The Golod Shafarevich counter-example without Hilbert series

TL;DR

This paper provides an elementary, Hilbert series-free exposition of the Golod-Shafarevich example of a finitely generated nil algebra over any field, emphasizing induction methods.

Contribution

It offers a simplified, induction-based construction and detailed explanation of the Golod-Shafarevich example without using Hilbert series techniques.

Findings

Elementary construction of the Golod-Shafarevich algebra

Demonstration of infinite dimensionality of the algebra

Clarification of the algebra's properties through detailed exposition

Abstract

Let $F$ be an arbitrary field. The Golod-Shafarevich example of a finitely generated nil $F$-algebra which is infinite dimensional -- is revisited. Here we offer a rather elementary treatment of that example, in which induction replaces Hilbert series techniques. This note also contains a detailed exposition of the construction of that example.