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 ▫$\Gamma$▫.
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