 
Summary: arXiv:1002.1385v1[math.RA]6Feb2010
GROUP GRADED PIALGEBRAS AND THEIR CODIMENSION
GROWTH
ELI ALJADEFF
Abstract. Let W be an associative PI algebra over a field F of characteristic
zero. Suppose W is Ggraded where G is a finite group. Let exp(W ) and
exp(We) denote the codimension growth of W and of the identity component
We, respectively. The following inequality had been conjectured by Bahturin
and Zaicev: exp(W ) G2 exp(We). The inequality is known in case the
algebra W is affine (i.e. finitely generated). Here we prove the conjecture in
general.
1. Introduction
Let W be any PIalgebra over a field of characteristic zero. Suppose W is G
graded where G is any finite group. Let exp(W) and exp(We) be the exponents of
the algebra W and of its ecomponent (determined by the Ggrading). The main
objective of this paper is to prove the following conjecture (see [8]):
Conjecture 1.1.
exp(W)  G 2
exp(We).
The case where W is affine was proved in [1]. Here we show that the conjecture
