Kemer's theory for H-module algebras with application to the PI exponent

Let H be a semisimple finite dimensional Hopf algebra over a field F of zero characteristic. We prove three major theorems. 1. The Representability theorem which states that every H-module (associative) F-algebra W satisfying an ordinary PI, has the same H  -identities as the Grassmann envelope of an H⊗(FZ/2Z)⁎H⊗(FZ/2Z)⁎-module algebra which is finite dimensional over a field extension of F. 2. The Specht problem for H  -module (ordinary) PI algebras. That is, every H–TH–T-ideal Γ which contains an ordinary PI contains H  -polynomials f1,…,fsf1,…,fs which generate Γ as an H–TH–T-ideal. 3. Amitsur's conjecture for H-module algebras, saying that the exponent of the H-codimension sequence of an ordinary PI H-module algebra is an integer.