## Algebras of GK dimension one

Posted: April 26, 2011 in Gelfand-Kirillov Dimension, Noncommutative Ring Theory Notes
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We have seen so far that the GK dimension of a $k$-algebra $A,$ where $k$ is a field, has possible values $0, 1$ and any real number $\geq 2.$ We showed that the GK dimension of $A$ is $0$ if and only if every finitely generated $k$-subalgebra of $A$ is finite dimensional as a $k$-vector subspace of $A.$ In particular, a finitely generated $k$-algebra $A$ has GK dimension $0$ if and only if $\dim_k A < \infty.$ Now what can we say about algebras of GK dimension one? First we show that they are not necessarily Noetherian.

Example. Let $A$ be the $k$-algebra generated by $x$ and $y$ with the relations $x^2=xyx=yxy=0.$ Then $A$ is not noetherian and ${\rm{GKdim}}(A)=1.$

Proof. It is not noetherian because it contains the infinite direct sum of ideals $\bigoplus_{n=2}^{\infty}kxy^nx.$ To see why the GK dimension of $A$ is one, consider the frame $V=k + kx + ky.$ Now, assuming that $n \geq 3$ and considering the relations on $A,$ we see that the only terms which appear in $V^n$ are

$1, y, \ldots , y^n, x, xy, \ldots , xy^{n-1}, yx, y^2x , \ldots , y^{n-1}x$ and $xy^2x, \ldots , xy^{n-2}x.$

Thus $\dim_k V^n = 4n-3$ and hence $\displaystyle {\rm{GKdim}}(A)= \lim_{n \to\infty} \log_n(4n-3)=1. \ \Box$

Next theorem shows that if $A$ is semiprime and has GK dimension 1, then $A$ is Noetherian. In fact it will be even more than just Noetherian.

Theorem 1. If $A$ is a finitely generated semiprime $k$-algebra, then ${\rm{GKdim}}(A)=1$ if and only if $A$ is finitely generated as a module over some polynomial algebra in one variable $k[x] \subseteq Z(A).$

Proof. If $A$ is finitely generated as a module over some polynomial algebra $k[x],$ then

${\rm{GKdim}}(A)={\rm{GKdim}}(k[x])=1,$

by this theorem and Corollary 2. Conversely, if ${\rm{GKdim}}(A)=1,$ then by a theorem of Small, Stafford and Warfield [2], $A$ is finitely genrated over its center $Z(A)$ and thus ${\rm{GKdim}}(Z(A))=1,$ by this theorem. We also have $k \subseteq Z(A) \subseteq A$ and we know that $A$ is both a finitely generated $k$-algebra and a finitely generated $Z(A)$-module. Thus, by Artin-Tate lemma, $Z(A)$ is a finitely generated $k$-algebra. Therefore ${\rm{tr.deg}}(Z(A)/k)=1,$ by the corollary in this post, and hence $Z(A)$ is a finitely generated module over some polynomial algebra $k[x],$ by the Noether normalization theorem. The result now follows because $A$ is a finitely generated $Z(A)$-module. $\Box$

Theorem 2. Let $k$ be an algebraically closed field and let $A$ be a $k$-algebra. If $A$ is a domain and ${\rm{GKdim}}(A) \leq 1,$ then $A$ is commutative.

Proof. First note that if $a,b \in A,$ then the $k$-subalgebra generated by $a,b$ has GK dimension at most one too and so we may assume that $A$ is finitely generated. The case ${\rm{GKdim}}(A)=0$ easily follows because then $A$ would be finite dimensional, and hence algebraic, over $k$ and therefore $A=k$ because $k$ is algebraically closed. Now, suppose that ${\rm{GKdim}}(A)=1.$ The algebra $A$ is PI, by [2], and thus $Q_Z(A),$ the central localization of $A,$ is a finite dimensional central simple algebra by Posner’s theorem [1]. Since $A$ is a domain, $Q_Z(A)$ is a domain and hence $Q_Z(A)=D$ is a finite dimensional division algebra over its center $F,$  which is the quotient field of $Z(A).$ Thus ${\rm{tr.deg}}(F/k)={\rm{tr.deg}}(Z(A)/k)=1,$ by the corollary in this post. Hence, by Tsen’s theorem [3], $Q_Z(A)=F.$ Thus $Q_Z(A),$ and so $A$ itself, is commutative. $\Box$

Remark. Theorem 2 does not hold if $k$ is not algebraically closed. For example, let $\mathbb{H}$ be the division ring of real quaternions. Then, as an $\mathbb{R}$-algebra, $\mathbb{H}$ is a noncommutative domain of GK dimension zero.  Similarly, if $x$ is a central variable over $\mathbb{H},$ then the polynomial ring $\mathbb{H}[x],$ as an $\mathbb{R}$-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).

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