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

TL;DR

This paper investigates the Golod-Shafarevich inequality for Hilbert series of quadratic algebras, proving the Anick conjecture for certain ranges of relations and confirming Vershik's conjecture in characteristic zero fields.

Contribution

It proves the Anick conjecture for a broad range of quadratic relations and confirms Vershik's conjecture over fields of characteristic zero.

Findings

Anick conjecture holds for d ≥ 4(n^2+n)/9 over any infinite field.

Vershik's conjecture is confirmed over fields of characteristic zero.

Provides asymptotic results related to the Golod-Shafarevich inequality.

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 $d \leq \frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, 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 $d \geq \frac{4(n^2+n)}{9}$ 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.