Bergman's gap theorem

Remarks

The gap between $0$ and $1$ is easy: if a finitely generated algebra is infinite-dimensional, then $V^n \subsetneq V^{n+1}$ for all $n$, so $\dim V^n \ge n + 1$. The gap between $1$ and $2$ is harder. One reduces to algebras given by monomial relations, whose growth is governed by counting words in the generators. Then a combinatorial argument on words applies. If the number of nonzero words of some length $n$ is at most $n$, then the number of nonzero words of each length stays bounded. So the growth is at most linear. Since the dimension of a general algebra is the supremum over its finitely generated subalgebras, the gaps pass to all algebras.

Example: $k[x]$ has Gelfand–Kirillov dimension $1$, and the polynomial ring $k[x, y]$ has dimension $2$. So both ends of the gap $(1, 2)$ are attained. No further gap exists: every real number $r \ge 2$ is the Gelfand–Kirillov dimension of some finitely generated algebra. This contrasts with the Krull dimension of commutative rings, which takes only integer values or $\infty$.

Going deeper

For a student

The theorem is a statement about the growth of algebras, and its heart is combinatorics on words. It shows that Gelfand–Kirillov dimension behaves like an integer-valued dimension at the bottom of the scale, while above $2$ every real value can occur. It is a basic input in the study of algebras of small Gelfand–Kirillov dimension, such as algebras of dimension $1$ and $2$, which are often compared with curves and surfaces.