Graded polynomial identities and exponential growth

Let $A$ be a finite dimensional algebra over a fieldof characteristic zero graded by a finite abelian group $G$. Here westudy a growth function related to the graded polynomial identitiessatisfied by $A$ by computing the exponential rate of growth of thesequence of graded codimensions of $A$. We prove that the$G$-exponent of $A$ exists and is an integer related in an explicitway to the dimension of a suitable semisimple subalgebra of $A$.