A Finiteness Criterion and Asymptotics for Codimensions of Generalized Identities

Let $A$ be an associative algebra over a field of characteristic zero. Then either all codimensions $\operatorname{gc}_n(A)$ of its generalized polynomial identities are infinite or $A$ is the sum of ideals $I$ and $J$ such that $\dim_FI<\infty$ and $J$ is nilpotent. In the latter case, there exist numbers $n_0\in\mathbb N$, $C\in\mathbb Q_+$, and $t\in\mathbb Z_+$ for which $\operatorname{gc}_n(A)<+\infty$ if $n\ge n_0$ and $\operatorname{gc}_n(A)\sim Cn^td^n$ as $n\to\infty$, where $d=\mathrm{PI}\exp(A)\in\mathbb Z_+$. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.