We defined the -th Weyl algebra over a ring in here. In this post we will find the GK dimension of in terms of the GK dimension of The result is similar to what we have already seen in commutative polynomial rings (see corollary 1 in here). We will assume that is a field and is a -algebra.
Proof. Suppose first that is finitely generated and let be a frame of Let Since we have
Let Clearly is a frame of and
for all because every element of commutes with every element of Therefore, since and for all we have and Thus and hence
Therefore by and we are done.
For the general case, let be any finitely generated – subalgebra of Then, by what we just proved,
and hence Now, let be a -subalgebra of generated by a finite set Let be the -subalgebra of generated by all the coefficients of Then and so
and the proof is complete.
Corollary. for all In particular,
Proof. It follows from the theorem and the fact that