---
tag: RC5A5T7YY2
kind: statement
role: theorem
name: Golod–Shafarevich theorem
msc: [16S15, 16W50, 20D15]
uses: [04C5ZBG1WT, 2P3Y2FRCSZ, 0CD87SFM2N, 8KB4SQGC0C, 033WGN00BF, 06QYRTX9KQ, WWCJ7CR2R3]
about: [2P3Y2FRCSZ]
sources:
  - url: https://en.wikipedia.org/wiki/Golod–Shafarevich_theorem
same_as:
  wikidata: Q17018210
  wikipedia: Golod–Shafarevich theorem
---

## Statement

Let $k$ be a [[field|04C5ZBG1WT]], let $d \ge 1$, and let $F = k\langle x_1, \ldots, x_d \rangle$ be the [[free algebra|2P3Y2FRCSZ]], graded by degree.
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] of $F$ generated by a set of homogeneous elements of degrees at least $2$.
For $i \ge 2$, let $r_i$ be the number of these generators of degree $i$, and assume each $r_i$ is finite.
Let $A = F/I$, which is a [[graded algebra|8KB4SQGC0C]] $A = \bigoplus_{n \ge 0} A_n$, and let $H_A(t) = \sum_{n \ge 0} (\dim_k A_n) t^n$ be its Hilbert series.
Let $R(t) = \sum_{i \ge 2} r_i t^i$.

1. Every coefficient of the [[formal power series|033WGN00BF]] $H_A(t) \big(1 - d t + R(t)\big) - 1$ is nonnegative.
2. If there is a real number $t_0$ with $0 < t_0 < 1$ such that $R(t_0)$ converges and $1 - d t_0 + R(t_0) \le 0$, then $A$ is infinite-dimensional.

## 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|06QYRTX9KQ]].
Let $d = \dim H^1(G, \F_p)$, its minimal number of generators, and $r = \dim H^2(G, \F_p)$; then $r > d^2/4$.

## Explanations

### L1 human

Start with a free algebra on $d$ letters, which is infinite-dimensional, and impose relations.
Each relation cuts down the size, but only by a limited amount in each degree.
The theorem says: if there are few relations, especially in low degrees, the quotient is still infinite-dimensional.

### L2 human

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|WWCJ7CR2R3]] ending in a field with class number $1$.