A finiteness criterion and asymptotics for codimensions of generalized identities

Abstract

Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gcn(A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dimFI < ∞ and J is nilpotent. In the latter case, there exist numbers n0 ∈ ℕ, C ∈ ℚ+, and t ∈ ℤ+ for which gcn(A) < +∞ if n ≥ n0 and gcn(A) ∼ Cntdn as n → ∞, where d = PIexp(A) ∈ ℤ+. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.