We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $\Gamma$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$ we prove the existence of the graded PI-exponent, provided that $\Gamma$ is a commutative semigroup. If $A$ is simple in a non-graded sense the existence of the graded PI-exponent is proved without any restrictions on $\Gamma$.
Categories: math.RA