Anomalies on codimension growth of algebras

This paper deals with the asymptotic behavior of the sequence of codimensions c n cn(A), n = 1, 2,.., n=1,2,\ldots, of an algebra A over a field of characteristic zero. It is shown that when such sequence is polynomially bounded, then lim sup n → ∞ e log n e cn (A) and lim inf n → ∞ e log n e cn (A) cn(A) can be arbitrarily distant. Also, in case the codimensions are exponentially bounded, we can construct an algebra A such that exp e (A) = 2 \exp(A)=2 and, for any q ≥ 1 q≥ 1, there are infinitely many integers n such that cn (A) > nq 2n.This gives counterexamples to a conjecture of Regev for both cases of polynomial and exponential codimension growth.