A remark on Golod--Shafarevich algebras

TL;DR

The paper proves that the direct limit of surjective sequences of (weak) Golod--Shafarevich algebras remains a weak Golod--Shafarevich algebra, which implies exponential growth, with specific conditions on filtrations.

Contribution

It establishes that under certain conditions, the direct limit of Golod--Shafarevich algebras preserves the weak Golod--Shafarevich property, extending understanding of their structural behavior.

Findings

Limit of surjective Golod--Shafarevich algebra sequences is weak Golod--Shafarevich.

The limit algebra exhibits exponential growth.

Assumptions are necessary; weakening them invalidates the result.

Abstract

We show that a direct limit of surjections of (weak) Golod--Shafarevich algebras is a weak Golod--Shafarevich algebra as well. This holds both for graded and for filtered algebras provided that the filtrations are induced by the filtration of the first entry of the sequence. It follows that the limit is an algebra of exponential growth. An example shows that the assumptions of this theorem cannot be directly weakened.