We have seen so far that the GK dimension of a -algebra where is a field, has possible values and any real number We showed that the GK dimension of is if and only if every finitely generated -subalgebra of is finite dimensional as a -vector subspace of In particular, a finitely generated -algebra has GK dimension if and only if Now what can we say about algebras of GK dimension one? First we show that they are not necessarily Noetherian.
Example. Let be the -algebra generated by and with the relations Then is not noetherian and
Proof. It is not noetherian because it contains the infinite direct sum of ideals To see why the GK dimension of is one, consider the frame Now, assuming that and considering the relations on we see that the only terms which appear in are
Thus and hence
Next theorem shows that if is semiprime and has GK dimension 1, then is Noetherian. In fact it will be even more than just Noetherian.
Theorem 1. If is a finitely generated semiprime -algebra, then if and only if is finitely generated as a module over some polynomial algebra in one variable
Proof. If is finitely generated as a module over some polynomial algebra then
by this theorem and Corollary 2. Conversely, if then by a theorem of Small, Stafford and Warfield , is finitely genrated over its center and thus by this theorem. We also have and we know that is both a finitely generated -algebra and a finitely generated -module. Thus, by Artin-Tate lemma, is a finitely generated -algebra. Therefore by the corollary in this post, and hence is a finitely generated module over some polynomial algebra by the Noether normalization theorem. The result now follows because is a finitely generated -module.
Theorem 2. Let be an algebraically closed field and let be a -algebra. If is a domain and then is commutative.
Proof. First note that if then the -subalgebra generated by has GK dimension at most one too and so we may assume that is finitely generated. The case easily follows because then would be finite dimensional, and hence algebraic, over and therefore because is algebraically closed. Now, suppose that The algebra is PI, by , and thus the central localization of is a finite dimensional central simple algebra by Posner’s theorem . Since is a domain, is a domain and hence is a finite dimensional division algebra over its center which is the quotient field of Thus by the corollary in this post. Hence, by Tsen’s theorem , Thus and so itself, is commutative.
Remark. Theorem 2 does not hold if is not algebraically closed. For example, let be the division ring of real quaternions. Then, as an -algebra, is a noncommutative domain of GK dimension zero. Similarly, if is a central variable over then the polynomial ring as an -algebra, is a noncommutative domain of GK dimension one.
1. E. C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. (1960) no. 2, 180-183.
2. L. W. Small, J. T. Stafford, and R. Warfield, Affine algebras of Gelfand Kirillov dimension one are PI, Math. Proc. Cambridge. Phil. Soc. (1984), 407-414.
3. C. Tsen, Divisionsalgebren uber Funktionenkorper, Nachr. Ges. Wiss. Gottingen (1933).