Golod–Shafarevich theorem

Remarks

Part 1 comes from a careful count of the degree $n$ part of $I$: it is spanned by products $a f b$ with $f$ a generator, and the count gives $\dim A_n \ge d \dim A_{n-1} - \sum_{i=2}^{n} r_i \dim A_{n-i}$. For part 2, if $A$ were finite-dimensional, then $H_A$ would be a polynomial with $H_A(t_0) > 0$. Evaluating the series of part 1 at $t_0$ would give $H_A(t_0)\big(1 - d t_0 + R(t_0)\big) \ge 1 > 0$, which contradicts $1 - d t_0 + R(t_0) \le 0$.

Example: take $d = 4$ and at most one relation in each degree $i \ge 2$. Then $R(t) \le t^2/(1 - t)$, and at $t_0 = 1/2$ one gets $1 - 2 + R(1/2) \le -1 + 1/2 < 0$. So every such quotient of $k\langle x_1, x_2, x_3, x_4 \rangle$ is infinite-dimensional.

Golod used this criterion to build finitely generated infinite-dimensional algebras in which every element is nilpotent. A version for groups concerns a nontrivial finite group $G$ of order a power of a prime $p$, that is, a p-group. Let $d = \dim H^1(G, \F_p)$, its minimal number of generators, and $r = \dim H^2(G, \F_p)$; then $r > d^2/4$.

Going deeper

For a student

The theorem is a tool for proving that objects are infinite. Through the group version, it shows that some groups with few relations are infinite, which gave infinite finitely generated periodic groups. In number theory, it shows that the class field tower of a number field can be infinite. Some imaginary quadratic fields have no finite tower of Hilbert class fields ending in a field with class number $1$.