The goal is to prove the following fundamental result: if an algebra is a finite module over some subalgebra then

**Lemma**. Let be a subalgebra of a -algebra Suppose that, as a (left) module, is finitely generated over Then

Proof. So for some Define by and let Let

Clearly is a subalgebra of Now, given define by where is any element of with Note that is well-defined because if for some other then and so Hence and so It is easy to see that Finally, define by Then is an -algebra onto homomorphism and hence

**Theorem**. Let be a subalgebra of a -algebra and suppose that, as a (left) module, is finitely generated over Then

Proof. The algebra has a natural embedding into and so Thus by the lemma.

An important consequence of the above theorem is the following result. It shows that for finitely generated commutative algebras, GK dimension is nothing but the transcendence degree of the algebra over the base field.

**Corollary**. If is a finitely generated commutative -algebra, then

*Proof*. Let Then contains a polynomial -algebra such that is a finitely generated -module, by the Noether normalization theorem. Thus, by Corollary 2 and this theorem,