Golod–Shafarevich theorem

In mathematics, the Golod–Shafarevich theorem was proved in 1964 by two Russian mathematicians, Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences in various branches of algebra.

The inequality

Let A = K<x₁, ..., x_n> be the free algebra over a field K in n = d + 1 non-commuting variables x_i.

Let J be the 2-sided ideal of A generated by homogeneous elements f_j of A of degree d_j with

2 ≤ d₁ ≤ d₂ ≤ ...

where d_j tends to infinity. Let r_i be the number of d_j equal to i.

Let B=A/J, a graded algebra. Let b_j = dim B_j.

The fundamental inequality of Golod and Shafarevich states that

$b_{j}\geq nb_{j-1}-\sum _{i=2}^{j}b_{j-i}r_{i}.$

As a consequence:

• B is infinite-dimensional if r_i ≤ d²/4 for all i

• if B is finite-dimensional, then r_i > d²/4 for some i.

Applications

This result has important applications in combinatorial group theory:

• If G is a nontrivial finite p-group, then r > d²/4 where d = dim H¹(G,Z/pZ) and r = dim H²(G,Z/pZ) (the mod p cohomology groups of G). In particular if G is a finite p-group with minimal number of generators d and has r relators in a given presentation, then r > d²/4.

• For each prime p, there is an infinite group G generated by three elements in which each element has order a power of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite torsion group, although there is no uniform bound on the order of its elements.

In class field theory, the class field tower of a number field K is created by iterating the Hilbert class field construction. Another consequence of the construction is that such towers may be infinite (in other words, do not always terminate in a field equal to its Hilbert class field). Specifically,

• Let K be an imaginary quadratic field whose discriminant has at least 6 prime factors. Then the maximal unramified 2-extension of K has infinite degree.

More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.

References

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• Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI.
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• Serre, J.-P. (2002), "Galois Cohomology," Springer-Verlag. ISBN 3-540-42192-0. See Appendix 2. (Translation of Cohomologie Galoisienne, Lecture Notes in Mathematics 5, 1973.)