Group graded PI-algebras and their codimension growth

TL;DR

This paper proves a conjecture relating the codimension growth of a G-graded associative PI-algebra to its identity component, confirming the inequality for all such algebras over a field of characteristic zero.

Contribution

The paper proves the conjectured inequality between the codimension growth of a G-graded PI-algebra and its identity component in full generality, extending previous results known for affine algebras.

Findings

Confirmed the inequality exp(W) ≤ |G|^2 exp(W_e) for all G-graded PI-algebras.

Extended the validity of the conjecture beyond affine algebras.

Provided a general proof applicable to all characteristic zero fields.

Abstract

Let W be an associative PI-algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. The following inequality had been conjectured by Bahturin and Zaicev: exp(W)\leq |G|^2 exp(W_e). The inequality is known in case the algebra W is affine (i.e. finitely generated). Here we prove the conjecture in general.