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.
Let J be the 2-sided ideal of A generated by homogeneous elements fj of A of degree dj with
- 2 ≤ d1 ≤ d2 ≤ ...
where dj tends to infinity. Let ri be the number of dj equal to i.
Let B=A/J, a graded algebra. Let bj = dim Bj.
The fundamental inequality of Golod and Shafarevich states that
As a consequence:
- B is infinite-dimensional if ri ≤ d2/4 for all i
- if B is finite-dimensional, then ri > d2/4 for some i.
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.
- Golod, E.S; Shafarevich, I.R. (1964), "On the class field tower", Izv. Akad. Nauk SSSSR 28: 261–272 (in Russian) MR0161852
- Golod, E.S (1964), "On nil-algebras and finitely approximable p-groups.", Izv. Akad. Nauk SSSSR 28: 273–276 (in Russian) MR0161878
- Herstein, I.N. (1968). Noncommutative rings. Carus Mathematical Monographs. MAA. ISBN 0-88385-039-7. See Chapter 8.
- Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI.
- Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 180. ISBN 3-540-63003-1. Zbl 0819.11044.
- Roquette, P. (1986). "On class field towers". In Cassels, J. W. S.; Fröhlich, A.. Algebraic number theory, Proceedings of the instructional conference held at the University of Sussex, Brighton, September 1–17, 1965 (Reprint of the 1967 original ed.). London: Academic Press. pp. 231–249. ISBN 0-12-163251-2.
- 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.)