The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

Abstract

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CW-complexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations $d\le \frac{n^2}{4}$ and $d\ge \frac{n^2}{2}$, and conjectured that this is the case for any number of quadratic relations. The particular point where the number of relations is equal to $\frac{n(n-1)}{2}$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We prove that over any infinite field, the Anick conjecture holds for $d\ge \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

17 pages, to appear in the Proceedings of the Royal Society Edinburgh A