GK dimension; definition and basic remarks

Posted: April 26, 2011 in Gelfand-Kirillov Dimension, Noncommutative Ring Theory Notes
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Throughout k is a field. Let A be a k-algebra. Let V be a k-vector subspace of A spanned by the set \{a_1, \ldots , a_m \}. For any integer n \geq 1 we will denote by V^n the the k-subspace of A spanned by all monomials of length n in a_1, \ldots , a_m. We will also define V^0=k. If A=k[a_1, \ldots , a_m], then V is called a generating subspace of A. If a generating subspace V of A contains 1, then V is called a frame of A. We will denote by V_n the subspace \sum_{i=0}^n V^i of A. Clearly if V is a generating subspace of A, then A=\bigcup_{n=0}^{\infty}V_n.

Remark 1. Let A be a finitely generated k-algebra and suppose that V and W are two generating subspaces of A. Then \displaystyle \limsup_{n\to\infty} \log_n (\dim V_n) = \limsup_{n\to\infty} \log_n (\dim W_n).

Proof. We have A=\bigcup_{n=0}^{\infty}V_n=\bigcup_{n=0}^{\infty}W_n. Since both V and W are finite dimensional, there exist integers r \geq 1 and s \geq 1 such that V \subseteq W_r and W \subseteq V_s. Thus V_n \subseteq W_{rn} and W_n \subseteq V_{sn} for all integers n \geq 0. Now, \dim V_n \leq \dim W_{rn} implies that

\log_n (\dim V_n) \leq \log_n(\dim W_{rn})=(1+\log_n r) \log_{rn}(\dim W_{rn}).

Taking limsup of both sides of the above inequality will give us

\displaystyle \limsup_{n\to\infty} \log_n(\dim V_n) \leq \limsup_{n\to\infty} \log_n(\dim W_n),

because \displaystyle \lim_{n\to\infty} (1+ \log_n r)=1 and

\displaystyle \limsup_{n\to\infty} \log_{rn}(\dim W_{rn}) \leq \limsup_{n\to\infty} \log_n (\dim W_n).

Similarly \dim W_n \leq \dim V_{sn} will imply

\displaystyle \limsup_{n\to\infty} \log_n(\dim W_n) \leq \limsup_{n\to\infty} \log_n(\dim V_n),

which completes the proof. \Box

So, by the above remark, the value \displaystyle \limsup_{n\to\infty} \log_n (\dim V_n) does not depend on the generaing subspace V and thus the following definition makes sense.

Definition. Let A be a finitely generated k-algebra and let V be a generating subspace of A. The Gelfand-Kirillov dimension, or GK dimension, of A, which we will  denote it by {\rm{GKdim}}(A), is defined by

\displaystyle {\rm{GKdim}}(A)=\limsup_{n\to\infty} \log_n (\dim V_n).

The GK dimension of a algebra somehow measures how far an algebra is from being finite dimensional. We will see later that if A is commutative, then the GK dimension of A is nothing but the transcendence degree of A.

Remark 2. In the above definition if V is a frame of A, then V_n=\sum_{i=0}^n V^i = V^n, because

k=V^0 \subseteq V \subseteq \ldots \subseteq V^n.

Thus we get this simple expression \displaystyle {\rm{GKdim}}(A) = \limsup_{n\to\infty} \log_n (\dim V^n).

  1. Alex Youcis says:

    Hey Yaghoub. Maybe it’s because I haven’t read enough of the after-this GK dimension posts, but what precisely is the point of it? I mean, I get roughly what it measures, but is there a typical place that GK dimension would come up?

    • Yaghoub says:

      Yes, GK dimension is a useful and important invariant of algebras, although it’s a relatively new subject in ring theory.
      For example, if k is an algebraically closed filed and A is a finitely generated k-algebra as well as a domain, then GKdim(A) = 1 implies that A is commutative. This is an important result.
      For commutative algebras GK dimension is not very interesting because it coincides with Krul dimension, which is a much more known invariant.

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