In algebra, the Gelfand–Kirillov dimension (or GK dimension) of a

right module In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modu ...

''M'' over a ''k''-algebra ''A'' is: :

\operatorname = \sup_ \limsup_ \log_n \dim_k M_0 V^n

where the

supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...

is taken over all finite-dimensional subspaces

V \subset A

and

M_0 \subset M

. 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

k_1, \dots, x_n /math> Is ''n''.
*(Warfield) For any real number''r'' ≥ 2, there exists a finitely generated algebra whose GK dimension is ''r''.

In the theory of D-Modules

Given a right module ''M'' over the Weyl algebra

A_n

, 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 In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...

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 In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from al ...

References

* * Coutinho: A primer of algebraic D-modules. Cambridge, 1995

Basic facts

In the theory of D-Modules

References

Further reading