Exercise 111.20.2. Let $k$ be a field. Let $K/k$ be a finitely generated extension of transcendence degree $d$. If $V, W \subset K$ are finite dimensional $k$-subvector spaces denote

This is a finite dimensional $k$-subvector space. Set $V^2 = VV$, $V^3 = V V^2$, etc.

Show you can find $V \subset K$ and $\epsilon > 0$ such that $\dim V^ n \geq \epsilon n^ d$ for all $n \geq 1$.

Conversely, show that for every finite dimensional $V \subset K$ there exists a $C > 0$ such that $\dim V^ n \leq C n^ d$ for all $n \geq 1$. (One possible way to proceed: First do this for subvector spaces of $k[x_1, \ldots , x_ d]$. Then do this for subvector spaces of $k(x_1, \ldots , x_ d)$. Finally, if $K/k(x_1, \ldots , x_ d)$ is a finite extension choose a basis of $K$ over $k(x_1, \ldots , x_ d)$ and argue using expansion in terms of this basis.)

Conclude that you can redefine the transcendence degree in terms of growth of powers of finite dimensional subvector spaces of $K$.

This is related to Gelfand-Kirillov dimension of (noncommutative) algebras over $k$.

## Comments (0)