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
and
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 [2],
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 [2], and thus
the central localization of
is a finite dimensional central simple algebra by Posner’s theorem [1]. 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 [3],
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.
Refferences:
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).