a finitely generated commutative algebra
*A*over a field is the Krull dimension of*A*(or equivalently the transcendence degree of the field of fractions of*A*over the base field.) - In particular, the GK dimension of the polynomial ring Is
*n*. - (Warfield) For any real number
*r*≥ 2, there exists a finitely generated algebra whose GK dimension is*r*.[1]

## In the theory of D-Modules

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.

## References

- Artin 1999, Theorem VI.2.1.

- Smith, S. Paul; Zhang, James J. (1998). "A remark on Gelfand–Kirillov dimension" (PDF).
*Proceedings of the American Mathematical Society*.**126**(2): 349–352. doi:10.1090/S0002-9939-98-04074-X. - Coutinho: A primer of algebraic D-modules. Cambridge, 1995